Polynomials in Bernstein form were first used by Bernstein in a constructive proof for the
Weierstrass approximation theorem. With the advent of computer graphics, Bernstein polynomials, restricted to the interval [0, 1], became important in the form of
Bézier curves.
The first few Bernstein basis polynomials for blending 1, 2, 3 or 4 values together are:
The Bernstein basis polynomials of degree n form a
basis for the
vector space of polynomials of degree at most n with real coefficients.
Bernstein polynomials
A linear combination of Bernstein basis polynomials
is called a Bernstein polynomial or polynomial in Bernstein form of degree n.[1] The coefficients are called Bernstein coefficients or Bézier coefficients.
The first few Bernstein basis polynomials from above in
monomial form are:
Properties
The Bernstein basis polynomials have the following properties:
Bernstein polynomials thus provide one way to prove the
Weierstrass approximation theorem that every real-valued continuous function on a real interval [a, b] can be uniformly approximated by polynomial functions over .[7]
A more general statement for a function with continuous kth derivative is
where additionally
is an
eigenvalue of Bn; the corresponding eigenfunction is a polynomial of degree k.
Probabilistic proof
This proof follows Bernstein's original proof of 1912.[8] See also Feller (1966) or Koralov & Sinai (2007).[9][5]
for every δ > 0. Moreover, this relation holds uniformly in x, which can be seen from its proof via
Chebyshev's inequality, taking into account that the variance of 1⁄nK, equal to 1⁄nx(1−x), is bounded from above by 1⁄(4n) irrespective of x.
Because ƒ, being continuous on a closed bounded interval, must be
uniformly continuous on that interval, one infers a statement of the form
uniformly in x. Taking into account that ƒ is bounded (on the given interval) one gets for the expectation
uniformly in x. To this end one splits the sum for the expectation in two parts. On one part the difference does not exceed ε; this part cannot contribute more than ε.
On the other part the difference exceeds ε, but does not exceed 2M, where M is an upper bound for |ƒ(x)|; this part cannot contribute more than 2M times the small probability that the difference exceeds ε.
Finally, one observes that the absolute value of the difference between expectations never exceeds the expectation of the absolute value of the difference, and
Elementary proof
The probabilistic proof can also be rephrased in an elementary way, using the underlying probabilistic ideas but proceeding by direct verification:[10][6][11][12][13]
The following identities can be verified:
("probability")
("mean")
("variance")
In fact, by the binomial theorem
and this equation can be applied twice to . The identities (1), (2), and (3) follow easily using the substitution .
Within these three identities, use the above basis polynomial notation
and let
Thus, by identity (1)
so that
Since f is uniformly continuous, given , there is a such that whenever
. Moreover, by continuity, . But then
The first sum is less than ε. On the other hand, by identity (3) above, and since , the second sum is bounded by times
It follows that the polynomials fn tend to f uniformly.
Generalizations to higher dimension
Bernstein polynomials can be generalized to k dimensions – the resulting polynomials have the form Bi1(x1) Bi2(x2) ... Bik(xk).[1] In the simplest case only products of the unit interval [0,1] are considered; but, using
affine transformations of the line, Bernstein polynomials can also be defined for products a1, b1] × [a2, b2] × ... × [ak, bk. For a continuous function f on the k-fold product of the unit interval, the proof that f(x1, x2, ... , xk) can be uniformly approximated by
is a straightforward extension of Bernstein's proof in one dimension.
[14]
^Koralov, L.; Sinai, Y. (2007). ""Probabilistic proof of the Weierstrass theorem"". Theory of probability and random processes (2nd ed.). Springer. p. 29.
Akhiezer, N. I. (1956),
Theory of approximation (in Russian), translated by Charles J. Hyman, Frederick Ungar, pp. 30–31, Russian edition first published in 1940
Caglar, Hakan; Akansu, Ali N. (July 1993). "A generalized parametric PR-QMF design technique based on Bernstein polynomial approximation". IEEE Transactions on Signal Processing. 41 (7): 2314–2321.
Bibcode:
1993ITSP...41.2314C.
doi:
10.1109/78.224242.
Zbl0825.93863.
Natanson, I.P. (1964). Constructive function theory. Volume I: Uniform approximation. Translated by Alexis N. Obolensky. New York: Frederick Ungar.
MR0196340.
Zbl0133.31101.
Feller, William (1966), An introduction to probability theory and its applications, Vol, II, John Wiley & Sons, pp. 149–150, 218–222
Farouki, Rida T. (2012). "The Bernstein polynomial basis: a centennial retrospective". Comp. Aid. Geom. Des. 29 (6): 379–419.
doi:
10.1016/j.cagd.2012.03.001.
