6.7. The Weierstrass Approximation Theorem


In part 4.5. we already used the Lagrange interpolating polynomials to connect given points in the xy-plane. The main issue then however was to capture finitely many values exactly. If these values are values of a function we normally have no idea how acurate its other values are hit..

This part now will prove that polynomials are able to match any continuous function on a closed interval with arbitrary acuracy within the whole range. K. Weierstrass proved this approximation behaviour in 1886, the constructive proof presented here however is due to S. N. Bernstein and originates from 1912.

Theorem (Weierstrass approximation theorem):  For any  f C 0 ([a,b]) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiabgIGiolaadoeadaahaaWcbeqaaiaaicdaaaGccaGGOaGaai4waiaadggacaGGSaGaamOyaiaac2facaGGPaaaaa@3FAA@ and any ε>0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTduMaeyOpa4JaaGimaaaa@3955@ there is a polynomial p such that

|f(x)p(x)|<ε  for all  x[a,b] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadAgacaGGOaGaamiEaiaacMcacqGHsislcaWGWbGaaiikaiaadIhacaGGPaGaaiiFaiabgYda8iabew7aLjaabAgacaqG8dGaaeOCaiaabccacaqGHbGaaeiBaiaabYgacaqGLbGaamiEaiabgIGiolaacUfacaWGHbGaaiilaiaadkgacaGGDbaaaa@5078@
[6.7.1]

At first we only consider the interval [0,1] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4waiaaicdacaGGSaGaaGymaiaac2faaaa@39D1@ . It will be sufficient to carry out the essential prove for this special interval. To this end we construct a sequence of polynomials ( B n f ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadkeadaWgaaWcbaGaamOBaaqabaGccaWGMbGaaiykaaaa@3A20@ which converges on [0,1] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4waiaaicdacaGGSaGaaGymaiaac2faaaa@39D1@ uniformly to  f, thus a sequence which allows an n 0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBaaaleaacaaIWaaabeaakiabgIGiolablwriLoaaCaaaleqabaGaey4fIOcaaaaa@3BDB@ for each ε>0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTduMaeyOpa4JaaGimaaaa@3955@ such that

|f(x) B n f (x)|<ε  for all  n n 0   and all  x[0,1] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadAgacaGGOaGaamiEaiaacMcacqGHsislcaWGcbWaaSbaaSqaaiaad6gaaeqaaOGaamOzaiaacIcacaWG4bGaaiykaiaacYhacqGH8aapcqaH1oqzcaqGMbGaaei=aiaabkhacaqGGaGaaeyyaiaabYgacaqGSbGaaeyzaiaad6gacqGHLjYScaWGUbWaaSbaaSqaaiaaicdaaeqaaOGaaeyDaiaab6gacaqGKbGaaeiiaiaabggacaqGSbGaaeiBaiaabwgacaWG4bGaeyicI4Saai4waiaaicdacaGGSaGaaGymaiaac2faaaa@5DBF@
[6.7.2]

B n 0 f MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOqamaaBaaaleaacaWGUbWaaSbaaWqaaiaaicdaaeqaaaWcbeaakiaadAgaaaa@39B9@ for example would then be a polynomial to prove the Weierstrass theorem.

We now introduce the approximating polynomials using the binomial coefficients  (T n k )T MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikauaabeqaceaaaeaacaWGUbaabaGaam4AaaaacaGGPaaaaa@3932@ .

Definition:  Let  f:[0,1] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacQdacaGGBbGaaGimaiaacYcacaaIXaGaaiyxaiabgkziUkabl2riHcaa@3ED7@ be any function. For an arbitrary n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabgIGiolablwriLoaaCaaaleqabaGaey4fIOcaaaaa@3AEB@ we call

B n f k=0 n f( k n ) (T n k )T X k (1X) nk MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOqamaaBaaaleaacaWGUbaabeaakiaadAgacqGH9aqpdaaeWbqaaiaadAgacaGGOaWaaSaaaeaacaWGRbaabaGaamOBaaaacaGGPaaaleaacaWGRbGaeyypa0JaaGimaaqaaiaad6gaa0GaeyyeIuoakiaacIcafaqabeGabaaabaGaamOBaaqaaiaadUgaaaGaaiykaiaadIfadaahaaWcbeqaaiaadUgaaaGccaGGOaGaaGymaiabgkHiTiaadIfacaGGPaWaaWbaaSqabeaacaWGUbGaeyOeI0Iaam4Aaaaaaaa@501B@
[6.7.3]

the n-th Bernstein polynomial of  f.

