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The article should say what the Hilbert transform is used for; currently is does not, and this is a significant omission. I suggest a section be added entitled 'Applications of the Hilbert transform' which gives practical examples of what the transform can be used for. —Preceding unsigned comment added by 81.168.113.121 ( talk) 08:53, 27 February 2008 (UTC)
It is not true that the integral defining the fourier transform of diverges. It is not even possible to consider as a tempered distribution and thereby get the result. On the other it is possible to define a tempered distribution out of by cutting off near 0 and taking a limit. But this is closely related to the fact we need a principle value. And it is correct to say the as an operator is the multiplier operator with multiplier . We may want to find a way to rephrase this part.
The notation should be removed. It is not standard across mathematics and signal processing. And conflicts with the notation more standard notation for the fourier transform. Seeing how the Hilbert transform almost always involves a discussion of the Fourier transform I believe this to be a poor choice of notation.
Thenub314 20:24, 17 July 2006 (UTC)
Math world has the plus and minus reversed in
The way it is now coincides with the MATLAB function when plotted, and also coincides with the diagram in my book (going from −∞ to +∞ during a positive side of the square wave), and the equation in my book for a pulse of width τ delayed by τ/2, so I'm leaving it this way. - Omegatron July 2, 2005 17:03 (UTC)
Very strange. In the mathworld we see that hilbert transform is the convolution with -1/t*pi function, not 1/t*pi. But the table with hilbert transform is nearly the same. Somebody just have written it down without thinking.
I will correct it as soon as I create an account.
-- 83.25.155.136 16:41, 18 September 2005 (UTC)
I'm trying to figure out what is going on with the discrete HT. That section now says that there is an ideal discrete HT, so and so, but this operation cannot be realized in the signal domain. Then, it presents a filter which seems to do the job, derived from the DFT. This seems contradictory. --KYN 21:41, 10 November 2005 (UTC)
I guess what is said is that from the ideal filter in the Z-domain it is not possible to derive a filter in the signal domain by means of the inverse Z-transform? Maybe then it is better not to involve the Z-domain in the discussion? Is it possible to present it in the following way? First define a "Hilbert filter" in the Fourier domain as
H(u)= -i for even integer < u < odd integer
H(u)= +i for odd integer < u < even integer
ie an oscillating square wave. The inverse DFT of this function will be precisely the discrete filter presented in the article. Then maybe continue to say that there is an ideal version of this filter in the Z-domain, with the given expression, but there is no formal relation to the discrete filter via the Z-transform or its inverse.
KYN 21:41, 10 November 2005 (UTC)
About the discrete algorithm I read a trick : it work better with a 0 for the first point.
So, thanks to the fast fourier transform the algorithm may be written
X(f) = FFT( x(t) ) if f == 0 : H(f) := 0 if f > 0 : H(f) := - i * X(f) if f < 0 : H(f) := +i * X(f) h(t) = iFFT( H(f) )
but with almost all the implementation of the FFT the spectrum unfolding (the negative part is stored after the positive one) imply one more 0 as you can see in this dirting matlab script :
len = length(wave_in); fft_in = fft(wave_in); fft_quad = [ 0 ; - 1i * fft_in(2 : len / 2); 0 ; 1i * fft_in(len / 2 + 2 :len)]; wave_out = real(ifft(fft_quad));
btw very useful to get the envelope ^^ enveloppe(t) = sqrt ( x(t)^2 * h(t)^2 ) for all t.
Forgor to write something in the "comment" but I noticed the comment about the discrete ideal filter being non-causal, and realized that this is a general propery of the HT, both continuous and discrete. I tried to formulate something about this.
I don't really have an experience with the DHT and therefore I must ask the following questions, some of which hopefully can work their way into the article to make it more understandable.
-- KYN 23:51, 3 January 2006 (UTC)
Now, I am still a bit concerned about the rest of that section. To me, it appears to discuss rather general principles of filter design, how to truncate and shift an infinitly extended filter, and how to implement the convolution operation in the frequency domain instead of the signal domain. The first part of this is more or less identical to the discussion for the continuous HT, isn't it? The latter, is something that relates to any discrete convolution operation, not just DHT. In that case, it could be moved to somewhere where the relation between signal domain convolution can be compared to "fast convolution" on a more general level. -- KYN 13:48, 4 January 2006 (UTC)
Did he develop and define this concept? Whaa? 21:31, 23 April 2006 (UTC)
This has undoubtedly been discussed somewhere else, so here we go again. If it comes to a vote, I prefer this convention:
to this one:
-- Bob K 14:50, 17 October 2006 (UTC)
If it comes to a vote I prefer
Thenub314 16:15, 17 October 2006 (UTC)
One thought on the subject is that the current notation is consistant with the Convolution article. Thenub314 23:36, 18 October 2006 (UTC)
I search in vain in each of several linked articles in this article including convolution, fourier transform, signal processing, etc., etc., etc., etc., finding the one thing all of these articles have in common is that not one defines ALL of the variables used. That is perfectly fine for those who are already experts, but makes it basically useless to those who are unfamiliar with the subject. If the unfamiliar reader isn't the targeted audience then what is the point of an encyclopedia anyway? Surely not just a book mark for all those equations you regularly use?
You don't need to break into the text, just add a section below where each variable is defined such as t =, omega =, theta=, etc.. I think that is not too much to ask. Drillerguy 14:42, 3 September 2007 (UTC)
What may be too much to ask is to describe the subject understandably to the general public in the introductory paragraph, before diving into the spagetti of equations. These readers may be very interested and very intelligent but still not familiar with the subject. Only the best technical writing achieves that and it may be too much to ask of Wiki but I hope is the goal. In the articles I have contributed to, I approach it as trying to explain the subject to my neighbor. Drillerguy 14:44, 3 September 2007 (UTC)
The Hilbert transform is not defined by convolution. The integral stated does not converge for reasonable functions (say ). I invite comments before I make any changes. Thenub314 ( talk) 17:27, 15 April 2008 (UTC)
As a PS to my comments about definitions, why does the introductory sentence demand the function be real valued? Thenub314 ( talk) 02:54, 17 April 2008 (UTC)
I have some issues with the sentence "Except for the component, which is lost..." For the discrete Hilbert transform it is certainly true. But in the case we are considering it doesn't follow. Thenub314 ( talk) 13:53, 20 April 2008 (UTC)
Sure, let's take for example then, let's denote the fourier transform by . We have , and if we calculate . But then .