Polynomials in Bernstein form were first used by Bernstein in a constructive proof for the
Weierstrass approximation theorem. With the advent of computer graphics, Bernstein polynomials, restricted to the interval [0, 1], became important in the form of
Bézier curves.
The first few Bernstein basis polynomials for blending 1, 2, 3 or 4 values together are:
The Bernstein basis polynomials of degree n form a
basis for the
vector space of polynomials of degree at most n with real coefficients.
Bernstein polynomials
A linear combination of Bernstein basis polynomials
is called a Bernstein polynomial or polynomial in Bernstein form of degree n.[1] The coefficients are called Bernstein coefficients or Bézier coefficients.
The first few Bernstein basis polynomials from above in
monomial form are:
Properties
The Bernstein basis polynomials have the following properties:
Bernstein polynomials thus provide one way to prove the
Weierstrass approximation theorem that every real-valued continuous function on a real interval [a, b] can be uniformly approximated by polynomial functions over .[7]
A more general statement for a function with continuous kth derivative is
where additionally
is an
eigenvalue of Bn; the corresponding eigenfunction is a polynomial of degree k.
Probabilistic proof
This proof follows Bernstein's original proof of 1912.[8] See also Feller (1966) or Koralov & Sinai (2007).[9][5]
for every δ > 0. Moreover, this relation holds uniformly in x, which can be seen from its proof via
Chebyshev's inequality, taking into account that the variance of 1⁄nK, equal to 1⁄nx(1−x), is bounded from above by 1⁄(4n) irrespective of x.
Because ƒ, being continuous on a closed bounded interval, must be
uniformly continuous on that interval, one infers a statement of the form
uniformly in x. Taking into account that ƒ is bounded (on the given interval) one gets for the expectation
uniformly in x. To this end one splits the sum for the expectation in two parts. On one part the difference does not exceed ε; this part cannot contribute more than ε.
On the other part the difference exceeds ε, but does not exceed 2M, where M is an upper bound for |ƒ(x)|; this part cannot contribute more than 2M times the small probability that the difference exceeds ε.
Finally, one observes that the absolute value of the difference between expectations never exceeds the expectation of the absolute value of the difference, and
Elementary proof
The probabilistic proof can also be rephrased in an elementary way, using the underlying probabilistic ideas but proceeding by direct verification:[10][6][11][12][13]
The following identities can be verified:
("probability")
("mean")
("variance")
In fact, by the binomial theorem
and this equation can be applied twice to . The identities (1), (2), and (3) follow easily using the substitution .
Within these three identities, use the above basis polynomial notation
and let
Thus, by identity (1)
so that
Since f is uniformly continuous, given , there is a such that whenever
. Moreover, by continuity, . But then
The first sum is less than ε. On the other hand, by identity (3) above, and since , the second sum is bounded by times
It follows that the polynomials fn tend to f uniformly.
Generalizations to higher dimension
Bernstein polynomials can be generalized to k dimensions – the resulting polynomials have the form Bi1(x1) Bi2(x2) ... Bik(xk).[1] In the simplest case only products of the unit interval [0,1] are considered; but, using
affine transformations of the line, Bernstein polynomials can also be defined for products a1, b1] × [a2, b2] × ... × [ak, bk. For a continuous function f on the k-fold product of the unit interval, the proof that f(x1, x2, ... , xk) can be uniformly approximated by
is a straightforward extension of Bernstein's proof in one dimension.
[14]
^Koralov, L.; Sinai, Y. (2007). ""Probabilistic proof of the Weierstrass theorem"". Theory of probability and random processes (2nd ed.). Springer. p. 29.
Akhiezer, N. I. (1956),
Theory of approximation (in Russian), translated by Charles J. Hyman, Frederick Ungar, pp. 30–31, Russian edition first published in 1940
Caglar, Hakan; Akansu, Ali N. (July 1993). "A generalized parametric PR-QMF design technique based on Bernstein polynomial approximation". IEEE Transactions on Signal Processing. 41 (7): 2314–2321.
Bibcode:
1993ITSP...41.2314C.
doi:
10.1109/78.224242.
Zbl0825.93863.
Natanson, I.P. (1964). Constructive function theory. Volume I: Uniform approximation. Translated by Alexis N. Obolensky. New York: Frederick Ungar.
MR0196340.
Zbl0133.31101.
Feller, William (1966), An introduction to probability theory and its applications, Vol, II, John Wiley & Sons, pp. 149–150, 218–222
Farouki, Rida T. (2012). "The Bernstein polynomial basis: a centennial retrospective". Comp. Aid. Geom. Des. 29 (6): 379–419.
doi:
10.1016/j.cagd.2012.03.001.