The first Bernstein polynomials for an arbitrary  f  are easily calculated:

  • B 1 f =f( 0 1 )(T 1 0 )T X 0 (1X) 1 +f( 1 1 )(T 1 1 )T X 1 (1X) 0 =(f(1)f(0))X+f(0) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@630F@

  • B 2 f =f( 0 2 )(T 2 0 )T X 0 (1X) 2 +f( 1 2 )(T 2 1 )T X 1 (1X) 1 +f( 2 2 )(T 2 2 )T X 2 (1X) 0 =(f(0)2f( 1 2 )+f(1)) X 2 +2(f(0)+f( 1 2 ))X+f(0) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@840C@
     

With the function |X 1 2 | MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadIfacqGHsisldaWcaaqaaiaaigdaaeaacaaIYaaaaiaacYhaaaa@3B3D@ e.g. we have  B 1 |X 1 2 |= 1 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOqamaaBaaaleaacaaIXaaabeaakiaacYhacaWGybGaeyOeI0YaaSaaaeaacaaIXaaabaGaaGOmaaaacaGG8bGaeyypa0ZaaSaaaeaacaaIXaaabaGaaGOmaaaaaaa@3F82@ and B 2 |X 1 2 |= X 2 X+ 1 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOqamaaBaaaleaacaaIYaaabeaakiaacYhacaWGybGaeyOeI0YaaSaaaeaacaaIXaaabaGaaGOmaaaacaGG8bGaeyypa0JaamiwamaaCaaaleqabaGaaGOmaaaakiabgkHiTiaadIfacqGHRaWkdaWcaaqaaiaaigdaaeaacaaIYaaaaaaa@43FF@ .
 

To prove [6.7.2] we need some conclusions from the generalized binomial theorem:

(a+b) n = k=0 n (T n k )T a nk b k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggacqGHRaWkcaWGIbGaaiykamaaCaaaleqabaGaamOBaaaakiabg2da9maaqahabaGaaiikauaabeqaceaaaeaacaWGUbaabaGaam4AaaaacaGGPaGaamyyamaaCaaaleqabaGaamOBaiabgkHiTiaadUgaaaGccaWGIbWaaWbaaSqabeaacaWGRbaaaaqaaiaadUgacqGH9aqpcaaIWaaabaGaamOBaaqdcqGHris5aaaa@4B35@
 

Proposition:  For all n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabgIGiolablwriLoaaCaaaleqabaGaey4fIOcaaaaa@3AEB@ and each x MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgIGiolabl2riHcaa@39DD@ we have

  1. k=0 n (T n k )T x k (1x) nk =1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabCaeaacaGGOaqbaeqabiqaaaqaaiaad6gaaeaacaWGRbaaaiaacMcacaWG4bWaaWbaaSqabeaacaWGRbaaaOGaaiikaiaaigdacqGHsislcaWG4bGaaiykamaaCaaaleqabaGaamOBaiabgkHiTiaadUgaaaaabaGaam4Aaiabg2da9iaaicdaaeaacaWGUbaaniabggHiLdGccqGH9aqpcaaIXaaaaa@49F9@

[6.7.4]
  1. k=0 n k(T n k )T x k (1x) nk =nx MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabCaeaacaWGRbGaaiikauaabeqaceaaaeaacaWGUbaabaGaam4AaaaacaGGPaGaamiEamaaCaaaleqabaGaam4AaaaakiaacIcacaaIXaGaeyOeI0IaamiEaiaacMcadaahaaWcbeqaaiaad6gacqGHsislcaWGRbaaaaqaaiaadUgacqGH9aqpcaaIWaaabaGaamOBaaqdcqGHris5aOGaeyypa0JaamOBaiaadIhaaaa@4C1E@

[6.7.5]
  1. k=0 n k(k1)(T n k )T x k (1x) nk =n(n1) x 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabCaeaacaWGRbGaaiikaiaadUgacqGHsislcaaIXaGaaiykaiaacIcafaqabeGabaaabaGaamOBaaqaaiaadUgaaaGaaiykaiaadIhadaahaaWcbeqaaiaadUgaaaGccaGGOaGaaGymaiabgkHiTiaadIhacaGGPaWaaWbaaSqabeaacaWGUbGaeyOeI0Iaam4AaaaaaeaacaWGRbGaeyypa0JaaGimaaqaaiaad6gaa0GaeyyeIuoakiabg2da9iaad6gacaGGOaGaamOBaiabgkHiTiaaigdacaGGPaGaamiEamaaCaaaleqabaGaaGOmaaaaaaa@54EC@