More generally, if for all then for all because changing the integrand at one point doesn't change the value of the integral. Thenub314 ( talk) 16:29, 20 April 2008 (UTC)
True, but you only run into this trouble when the DC component is infinite. (Infact, that is not really enough, you really need to be dealing with something like the delta function.) It is a theorem that the square of the Hilbert transform is the the negative of the identity on the space of functions. (Yuck what an ugly sentence. I just mean on ) So it is not the case the DC component is necessarily lost. Thenub314 ( talk) 19:34, 20 April 2008 (UTC)
I like the change. Just what I would have done. Thenub314 ( talk) 20:29, 20 April 2008 (UTC)
The recent edits by 69.247.68.82 are good information to have in the article. Unfortunately it calls attention to a bit of a conflict in notation. In this article (as is common in many applications) the Hilbert transfrom of s(t)is dentoted by a . This conflicts with the notation used for the Fourier transform by most professional mathematicians. I suggest we denote the Hilbert transform by which is recognized by both groups, and appears in the definition. Thenub314 ( talk) 15:12, 29 April 2008 (UTC)
I did some reading about the history and tried to include some references. I hope people like this version, there are admittedly some omissions (generalizations of the H.T. for example). And places that are very awkward (discussion of other types of discrete Hilbert transforms.) But I think the info should make a nice addition. Thenub314 ( talk) 15:36, 12 May 2008 (UTC)
{{
cite book}}
: Cite has empty unknown parameter: |coauthors=
(
help)The exact same text seems to have been added in the page Gianfelici Transform. This article has been nominated for deletion because it is not clear if this transform is notable. Maybe we should make sure this section belongs here. There doesn't seem to be a lot that comes up about it on Google besides the reference given. Thenub314 ( talk) 17:33, 12 May 2008 (UTC)
Does the phrase "Improper Integration" imply that we are dealing with something other then the Lebesgue integral? Being an encyclopedia article I don't want to get too technical. Though if you follow the link to the Improper integration page it says something like: "For the Lebesgue integral, deals differently with unbounded domains and unbounded functions, and one does not distinguish an 'improper Lebesgue integral'...", but then it follows this with something (I think) is false so I don't know what to make of that statement. Anyways, it is a fine expression to have, maybe we could write the limit directly? Thenub314 ( talk) 12:36, 16 May 2008 (UTC)
Thanks for fixing that example, I didn't quite have enough time before I left for work. It was mostly the phrase "improper integral" I was asking about. I thought it simply wasn't used when discussing the Lebesgue integrals. I am not sure why, it seems natural enough. This is what I thought the page improper integral meant by "...one does not distinguish...". Thenub314 ( talk) 16:42, 16 May 2008 (UTC)
It is not quite true that "The Hilbert transform of the sin and cos functions are defined using the periodic form of the transform, since the integral defining them otherwise fails to converge absolutely." At some point I was bothered by these two being in the table aswell, but when I start with the definition (instead of thinking about it as an operator) this is what I get, if there are any mistakes please let me know.
As an operator it does map L∞ to BMO, so the statement could also make sense in that way. But I feel those the above calculation justifies the entries in the table. Thenub314 ( talk) 00:22, 18 May 2008 (UTC)
Just to make sure I am correct, consider the sentance... "However, a priori this may only be defined for u a distribution of compact support. It is possible to work somewhat rigorously with this since compactly supported distributions are dense in Lp."
Shouldn't this be functions of compact support? Distributions of compact support do not sit inside Lp, if they are dense you'd first want to intersect with Lp. Thenub314 ( talk) 13:53, 18 May 2008 (UTC)
Ok, cool, you can allow Schwartz functions if you really want, as you point out in the line above that is a tempered distribution. Thenub314 ( talk) 14:06, 18 May 2008 (UTC)
Would anyone object to me changing the current notation for the Hilbert transform of a function from to ? This is more consistent with the way Fourier transforms are defined in signal processing (in my experience), and less confusing since it's currently easy to confuse these expressions for a multiplication. The only potential downside is that the Hilbert domain variable isn't immediately obvious, but that isn't really a big deal. -Roger ( talk) 16:54, 15 May 2009 (UTC)
I have added a sentence on the development of Hilbert spectroscopy to the History section, as I could not see a better place. I have added the redlinked topic at Wikipedia:Requested articles/Natural sciences#Physical chemistry because I know next to nothing about it outside of reading the topic on BBC news, a couple of paper abstracts and the Terahertz article. My experience ended at narrowband spectral analysis for specific chemical bonds using infrared. If this is the wrong place, please move it to where it fits better. - 84user ( talk) 17:00, 20 October 2009 (UTC)
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Reviewer: Sławomir Biały ( talk · contribs) 13:21, 22 November 2011 (UTC)
After a quick scan, I have the following comments, in no particular order. I will add more as I read the article more carefully. Sławomir Biały ( talk) 13:21, 22 November 2011 (UTC)
A few remarks (after looking through the first half):
What's the status of this review? Comments appear to have been addressed weeks ago. Wizardman Operation Big Bear 04:21, 18 December 2011 (UTC)
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Reviewer: Dolphin51 ( talk · contribs) 03:54, 29 December 2011 (UTC)
I am aware of the guidance about citation of sources in articles on math and science found at WP:SCG but my view is that Hilbert transform presently lacks in-line citations sufficient to allow independent verification of the material. It contains about 27 references and a similar number of Harvard-style citations; but only four in-line citations and they all apply to the same sentence. Mostly the Harvard-style citations consist of a document title but no detail regarding chapter, verse or page. There are many statements in the article that could, and should, be supported by an in-line citation. For example, as I finished perusing the article I was struck by the last sentence:
In the absence of an in-line citation, statements like this look like original research.
Articles on mathematical subjects are not within my field of expertise so I will ask for a second opinion. Dolphin ( t) 04:28, 29 December 2011 (UTC)
The citation style used extensively in the article is called parenthetical referencing and detailed guidance is available at Wikipedia:Parenthetical referencing. Many of the citations presented using this method are presented only as author and date - no detailed information about chapter, verse or page number is provided. This is a significant impediment to the task of independently verifying the statement to which the citation is attached. Parenthetical referencing is not a means for Users to avoid the tedious task of supplying details such as page numbers - see WP:Parenthetical referencing#Page numbers
There are now a number of Unreferenced section banners in this article, and there is the problem of citations that lack appropriate detail. I believe these things need to be repaired before Hilbert transform can be promoted to Good Article. Dolphin ( t) 07:25, 30 December 2011 (UTC)
Most of these do not require page numbers. For instance, Bitsadze (2001) is a very short EoM article as is Khvedelidze (2001). Bedrosian (1962) is a technical report that is entirely devoted to the topic that it is being referenced for. I'm not sure how the Hilbert reference is being used, I think it's as a primary source. I've added a secondary source to back it up, feeling that's ultimately more important than giving a specific page number to the Hilbert reference. I agree with the other two (but I have already flagged these as needing page numbers). Anything else? Sławomir Biały ( talk) 12:26, 30 December 2011 (UTC)
Hilbert transform was nominated for Good Article status in late November by User:Thenub314. On two occasions, reviewers have volunteered to review the article as part of the GA process. (The previous occasion is recorded at Talk:Hilbert transform/GA1.) On neither occasion has Thenub314 participated in the discussion on the Talk pages, or contributed to the Talk pages in any way.
