MATH 174: Numerical Analysis I

Lecturer: Jomar Fajardo Rabajante1st Sem AY 2018-2019

APPROXIMATION OF DATA SETS AND FUNCTIONS

CHEBYSHEV, 1853Given a continuous function f defined on a closed interval [a,b] and a positive integer n, can we “represent” f by a polynomial p(x), of degree at most n, in such a way that the maximum error at any point x in [a,b] is controlled?

In particular, is it possible to construct p so that the error

is minimized?

)()(max xpxfbxa -££

QUESTIONS?• Why should such a polynomial even exist?• If it does, can we hope to construct it?• If it exists, is it also unique?• What happens if we change the measure of

error to, say,

• Can we approximate f by a rational function? What about by a trigonometric function?

ò -b

adxxpxf 2)()(

APPROXIMATION THEORY

• In mathematics, approximation theory is concerned with how functions can best be approximated with simpler functions, and with quantitatively characterizing the errorsintroduced thereby. Note that what is meant by best and simpler will depend on the application.

WEIERSTRASS (FIRST) APPROXIMATION THEOREM, 1885

THEOREM: Let

For every

there exists a polynomial p such that

].,[ baCf Î

,0>e

. ,)()( bxaxpxf ££<- e

For Math 174, suppose f and p are real-valued

Karl Weierstrass

In other words: Every continuous real-valued function defined on an interval [a,b] can be uniformly approximated as closely as desired by a real-valued polynomial function.

In other words: For every continuous function f:[a,b]àR there exists a sequence of real-valued polynomials p1, p2, p3,… uniformly converging to f on the [a,b].

WEIERSTRASS (FIRST) APPROXIMATION THEOREM, 1885

A nice GIF from wiki

Bernstein polynomials approximating a curve

Why polynomials?

Because polynomials are the simplest functions, and computers can directly evaluate polynomials, this theorem has both practical and theoretical relevance.

WEIERSTRASS (FIRST) APPROXIMATION THEOREM, 1885

The Weierstrass approximation theorem was generalized by Marshall H. Stone in the

STONE-WEIERSTRASS THEOREM (1937).

There are hundreds of proofs of the WeierstrassApproximation Theorem. One is the constructive proof using the Bernstein Polynomials.

Bernstein polynomial is not the only approximating polynomial. We just need this to prove existence.

WEIERSTRASS (FIRST) APPROXIMATION THEOREM, 1885

Bernstein Basis Polynomials

vnvnv xx

vn

xb --÷÷ø

öççè

æ= )1()(,

( )( )

44,4

34,3

23,1

1,1

)(

14)(

13)(

)(

xxb

xxxb

xxxb

xxb

=

-=

-=

=

( )22,0

1,0

0,0

1)(

1)(1)(

xxb

xxbxb

-=

-=

=

EXAMPLES:

Bernstein Basis Polynomials

)(8, xbi0

12 3

8

Bernstein Polynomials

åå=

-

=

-÷÷ø

öççè

æ==

n

v

vnvv

n

vnvvn xx

vn

xbxB00

, )1()();( bbb

Bernstein coefficients or Bézier coefficients

INSIGHT ON A PROOF OF THE WEIERSTRASS APPROXIMATION THEOREM, 1912

Why create “new” polynomials instead of using Taylor polynomials?

Taylor polynomials are not appropriate, they are applicable only to functions that are infinitely differentiable and not to all continuous functions.

WLOG, assume [a,b]=[0,1]. Change of variable:

å=

--÷÷ø

öççè

æ÷øö

çèæ=

NÎ÷øö

çèæ=

n

v

vnvn

v

xxvn

nvfxfB

nvf

0)1();(

n ,Let b

abaxx old

new --

=

INSIGHT ON A PROOF OF THE WEIERSTRASS APPROXIMATION THEOREM, 1912

Notice the similarity to binomial distribution:

1)1(0

=å -÷÷ø

öççè

æ=

-n

v

vnv ppvn

np=value expected [ ]1,0Îp

INSIGHT ON A PROOF OF THE WEIERSTRASS APPROXIMATION THEOREM, 1912

Also the median and mode

INSIGHT ON A PROOF OF THE WEIERSTRASS APPROXIMATION THEOREM, 1912

v=Number of successes

b v,n

(x)

)()}(max{, and Fix

,, xbxbxn

nnxnv =

)()}(max{, and Fix

,, xbxbxn

nnxnv =

INSIGHT ON A PROOF OF THE WEIERSTRASS APPROXIMATION THEOREM, 1912

negligible if n is large

INSIGHT ON A PROOF OF THE WEIERSTRASS APPROXIMATION THEOREM, 1912

Recall: Weak law of large numbers

INSIGHT ON HOW BERNSTEIN PROVED THE WEIERSTRASS APPROXIMATION THEOREM, 1912

úû

ùêë

é=÷

øö

çèæ=

å -÷÷ø

öççè

æ÷øö

çèæ=

=

-

nvnvfE

xxvn

nvfxfB

n

v

vnvn

,...,1,0,

)1();(0

We can treat Bn(f;x) as the expected value of the numbers f(v/n), v=0,1,…,n.

INSIGHT ON HOW BERNSTEIN PROVED THE WEIERSTRASS APPROXIMATION THEOREM, 1912

( )xfnnxf

nnxfEnv

nvfE

xxvn

nvfxfB

n

v

vnvn

=÷øö

çèæ=

úû

ùêë

é÷øö

çèæ»ú

û

ùêë

é=÷

øö

çèæ=

å -÷÷ø

öççè

æ÷øö

çèæ=

=

-

,...,1,0,

)1();(0

If n is large…

In summary, for large n

)();( xfxfBn »

INSIGHT ON HOW BERNSTEIN PROVED THE WEIERSTRASS APPROXIMATION THEOREM, 1912

What we have done is a naïve way of suspecting that

Actually, the above limit is true for any point x in [a,b].

INSIGHT ON HOW BERNSTEIN PROVED THE WEIERSTRASS APPROXIMATION THEOREM, 1912

( ) ¥®® nxfxfBn as ,);(

A nice GIF from wiki

Bernstein polynomials approximating a curve