[6.7.6]
  1. k=0 n k 2 (T n k )T x k (1x) nk =nxn x 2 + n 2 x 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabCaeaacaWGRbWaaWbaaSqabeaacaaIYaaaaOGaaiikauaabeqaceaaaeaacaWGUbaabaGaam4AaaaacaGGPaGaamiEamaaCaaaleqabaGaam4AaaaakiaacIcacaaIXaGaeyOeI0IaamiEaiaacMcadaahaaWcbeqaaiaad6gacqGHsislcaWGRbaaaaqaaiaadUgacqGH9aqpcaaIWaaabaGaamOBaaqdcqGHris5aOGaeyypa0JaamOBaiaadIhacqGHsislcaWGUbGaamiEamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaad6gadaahaaWcbeqaaiaaikdaaaGccaWG4bWaaWbaaSqabeaacaaIYaaaaaaa@558F@

[6.7.7]
  1. k=0 n (knx) 2 (T n k )T x k (1x) nk n 4 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabCaeaacaGGOaGaam4AaiabgkHiTiaad6gacaWG4bGaaiykamaaCaaaleqabaGaaGOmaaaakiaacIcafaqabeGabaaabaGaamOBaaqaaiaadUgaaaGaaiykaiaadIhadaahaaWcbeqaaiaadUgaaaGccaGGOaGaaGymaiabgkHiTiaadIhacaGGPaWaaWbaaSqabeaacaWGUbGaeyOeI0Iaam4AaaaaaeaacaWGRbGaeyypa0JaaGimaaqaaiaad6gaa0GaeyyeIuoakiabgsMiJoaalaaabaGaamOBaaqaaiaaisdaaaaaaa@51C7@

[6.7.8]

Proof:  
1.  The assertion results immediately from the generalized binomial theorem:

k=0 n (T n k )T x k (1x) nk = (x+1x) n =1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabCaeaacaGGOaqbaeqabiqaaaqaaiaad6gaaeaacaWGRbaaaiaacMcacaWG4bWaaWbaaSqabeaacaWGRbaaaOGaaiikaiaaigdacqGHsislcaWG4bGaaiykamaaCaaaleqabaGaamOBaiabgkHiTiaadUgaaaaabaGaam4Aaiabg2da9iaaicdaaeaacaWGUbaaniabggHiLdGccqGH9aqpcaGGOaGaamiEaiabgUcaRiaaigdacqGHsislcaWG4bGaaiykamaaCaaaleqabaGaamOBaaaakiabg2da9iaaigdaaaa@5206@

2.  If n=1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabg2da9iaaigdaaaa@38A0@ there is nothing to show in essence. If n>1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabg6da+iaaigdaaaa@38A2@ we quote [6.7.4] with n1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabgkHiTiaaigdaaaa@3887@ and calculate as follows (note the index shift):

nx =nx k=0 n1 (T n1 k )T x k (1x) n1k = k=0 n1 n! k!(n1k)! x k+1 (1x) nk1 = k=1 n n! (k1)!(nk)! x k (1x) nk = k=1 n k n! k!(nk)! x k (1x) nk = k=0 n k(T n k )T x k (1x) nk MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@B92A@

3.  Only n>2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabg6da+iaaikdaaaa@38A3@ is non trivial and in that case we proceed similar as before:

n(n1) x 2 =n(n1) x 2 k=0 n2 (T n2 k )T x k (1x) n2k = k=0 n2 n! k!(n2k)! x k+2 (1x) n2k = k=2 n n! (k2)!(nk)! x k (1x) nk = k=2 n k(k1) n! k!(nk)! x k (1x) nk = k=0 n k(k1)(T n k )T x k (1x) nk MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@CAE4@

4.  We just need to join [6.7.5] and [6.7.6]:

nxn x 2 + n 2 x 2 =nx+n(n1) x 2 = k=0 n k(T n k )T x k (1x) nk + k=0 n k(k1)(T n k )T x k (1x) nk = k=0 n (k+k(k1))(T n k )T x k (1x) nk = k=0 n k 2 (T n k )T x k (1x) nk MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@A802@