Almost a week ago I left a message at Thenub314's User talk page, asking that he participate in the discussion on the Talk pages or let us know what his intentions are regarding his GA nomination of Hilbert transform - see my diff. Thenub314 has edited Wikipedia only once since I left my message, and that was not an edit related to the GA nomination of Hilbert transform.
It seems Thenub314 is busy with other activities at present, and is unable to spare the time to participate in the GA nomination of Hilbert transform. Within the next 24 hours it is my intention to close the current GA review of Hilbert transform, but Thenub314, or anyone else, will be most welcome to re-nominate the article for GA status at any time in the future. Dolphin ( t) 01:54, 12 January 2012 (UTC)
The introduction says: "[..] if f(z) is analytic in the plane Im z > 0 and u(t) = Re f(t + 0·i ) then Im f(t + 0·i ) = H(u)(t) up to an additive constant, provided this Hilbert transform exists." That can't be right, as it implies that both and are Hilbert transforms of the cos(t) (since cos(z) and e^iz are both holomorphic on the plane). Basically the same claim is repeated under "Conjugate functions". I'm not sure what would be a correct version of the statement since I don't know much about Hilbert transforms (that's why I came here..), and don't have easy access to the cited source. I guess some condition like (or perhaps ) is missing? -- 132.68.56.67 ( talk) 13:42, 5 January 2012 (UTC)
In sect 6.6 it says H = i (2P - I), but P has not been mantioned as far as I can see, Presumably the projection on the one component of the splitting mentioned above? Billlion ( talk) 12:31, 12 April 2013 (UTC)
In the history section it says the Hilbert transform may be defined for Lp with 1 ≤ p ≤ infinity. I think this is true, but the following sentence says that the Hilbert transform is bounded on the same range of p; I think the actual statement should exclude p=1 and p=infinity. — Preceding unsigned comment added by Mdn33 ( talk • contribs) 22:04, 15 July 2014 (UTC)
Hello Sławomir,
Regarding your 2 reverts, with these annotations:
I was unable to find your reference to "Under default settings (i.e., no mathjax)", but after consulting WP:MSM, my conclusion is that we are talking about a difference of personal preferences, not policy. And that conclusion is supported at Help:Displaying_a_formula#TeX_vs_HTML, as you probably know. I in particular dislike the use of H(u) vs or (even better) to denote the Hilbert transform operator (on function u).
Therefore, I would like to hear some other readers' comments regarding which "looks worse":
That is why I am posting this new section. Thanks in advance for any ensuing comments.
-- Bob K ( talk) 21:09, 1 March 2015 (UTC)
<math>
is acceptable for inline; it is not. By default, <math>
generates PNG images which are twice as big as the surrounding text (at default text sizes) and it is unsightly. Since we cannot force MathJax, {{
math}} is the only suitable alternative for inline math. It was crafted for this very purpose (as far as simply formulas and variables are concerned), and designed to match the MathJax font (Times-based). HTML also scales with the text, where <math />
does not. So my rule of thumb has always been: <math />
for display, {{
math}} for inline. So let me add a third option:
-- [[
User:Edokter]] {{
talk}}
16:18, 2 March 2015 (UTC)<i>
. (Mvar uses CSS font-style to display italics, math uses <i>
.) -- [[
User:Edokter]] {{
talk}}
21:42, 5 March 2015 (UTC)In my browsers, (and ) is more clearly an operation on a function than is H(u), which looks like a function of variable u. Does anyone agree? If so, perhaps that should be the starting point for this discussion. -- Bob K ( talk) 06:35, 3 March 2015 (UTC)
-- [[
User:Edokter]] {{
talk}}
16:11, 5 March 2015 (UTC)-- [[
User:Edokter]] {{
talk}}
21:03, 5 March 2015 (UTC)
-- [[
User:Edokter]] {{
talk}}
21:39, 5 March 2015 (UTC)
-- [[
User:Edokter]] {{
talk}}
09:17, 6 March 2015 (UTC)Some of you are missing the point. The issue here isn't the exact correct typesetting of H(u) or whatever. It is the fact that inline, compromises have to be made that clash with perfection. The presence of a single ruins the appearance of the whole article. I am not "pro-HTML". I'd love to see TeX supported 100% and could happily accept the whole articles being written in TeX. But as it stands now, we have to mix. Mixing with inline TeX simply ruins the appearance of the whole article on some systems. Mixing with inline HTML does not. It's as simple as that. YohanN7 ( talk) 11:39, 6 March 2015 (UTC)
In the Introduction/Notation section, we have references to Brandwood 2003 and Bracewell 2000. No such reference is listed in the references section. Does anyone have any idea what these references are supposed to refer to? There is a Bracewell 1986, possibly a different edition of the same book, but if so that would need to be checked to make sure the page numbers are still correct. — David Eppstein ( talk) 21:16, 5 March 2015 (UTC)
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It seems to me that the introductory paragraph of this article is not adequate. The first paragraph ought to be almost a dictionary definition, but this is only a very generic description:
In mathematics and in signal processing, the Hilbert transform is a linear operator that takes a function, u(t) of a real variable and produces another function of a real variable H(u)(t).
I'm not qualified to do so, but isn't there some two-sentence summary that describes what the transform actually is? Even if we changed it to "...another function of a real variable H(u)(t), given by the improper integral X," or ", essentially the convolution of u with 1/πt" it would be a better summary.
Maybe that's just not possible. In almost every online citation where people try to define the transform, they actually say what it's like, or what it can be used for. Does it simply defy definition?
Timrprobocom ( talk) 19:20, 30 January 2018 (UTC)
I've tried to tweak the latest to include both the integral expression and the phase shift. Sławomir Biały ( talk) 02:19, 4 February 2018 (UTC)
What purpose is served by including and in the
Hilbert_transform#Table_of_selected_Hilbert_transforms? First of all, the transform of a complex function is the transforms of its real and imaginary parts, recombined as a complex function. But also, what is the application we have in mind?
--
Bob K (
talk) 13:58, 11 February 2018 (UTC)
The following comment was inserted after the section heading of the Titchmarsh's theorem section of this article, by User:24.233.245.156 ( talk • contribs) 08:23, 4 May 2018 (UTC):
Warning: This entry needs corrections. The results below are due to Paley-Wiener (Fourier transform part) and Marcel Riesz (Hilbert transform part). See e.g. https://link.springer.com/article/10.1140/epjh/e2014-50021-1.
-- Pipetricker ( talk) 09:45, 4 May 2018 (UTC)
Here is a partial list of problems with the section ‘Titchmarsh Theorem’
1. As it stands now, the anonymous author(s) XX of this Wikipedia entry refer(s) to some results, attributing them to Titchmarsh in a 1948 book. In that case, a reference to the original 1937 edition should have been used. The 2nd edition is the same as first save some added (mostly irrelevant today) positions to the bibliography.
2. The review of 1937 edition in Zentralblatt fuer Mathematik is by Hille. In the recently (added by XX after my intervention) sentence with a citation to King’s book, Hille is (particularly! according to King) the author of some results in ‘Titchmarsh Theorem Section’ (Chapter V in the book). Here is the beginning of Hille’s review of the book: “A couple of introductory chapters bring the formal theory of Fourier, Laplace and Mellin transforms and the convergence and summability theory of the Fourier simple and double integrals. This is followed by a thorough discussion of the theory of Fourier transforms in the class L, L_2 (Plancherel), and L_p (Titchmarsh). In Chapter V we find the theory of conjugate functions and Hilbert transforms for the classes L_p, p=1 or 1< p, where the central theorems are due to M. Riesz. The rest of review follows (but is here irrelevant).
3. In the so-called Titchmarsh Theorem, as written by XX, there are three dotted sentences (1) (2) (3) concerning H_2 Hardy space, and the equivalence of all three is claimed. Sentence (1) has no meaning, because it does not say in what convergence the limit is meant to be. The intended meaning is almost everywhere convergence, but then (1), as written, becomes false. The sentence (2) is obviously false.
4. Both (1) and (2) can, of course, be corrected. Let us assume we did it. Then (1), after a minimal rephrasing, contains what is called ‘the upper half-plane analogue of Fatou’s theorem on radial limits’. Knowing that we are in H_2 and that (1) holds, the equivalence of (1) and (2) is due to Marcel Riesz and it is referred to Marcel Riesz in every book on the subject written by a mathematician. Similarly, the equivalence of (1) and (3) is then a theorem due to Paley and Wiener (Theorem 5 in their 1934 book) and, strictly speaking, has nothing to do with the Hilbert transform.
5. The purpose of citation to books (as opposed to papers) in the Wikipedia articles is (or should be) to facilitate further study by the reader, who might be, for instance, interested in a proof. Here is Titchmarsh’s proof of the equivalence of (1) and (2): “The equivalence of (1) and (2) follows at once from the above theorems”. We are on the page 128 of the book!
6. This is not the end of Titchmarsh’s pedagogical ‘achievements’. The title of Chapter V is “Conjugate integrals, Hilbert Transforms”. He never defines Hilbert transform. There are no ‘conjugate integrals’ in the text. He speaks about ‘conjugate functions’ instead, but there is no definition either. He never says what ‘regular analytic function’ (a term he uses for holomorphic function) is. There is quite a bit more, but I think it is already clear that this is not a book to be used for citations.
7. So, corrections are needed. But before doing anything, I am awaiting further comments. Math45-oxford ( talk) 15:20, 10 June 2018 (UTC)
Riemann-Hilbert problem. I also have a question concerning the Riemann-Hilbert problem section. What is it supposed to mean that F_+ and F_ solve the problem ‘formally’? Further, let us assume the function f indeed solves the problem (i.e. really as opposed to formally). Where precisely Pandey states the quoted result in Chapter 2? I’d like a precise reference because I just read that chapter twice and cannot find the result. Math45-oxford ( talk) 21:17, 14 June 2018 (UTC)
Precise conditions would only be about a solution of the Riemann-Hilbert problem. But I already assumed that $F plus, minus$ form the solution. Where is the statement about the Hilbert transform? I do not believe it is true, so if I am right it might be rather difficult to find a reference (if I am wrong, the author XXX must have seen it in Pandey – so, again, where is it in Pandey?). By the way, (Titchmarsh’s version of) the Kolmogorov theorem is not correctly copied from Titchmarsh’s book and, as stated, is wrong. The previous Theorem, the one for $p>1$ is also not correct with our definition of Hilbert transform. Titchmarsh had the minus sign, because his Hilbert transform (that he never openly defined) was what for us in ‘inverse Hilbert transform’. Math45-oxford ( talk) 16:24, 15 June 2018 (UTC)
Some more remarks about Section 8.2. Riemann-Hilbert problem. It is stated that if , where is holomorphic in the upper half-plane and is holomorphic in the lower half-plane, then FORMALLY . The reference is to Pandey’s book Chapter 2. Chapter 2 in Pandey’s book gives a sketch of the ‘classical’ (i.e. no distributions) solution of the HP. This is done before Hilbert transform was even introduced. So it looks like the author of this entry might have thought about a reference to Section 6.2 in Pandey’s book. This, however, would not be very helpful, because Pandey’s discussion there is very imprecise (with wrong references to his own papers and to non-existing formulas, in particular). But the main issue is that "FORMALLY" presumably refers to some distributional equality. This would need to be stated in a precise way, otherwise the claimed equality is meaningless. Finally, even assuming things are corrected: Hilbert Problem is not about Hilbert Transform and has its own Wikipedia entry. Is this Section 8.2 needed at all? Math45-oxford ( talk) — Preceding unsigned comment added by Math45-oxford ( talk • contribs) 19:47, 22 July 2018 (UTC)
Section 5. Domain of definition.
1. In the two formulas there the minus sign in front of the integrals is an effect of Zygmund’s use of –H as the definition of the Hilbert transform and should be corrected into +. Further, what would be the reason for using this formula, instead of the defining one, at all? By the way, the formula is also used (and should be corrected) in the Introduction. There is no clear reason to state it in the Introduction either, because that formula is not more explicit than the defining one, once one understands what the principal value is.
2. The almost everywhere convergence happens to be true. As for the norm convergence, I do not know. At any rate, I did not see it stated for the norm convergence this way anywhere. Now, why the claimed convergences would be a consequence of Titchmarsh Theorem? What Titchmarsh Theorem? A few subsections later a so-called Titchmarsh Theorem is stated. It refers to p=2 only and does not provide the claimed implications even in the case of p=2. Math45-oxford ( talk) 20:41, 22 July 2018 (UTC)
The Hilbert transform can be defined in various ways, one of which leads to a change in sign of . The current definition with actually leads to a Fourier multiplier of , not . At least this is my understanding from Simon's "A comprehensive course in analysis." I do not know if it is easier to change the original definition or the Fourier multiplier later, since I think the original definition is actually the nonstandard one and the Fourier multiplier given here is the standard multiplier.
Jupsal ( talk) 00:36, 26 December 2018 (UTC)
Maybe it's just my ignorance, but the sentence that doesn't seem very helpful is:
Specifically, the Hilbert transform of u is its harmonic conjugate v, a function of the real variable t such that the complex-valued function u + i v admits an extension to the complex upper half-plane satisfying the Cauchy–Riemann equations.
Perhaps it can be re-phrased or re-located to a more mathematical section.