5.  For all x we have  0 (2x1) 2 =4 x 2 4x+14x(1x)1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiabgsMiJkaacIcacaaIYaGaamiEaiabgkHiTiaaigdacaGGPaWaaWbaaSqabeaacaaIYaaaaOGaeyypa0JaaGinaiaadIhadaahaaWcbeqaaiaaikdaaaGccqGHsislcaaI0aGaamiEaiabgUcaRiaaigdacaaMf8Uaeyi1HSTaaGzbVlaaisdacaWG4bGaaiikaiaaigdacqGHsislcaWG4bGaaiykaiabgsMiJkaaigdaaaa@53A2@ . Thus:

x(1x) 1 4 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiaacIcacaaIXaGaeyOeI0IaamiEaiaacMcacqGHKjYOdaWcaaqaaiaaigdaaeaacaaI0aaaaaaa@3E25@

This estimate and the results obtained so far now yield:

k=0 n (knx) 2 (T n k )T x k (1x) nk = k=0 n k 2 (T n k )T x k (1x) nk 2nx k=0 n k(T n k )T x k (1x) nk + n 2 x 2 k=0 n (T n k )T x k (1x) nk = nxn x 2 + n 2 x 2 2nxnx+ n 2 x 2 = nx(1x) n 4 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@B3EA@

Now we are prepared to prove version [6.7.2] of the Weierstrass theorem. For any  f C 0 ([a,b]) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiabgIGiolaadoeadaahaaWcbeqaaiaaicdaaaGccaGGOaGaai4waiaadggacaGGSaGaamOyaiaac2facaGGPaaaaa@3FAA@ and any ε>0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTduMaeyOpa4JaaGimaaaa@3955@ we have to find an n 0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBaaaleaacaaIWaaabeaakiabgIGiolablwriLoaaCaaaleqabaGaey4fIOcaaaaa@3BDB@ such that

|f(x) B n f(x)|<ε  for all  n n 0   and every  x[0,1] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadAgacaGGOaGaamiEaiaacMcacqGHsislcaWGcbWaaSbaaSqaaiaad6gaaeqaaOGaamOzaiaacIcacaWG4bGaaiykaiaacYhacqGH8aapcqaH1oqzcaqGMbGaaei=aiaabkhacaqGGaGaaeyyaiaabYgacaqGSbGaaeyzaiaad6gacqGHLjYScaWGUbWaaSbaaSqaaiaaicdaaeqaaOGaaeyDaiaab6gacaqGKbGaaeiiaiaabggacaqGSbGaaeiBaiaabwgacaWG4bGaeyicI4Saai4waiaaicdacaGGSaGaaGymaiaac2faaaa@5DBF@

We may abbreviate  msup{f(x)|x[0,1]} MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBaiabg2da9iGacohacaGG1bGaaiiCaiaacUhacaWGMbGaaiikaiaadIhacaGGPaGaaiiFaiaadIhacqGHiiIZcaGGBbGaaGimaiaacYcacaaIXaGaaiyxaiaac2haaaa@4771@ as  f is bounded on [0,1] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4waiaaicdacaGGSaGaaGymaiaac2faaaa@39D1@ according to [6.6.4]. As  f is uniformly continuous on [0,1] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4waiaaicdacaGGSaGaaGymaiaac2faaaa@39D1@ (see [6.5.5]) there is a δ>0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdqMaeyOpa4JaaGimaaaa@3953@ such that

|f(x)f(y)|< ε 2   for all  x[0,1]  satisfying |xy|<δ MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadAgacaGGOaGaamiEaiaacMcacqGHsislcaWGMbGaaiikaiaadMhacaGGPaGaaiiFaiabgYda8maalaaabaGaeqyTdugabaGaaGOmaaaacaqGMbGaaei=aiaabkhacaqGGaGaaeyyaiaabYgacaqGSbGaaeyzaiaadIhacqGHiiIZcaGGBbGaaGimaiaacYcacaaIXaGaaiyxaiaab2gacaqGPbGaaeiDaiaacYhacaWG4bGaeyOeI0IaamyEaiaacYhacqGH8aapcqaH0oazaaa@5B47@ [1]