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The article should say what the Hilbert transform is used for; currently is does not, and this is a significant omission. I suggest a section be added entitled 'Applications of the Hilbert transform' which gives practical examples of what the transform can be used for. —Preceding unsigned comment added by 81.168.113.121 ( talk) 08:53, 27 February 2008 (UTC)
It is not true that the integral defining the fourier transform of diverges. It is not even possible to consider as a tempered distribution and thereby get the result. On the other it is possible to define a tempered distribution out of by cutting off near 0 and taking a limit. But this is closely related to the fact we need a principle value. And it is correct to say the as an operator is the multiplier operator with multiplier . We may want to find a way to rephrase this part.
The notation should be removed. It is not standard across mathematics and signal processing. And conflicts with the notation more standard notation for the fourier transform. Seeing how the Hilbert transform almost always involves a discussion of the Fourier transform I believe this to be a poor choice of notation.
Thenub314 20:24, 17 July 2006 (UTC)
Math world has the plus and minus reversed in
The way it is now coincides with the MATLAB function when plotted, and also coincides with the diagram in my book (going from −∞ to +∞ during a positive side of the square wave), and the equation in my book for a pulse of width τ delayed by τ/2, so I'm leaving it this way. - Omegatron July 2, 2005 17:03 (UTC)
Very strange. In the mathworld we see that hilbert transform is the convolution with -1/t*pi function, not 1/t*pi. But the table with hilbert transform is nearly the same. Somebody just have written it down without thinking.
I will correct it as soon as I create an account.
-- 83.25.155.136 16:41, 18 September 2005 (UTC)
I'm trying to figure out what is going on with the discrete HT. That section now says that there is an ideal discrete HT, so and so, but this operation cannot be realized in the signal domain. Then, it presents a filter which seems to do the job, derived from the DFT. This seems contradictory. --KYN 21:41, 10 November 2005 (UTC)
I guess what is said is that from the ideal filter in the Z-domain it is not possible to derive a filter in the signal domain by means of the inverse Z-transform? Maybe then it is better not to involve the Z-domain in the discussion? Is it possible to present it in the following way? First define a "Hilbert filter" in the Fourier domain as
H(u)= -i for even integer < u < odd integer
H(u)= +i for odd integer < u < even integer
ie an oscillating square wave. The inverse DFT of this function will be precisely the discrete filter presented in the article. Then maybe continue to say that there is an ideal version of this filter in the Z-domain, with the given expression, but there is no formal relation to the discrete filter via the Z-transform or its inverse.
KYN 21:41, 10 November 2005 (UTC)
About the discrete algorithm I read a trick : it work better with a 0 for the first point.
So, thanks to the fast fourier transform the algorithm may be written
X(f) = FFT( x(t) ) if f == 0 : H(f) := 0 if f > 0 : H(f) := - i * X(f) if f < 0 : H(f) := +i * X(f) h(t) = iFFT( H(f) )
but with almost all the implementation of the FFT the spectrum unfolding (the negative part is stored after the positive one) imply one more 0 as you can see in this dirting matlab script :
len = length(wave_in); fft_in = fft(wave_in); fft_quad = [ 0 ; - 1i * fft_in(2 : len / 2); 0 ; 1i * fft_in(len / 2 + 2 :len)]; wave_out = real(ifft(fft_quad));
btw very useful to get the envelope ^^ enveloppe(t) = sqrt ( x(t)^2 * h(t)^2 ) for all t.
Forgor to write something in the "comment" but I noticed the comment about the discrete ideal filter being non-causal, and realized that this is a general propery of the HT, both continuous and discrete. I tried to formulate something about this.
I don't really have an experience with the DHT and therefore I must ask the following questions, some of which hopefully can work their way into the article to make it more understandable.
-- KYN 23:51, 3 January 2006 (UTC)
Now, I am still a bit concerned about the rest of that section. To me, it appears to discuss rather general principles of filter design, how to truncate and shift an infinitly extended filter, and how to implement the convolution operation in the frequency domain instead of the signal domain. The first part of this is more or less identical to the discussion for the continuous HT, isn't it? The latter, is something that relates to any discrete convolution operation, not just DHT. In that case, it could be moved to somewhere where the relation between signal domain convolution can be compared to "fast convolution" on a more general level. -- KYN 13:48, 4 January 2006 (UTC)
Did he develop and define this concept? Whaa? 21:31, 23 April 2006 (UTC)
This has undoubtedly been discussed somewhere else, so here we go again. If it comes to a vote, I prefer this convention:
to this one:
-- Bob K 14:50, 17 October 2006 (UTC)
If it comes to a vote I prefer
Thenub314 16:15, 17 October 2006 (UTC)
One thought on the subject is that the current notation is consistant with the Convolution article. Thenub314 23:36, 18 October 2006 (UTC)
I search in vain in each of several linked articles in this article including convolution, fourier transform, signal processing, etc., etc., etc., etc., finding the one thing all of these articles have in common is that not one defines ALL of the variables used. That is perfectly fine for those who are already experts, but makes it basically useless to those who are unfamiliar with the subject. If the unfamiliar reader isn't the targeted audience then what is the point of an encyclopedia anyway? Surely not just a book mark for all those equations you regularly use?
You don't need to break into the text, just add a section below where each variable is defined such as t =, omega =, theta=, etc.. I think that is not too much to ask. Drillerguy 14:42, 3 September 2007 (UTC)
What may be too much to ask is to describe the subject understandably to the general public in the introductory paragraph, before diving into the spagetti of equations. These readers may be very interested and very intelligent but still not familiar with the subject. Only the best technical writing achieves that and it may be too much to ask of Wiki but I hope is the goal. In the articles I have contributed to, I approach it as trying to explain the subject to my neighbor. Drillerguy 14:44, 3 September 2007 (UTC)
The Hilbert transform is not defined by convolution. The integral stated does not converge for reasonable functions (say ). I invite comments before I make any changes. Thenub314 ( talk) 17:27, 15 April 2008 (UTC)
As a PS to my comments about definitions, why does the introductory sentence demand the function be real valued? Thenub314 ( talk) 02:54, 17 April 2008 (UTC)
I have some issues with the sentence "Except for the component, which is lost..." For the discrete Hilbert transform it is certainly true. But in the case we are considering it doesn't follow. Thenub314 ( talk) 13:53, 20 April 2008 (UTC)
Sure, let's take for example then, let's denote the fourier transform by . We have , and if we calculate . But then .
More generally, if for all then for all because changing the integrand at one point doesn't change the value of the integral. Thenub314 ( talk) 16:29, 20 April 2008 (UTC)
True, but you only run into this trouble when the DC component is infinite. (Infact, that is not really enough, you really need to be dealing with something like the delta function.) It is a theorem that the square of the Hilbert transform is the the negative of the identity on the space of functions. (Yuck what an ugly sentence. I just mean on ) So it is not the case the DC component is necessarily lost. Thenub314 ( talk) 19:34, 20 April 2008 (UTC)
I like the change. Just what I would have done. Thenub314 ( talk) 20:29, 20 April 2008 (UTC)
The recent edits by 69.247.68.82 are good information to have in the article. Unfortunately it calls attention to a bit of a conflict in notation. In this article (as is common in many applications) the Hilbert transfrom of s(t)is dentoted by a . This conflicts with the notation used for the Fourier transform by most professional mathematicians. I suggest we denote the Hilbert transform by which is recognized by both groups, and appears in the definition. Thenub314 ( talk) 15:12, 29 April 2008 (UTC)
I did some reading about the history and tried to include some references. I hope people like this version, there are admittedly some omissions (generalizations of the H.T. for example). And places that are very awkward (discussion of other types of discrete Hilbert transforms.) But I think the info should make a nice addition. Thenub314 ( talk) 15:36, 12 May 2008 (UTC)
{{
cite book}}
: Cite has empty unknown parameter: |coauthors=
(
help)The exact same text seems to have been added in the page Gianfelici Transform. This article has been nominated for deletion because it is not clear if this transform is notable. Maybe we should make sure this section belongs here. There doesn't seem to be a lot that comes up about it on Google besides the reference given. Thenub314 ( talk) 17:33, 12 May 2008 (UTC)
Does the phrase "Improper Integration" imply that we are dealing with something other then the Lebesgue integral? Being an encyclopedia article I don't want to get too technical. Though if you follow the link to the Improper integration page it says something like: "For the Lebesgue integral, deals differently with unbounded domains and unbounded functions, and one does not distinguish an 'improper Lebesgue integral'...", but then it follows this with something (I think) is false so I don't know what to make of that statement. Anyways, it is a fine expression to have, maybe we could write the limit directly? Thenub314 ( talk) 12:36, 16 May 2008 (UTC)
Thanks for fixing that example, I didn't quite have enough time before I left for work. It was mostly the phrase "improper integral" I was asking about. I thought it simply wasn't used when discussing the Lebesgue integrals. I am not sure why, it seems natural enough. This is what I thought the page improper integral meant by "...one does not distinguish...". Thenub314 ( talk) 16:42, 16 May 2008 (UTC)
It is not quite true that "The Hilbert transform of the sin and cos functions are defined using the periodic form of the transform, since the integral defining them otherwise fails to converge absolutely." At some point I was bothered by these two being in the table aswell, but when I start with the definition (instead of thinking about it as an operator) this is what I get, if there are any mistakes please let me know.
As an operator it does map L∞ to BMO, so the statement could also make sense in that way. But I feel those the above calculation justifies the entries in the table. Thenub314 ( talk) 00:22, 18 May 2008 (UTC)
Just to make sure I am correct, consider the sentance... "However, a priori this may only be defined for u a distribution of compact support. It is possible to work somewhat rigorously with this since compactly supported distributions are dense in Lp."
Shouldn't this be functions of compact support? Distributions of compact support do not sit inside Lp, if they are dense you'd first want to intersect with Lp. Thenub314 ( talk) 13:53, 18 May 2008 (UTC)
Ok, cool, you can allow Schwartz functions if you really want, as you point out in the line above that is a tempered distribution. Thenub314 ( talk) 14:06, 18 May 2008 (UTC)
Would anyone object to me changing the current notation for the Hilbert transform of a function from to ? This is more consistent with the way Fourier transforms are defined in signal processing (in my experience), and less confusing since it's currently easy to confuse these expressions for a multiplication. The only potential downside is that the Hilbert domain variable isn't immediately obvious, but that isn't really a big deal. -Roger ( talk) 16:54, 15 May 2009 (UTC)
I have added a sentence on the development of Hilbert spectroscopy to the History section, as I could not see a better place. I have added the redlinked topic at Wikipedia:Requested articles/Natural sciences#Physical chemistry because I know next to nothing about it outside of reading the topic on BBC news, a couple of paper abstracts and the Terahertz article. My experience ended at narrowband spectral analysis for specific chemical bonds using infrared. If this is the wrong place, please move it to where it fits better. - 84user ( talk) 17:00, 20 October 2009 (UTC)
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Reviewer: Sławomir Biały ( talk · contribs) 13:21, 22 November 2011 (UTC)
After a quick scan, I have the following comments, in no particular order. I will add more as I read the article more carefully. Sławomir Biały ( talk) 13:21, 22 November 2011 (UTC)
A few remarks (after looking through the first half):
What's the status of this review? Comments appear to have been addressed weeks ago. Wizardman Operation Big Bear 04:21, 18 December 2011 (UTC)
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Reviewer: Dolphin51 ( talk · contribs) 03:54, 29 December 2011 (UTC)
I am aware of the guidance about citation of sources in articles on math and science found at WP:SCG but my view is that Hilbert transform presently lacks in-line citations sufficient to allow independent verification of the material. It contains about 27 references and a similar number of Harvard-style citations; but only four in-line citations and they all apply to the same sentence. Mostly the Harvard-style citations consist of a document title but no detail regarding chapter, verse or page. There are many statements in the article that could, and should, be supported by an in-line citation. For example, as I finished perusing the article I was struck by the last sentence:
In the absence of an in-line citation, statements like this look like original research.
Articles on mathematical subjects are not within my field of expertise so I will ask for a second opinion. Dolphin ( t) 04:28, 29 December 2011 (UTC)
The citation style used extensively in the article is called parenthetical referencing and detailed guidance is available at Wikipedia:Parenthetical referencing. Many of the citations presented using this method are presented only as author and date - no detailed information about chapter, verse or page number is provided. This is a significant impediment to the task of independently verifying the statement to which the citation is attached. Parenthetical referencing is not a means for Users to avoid the tedious task of supplying details such as page numbers - see WP:Parenthetical referencing#Page numbers
There are now a number of Unreferenced section banners in this article, and there is the problem of citations that lack appropriate detail. I believe these things need to be repaired before Hilbert transform can be promoted to Good Article. Dolphin ( t) 07:25, 30 December 2011 (UTC)
Most of these do not require page numbers. For instance, Bitsadze (2001) is a very short EoM article as is Khvedelidze (2001). Bedrosian (1962) is a technical report that is entirely devoted to the topic that it is being referenced for. I'm not sure how the Hilbert reference is being used, I think it's as a primary source. I've added a secondary source to back it up, feeling that's ultimately more important than giving a specific page number to the Hilbert reference. I agree with the other two (but I have already flagged these as needing page numbers). Anything else? Sławomir Biały ( talk) 12:26, 30 December 2011 (UTC)
Hilbert transform was nominated for Good Article status in late November by User:Thenub314. On two occasions, reviewers have volunteered to review the article as part of the GA process. (The previous occasion is recorded at Talk:Hilbert transform/GA1.) On neither occasion has Thenub314 participated in the discussion on the Talk pages, or contributed to the Talk pages in any way.
Almost a week ago I left a message at Thenub314's User talk page, asking that he participate in the discussion on the Talk pages or let us know what his intentions are regarding his GA nomination of Hilbert transform - see my diff. Thenub314 has edited Wikipedia only once since I left my message, and that was not an edit related to the GA nomination of Hilbert transform.
It seems Thenub314 is busy with other activities at present, and is unable to spare the time to participate in the GA nomination of Hilbert transform. Within the next 24 hours it is my intention to close the current GA review of Hilbert transform, but Thenub314, or anyone else, will be most welcome to re-nominate the article for GA status at any time in the future. Dolphin ( t) 01:54, 12 January 2012 (UTC)
The introduction says: "[..] if f(z) is analytic in the plane Im z > 0 and u(t) = Re f(t + 0·i ) then Im f(t + 0·i ) = H(u)(t) up to an additive constant, provided this Hilbert transform exists." That can't be right, as it implies that both and are Hilbert transforms of the cos(t) (since cos(z) and e^iz are both holomorphic on the plane). Basically the same claim is repeated under "Conjugate functions". I'm not sure what would be a correct version of the statement since I don't know much about Hilbert transforms (that's why I came here..), and don't have easy access to the cited source. I guess some condition like (or perhaps ) is missing? -- 132.68.56.67 ( talk) 13:42, 5 January 2012 (UTC)
In sect 6.6 it says H = i (2P - I), but P has not been mantioned as far as I can see, Presumably the projection on the one component of the splitting mentioned above? Billlion ( talk) 12:31, 12 April 2013 (UTC)
In the history section it says the Hilbert transform may be defined for Lp with 1 ≤ p ≤ infinity. I think this is true, but the following sentence says that the Hilbert transform is bounded on the same range of p; I think the actual statement should exclude p=1 and p=infinity. — Preceding unsigned comment added by Mdn33 ( talk • contribs) 22:04, 15 July 2014 (UTC)
Hello Sławomir,
Regarding your 2 reverts, with these annotations:
I was unable to find your reference to "Under default settings (i.e., no mathjax)", but after consulting WP:MSM, my conclusion is that we are talking about a difference of personal preferences, not policy. And that conclusion is supported at Help:Displaying_a_formula#TeX_vs_HTML, as you probably know. I in particular dislike the use of H(u) vs or (even better) to denote the Hilbert transform operator (on function u).
Therefore, I would like to hear some other readers' comments regarding which "looks worse":
That is why I am posting this new section. Thanks in advance for any ensuing comments.
-- Bob K ( talk) 21:09, 1 March 2015 (UTC)
<math>
is acceptable for inline; it is not. By default, <math>
generates PNG images which are twice as big as the surrounding text (at default text sizes) and it is unsightly. Since we cannot force MathJax, {{
math}} is the only suitable alternative for inline math. It was crafted for this very purpose (as far as simply formulas and variables are concerned), and designed to match the MathJax font (Times-based). HTML also scales with the text, where <math />
does not. So my rule of thumb has always been: <math />
for display, {{
math}} for inline. So let me add a third option:
-- [[
User:Edokter]] {{
talk}}
16:18, 2 March 2015 (UTC)<i>
. (Mvar uses CSS font-style to display italics, math uses <i>
.) -- [[
User:Edokter]] {{
talk}}
21:42, 5 March 2015 (UTC)In my browsers, (and ) is more clearly an operation on a function than is H(u), which looks like a function of variable u. Does anyone agree? If so, perhaps that should be the starting point for this discussion. -- Bob K ( talk) 06:35, 3 March 2015 (UTC)
-- [[
User:Edokter]] {{
talk}}
16:11, 5 March 2015 (UTC)-- [[
User:Edokter]] {{
talk}}
21:03, 5 March 2015 (UTC)
-- [[
User:Edokter]] {{
talk}}
21:39, 5 March 2015 (UTC)
-- [[
User:Edokter]] {{
talk}}
09:17, 6 March 2015 (UTC)Some of you are missing the point. The issue here isn't the exact correct typesetting of H(u) or whatever. It is the fact that inline, compromises have to be made that clash with perfection. The presence of a single ruins the appearance of the whole article. I am not "pro-HTML". I'd love to see TeX supported 100% and could happily accept the whole articles being written in TeX. But as it stands now, we have to mix. Mixing with inline TeX simply ruins the appearance of the whole article on some systems. Mixing with inline HTML does not. It's as simple as that. YohanN7 ( talk) 11:39, 6 March 2015 (UTC)
In the Introduction/Notation section, we have references to Brandwood 2003 and Bracewell 2000. No such reference is listed in the references section. Does anyone have any idea what these references are supposed to refer to? There is a Bracewell 1986, possibly a different edition of the same book, but if so that would need to be checked to make sure the page numbers are still correct. — David Eppstein ( talk) 21:16, 5 March 2015 (UTC)
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It seems to me that the introductory paragraph of this article is not adequate. The first paragraph ought to be almost a dictionary definition, but this is only a very generic description:
In mathematics and in signal processing, the Hilbert transform is a linear operator that takes a function, u(t) of a real variable and produces another function of a real variable H(u)(t).
I'm not qualified to do so, but isn't there some two-sentence summary that describes what the transform actually is? Even if we changed it to "...another function of a real variable H(u)(t), given by the improper integral X," or ", essentially the convolution of u with 1/πt" it would be a better summary.
Maybe that's just not possible. In almost every online citation where people try to define the transform, they actually say what it's like, or what it can be used for. Does it simply defy definition?
Timrprobocom ( talk) 19:20, 30 January 2018 (UTC)
I've tried to tweak the latest to include both the integral expression and the phase shift. Sławomir Biały ( talk) 02:19, 4 February 2018 (UTC)
What purpose is served by including and in the
Hilbert_transform#Table_of_selected_Hilbert_transforms? First of all, the transform of a complex function is the transforms of its real and imaginary parts, recombined as a complex function. But also, what is the application we have in mind?
--
Bob K (
talk) 13:58, 11 February 2018 (UTC)
The following comment was inserted after the section heading of the Titchmarsh's theorem section of this article, by User:24.233.245.156 ( talk • contribs) 08:23, 4 May 2018 (UTC):
Warning: This entry needs corrections. The results below are due to Paley-Wiener (Fourier transform part) and Marcel Riesz (Hilbert transform part). See e.g. https://link.springer.com/article/10.1140/epjh/e2014-50021-1.
-- Pipetricker ( talk) 09:45, 4 May 2018 (UTC)
Here is a partial list of problems with the section ‘Titchmarsh Theorem’
1. As it stands now, the anonymous author(s) XX of this Wikipedia entry refer(s) to some results, attributing them to Titchmarsh in a 1948 book. In that case, a reference to the original 1937 edition should have been used. The 2nd edition is the same as first save some added (mostly irrelevant today) positions to the bibliography.
2. The review of 1937 edition in Zentralblatt fuer Mathematik is by Hille. In the recently (added by XX after my intervention) sentence with a citation to King’s book, Hille is (particularly! according to King) the author of some results in ‘Titchmarsh Theorem Section’ (Chapter V in the book). Here is the beginning of Hille’s review of the book: “A couple of introductory chapters bring the formal theory of Fourier, Laplace and Mellin transforms and the convergence and summability theory of the Fourier simple and double integrals. This is followed by a thorough discussion of the theory of Fourier transforms in the class L, L_2 (Plancherel), and L_p (Titchmarsh). In Chapter V we find the theory of conjugate functions and Hilbert transforms for the classes L_p, p=1 or 1< p, where the central theorems are due to M. Riesz. The rest of review follows (but is here irrelevant).
3. In the so-called Titchmarsh Theorem, as written by XX, there are three dotted sentences (1) (2) (3) concerning H_2 Hardy space, and the equivalence of all three is claimed. Sentence (1) has no meaning, because it does not say in what convergence the limit is meant to be. The intended meaning is almost everywhere convergence, but then (1), as written, becomes false. The sentence (2) is obviously false.
4. Both (1) and (2) can, of course, be corrected. Let us assume we did it. Then (1), after a minimal rephrasing, contains what is called ‘the upper half-plane analogue of Fatou’s theorem on radial limits’. Knowing that we are in H_2 and that (1) holds, the equivalence of (1) and (2) is due to Marcel Riesz and it is referred to Marcel Riesz in every book on the subject written by a mathematician. Similarly, the equivalence of (1) and (3) is then a theorem due to Paley and Wiener (Theorem 5 in their 1934 book) and, strictly speaking, has nothing to do with the Hilbert transform.
5. The purpose of citation to books (as opposed to papers) in the Wikipedia articles is (or should be) to facilitate further study by the reader, who might be, for instance, interested in a proof. Here is Titchmarsh’s proof of the equivalence of (1) and (2): “The equivalence of (1) and (2) follows at once from the above theorems”. We are on the page 128 of the book!
6. This is not the end of Titchmarsh’s pedagogical ‘achievements’. The title of Chapter V is “Conjugate integrals, Hilbert Transforms”. He never defines Hilbert transform. There are no ‘conjugate integrals’ in the text. He speaks about ‘conjugate functions’ instead, but there is no definition either. He never says what ‘regular analytic function’ (a term he uses for holomorphic function) is. There is quite a bit more, but I think it is already clear that this is not a book to be used for citations.
7. So, corrections are needed. But before doing anything, I am awaiting further comments. Math45-oxford ( talk) 15:20, 10 June 2018 (UTC)
Riemann-Hilbert problem. I also have a question concerning the Riemann-Hilbert problem section. What is it supposed to mean that F_+ and F_ solve the problem ‘formally’? Further, let us assume the function f indeed solves the problem (i.e. really as opposed to formally). Where precisely Pandey states the quoted result in Chapter 2? I’d like a precise reference because I just read that chapter twice and cannot find the result. Math45-oxford ( talk) 21:17, 14 June 2018 (UTC)
Precise conditions would only be about a solution of the Riemann-Hilbert problem. But I already assumed that $F plus, minus$ form the solution. Where is the statement about the Hilbert transform? I do not believe it is true, so if I am right it might be rather difficult to find a reference (if I am wrong, the author XXX must have seen it in Pandey – so, again, where is it in Pandey?). By the way, (Titchmarsh’s version of) the Kolmogorov theorem is not correctly copied from Titchmarsh’s book and, as stated, is wrong. The previous Theorem, the one for $p>1$ is also not correct with our definition of Hilbert transform. Titchmarsh had the minus sign, because his Hilbert transform (that he never openly defined) was what for us in ‘inverse Hilbert transform’. Math45-oxford ( talk) 16:24, 15 June 2018 (UTC)
Some more remarks about Section 8.2. Riemann-Hilbert problem. It is stated that if , where is holomorphic in the upper half-plane and is holomorphic in the lower half-plane, then FORMALLY . The reference is to Pandey’s book Chapter 2. Chapter 2 in Pandey’s book gives a sketch of the ‘classical’ (i.e. no distributions) solution of the HP. This is done before Hilbert transform was even introduced. So it looks like the author of this entry might have thought about a reference to Section 6.2 in Pandey’s book. This, however, would not be very helpful, because Pandey’s discussion there is very imprecise (with wrong references to his own papers and to non-existing formulas, in particular). But the main issue is that "FORMALLY" presumably refers to some distributional equality. This would need to be stated in a precise way, otherwise the claimed equality is meaningless. Finally, even assuming things are corrected: Hilbert Problem is not about Hilbert Transform and has its own Wikipedia entry. Is this Section 8.2 needed at all? Math45-oxford ( talk) — Preceding unsigned comment added by Math45-oxford ( talk • contribs) 19:47, 22 July 2018 (UTC)
Section 5. Domain of definition.
1. In the two formulas there the minus sign in front of the integrals is an effect of Zygmund’s use of –H as the definition of the Hilbert transform and should be corrected into +. Further, what would be the reason for using this formula, instead of the defining one, at all? By the way, the formula is also used (and should be corrected) in the Introduction. There is no clear reason to state it in the Introduction either, because that formula is not more explicit than the defining one, once one understands what the principal value is.
2. The almost everywhere convergence happens to be true. As for the norm convergence, I do not know. At any rate, I did not see it stated for the norm convergence this way anywhere. Now, why the claimed convergences would be a consequence of Titchmarsh Theorem? What Titchmarsh Theorem? A few subsections later a so-called Titchmarsh Theorem is stated. It refers to p=2 only and does not provide the claimed implications even in the case of p=2. Math45-oxford ( talk) 20:41, 22 July 2018 (UTC)
The Hilbert transform can be defined in various ways, one of which leads to a change in sign of . The current definition with actually leads to a Fourier multiplier of , not . At least this is my understanding from Simon's "A comprehensive course in analysis." I do not know if it is easier to change the original definition or the Fourier multiplier later, since I think the original definition is actually the nonstandard one and the Fourier multiplier given here is the standard multiplier.
Jupsal ( talk) 00:36, 26 December 2018 (UTC)
Maybe it's just my ignorance, but the sentence that doesn't seem very helpful is:
Specifically, the Hilbert transform of u is its harmonic conjugate v, a function of the real variable t such that the complex-valued function u + i v admits an extension to the complex upper half-plane satisfying the Cauchy–Riemann equations.
Perhaps it can be re-phrased or re-located to a more mathematical section.