Using the equality  f(x)=f(x) k=0 n (T n k )T x k (1x) nk = k=0 n f(x)(T n k )T x k (1x) nk MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@6649@ (see [6.7.4]) as well as the triangle inequality we see that every n holds the estimate

|f(x) B n f(x)| k=0 n |f(x)f( k n )|(T n k )T x k (1x) nk MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadAgacaGGOaGaamiEaiaacMcacqGHsislcaWGcbWaaSbaaSqaaiaad6gaaeqaaOGaamOzaiaacIcacaWG4bGaaiykaiaacYhacqGHKjYOdaaeWbqaaiaacYhacaWGMbGaaiikaiaadIhacaGGPaGaeyOeI0IaamOzaiaacIcadaWcaaqaaiaadUgaaeaacaWGUbaaaiaacMcacaGG8bGaaiikauaabeqaceaaaeaacaWGUbaabaGaam4AaaaacaGGPaGaamiEamaaCaaaleqabaGaam4AaaaakiaacIcacaaIXaGaeyOeI0IaamiEaiaacMcadaahaaWcbeqaaiaad6gacqGHsislcaWGRbaaaaqaaiaadUgacqGH9aqpcaaIWaaabaGaamOBaaqdcqGHris5aaaa@5FA7@ [2]

For a fixed x we now split the addends involved into two disjoint lots:

A{k{0,,n}||x k n |<δ} B{k{0,,n}||x k n |δ} MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeaabiqaaaqaaiaadgeacqGH9aqpcaGG7bGaam4AaiabgIGiolaacUhacaaIWaGaaiilaiablAciljaacYcacaWGUbGaaiyFaiaacYhacaGG8bGaamiEaiabgkHiTmaalaaabaGaam4Aaaqaaiaad6gaaaGaaiiFaiabgYda8iabes7aKjaac2haaeaacaWGcbGaeyypa0Jaai4EaiaadUgacqGHiiIZcaGG7bGaaGimaiaacYcacqWIMaYscaGGSaGaamOBaiaac2hacaGG8bGaaiiFaiaadIhacqGHsisldaWcaaqaaiaadUgaaeaacaWGUbaaaiaacYhacqGHLjYScqaH0oazcaGG9baaaaaa@62A5@

If kA MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4AaiabgIGiolaadgeaaaa@3926@ we have |f(x)f( k n )|< ε 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadAgacaGGOaGaamiEaiaacMcacqGHsislcaWGMbGaaiikamaalaaabaGaam4Aaaqaaiaad6gaaaGaaiykaiaacYhacqGH8aapdaWcaaqaaiabew7aLbqaaiaaikdaaaaaaa@43C8@ due to [1]  For A MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaiabgcMi5kabgwGigdaa@39F2@ we thus may estimate as follows (In case A= MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaiabg2da9iabgwGigdaa@3931@  [3] is valid as well as the empty sum's value equals 0):

kA |f(x)f( k n )|(T n k )T x k (1x) nk < kA ε 2 (T n k )T x k (1x) nk = ε 2 kA (T n k )T x k (1x) nk = ε 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@8018@ [3]

If kB MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4AaiabgIGiolaadkeaaaa@3927@ the equivalence | k n x|δ (knx) 2 n 2 δ 2 (knx) 2 n 2 δ 2 1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFamaalaaabaGaam4Aaaqaaiaad6gaaaGaeyOeI0IaamiEaiaacYhacqGHLjYScqaH0oazcaaMf8Uaeyi1HSTaaGzbVpaalaaabaGaaiikaiaadUgacqGHsislcaWGUbGaamiEaiaacMcadaahaaWcbeqaaiaaikdaaaaakeaacaWGUbWaaWbaaSqabeaacaaIYaaaaaaakiabgwMiZkabes7aKnaaCaaaleqabaGaaGOmaaaakiaaywW7cqGHuhY2caaMf8+aaSaaaeaacaGGOaGaam4AaiabgkHiTiaad6gacaWG4bGaaiykamaaCaaaleqabaGaaGOmaaaaaOqaaiaad6gadaahaaWcbeqaaiaaikdaaaGccqaH0oazdaahaaWcbeqaaiaaikdaaaaaaOGaeyyzImRaaGymaaaa@63B9@ allows to employ [6.7.8]:

kB |f(x)f( k n )|(T n k )T x k (1x) nk 2m kB (T n k )T x k (1x) nk 2m kB (knx) 2 n 2 δ 2 (T n k )T x k (1x) nk = 2m n 2 δ 2 kB (knx) 2 (T n k )T x k (1x) nk 2m n 2 δ 2 k=0 n (knx) 2 (T n k )T x k (1x) nk 2m n 2 δ 2 n 4 = m 2n δ 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@D8DF@ [4]

Choosing now a natural number n 0 > m ε δ 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBaaaleaacaaIWaaabeaakiabg6da+maalaaabaGaamyBaaqaaiabew7aLjabes7aKnaaCaaaleqabaGaaGOmaaaaaaaaaa@3E0E@ and employing [3] and [4] we may, for n n 0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabgwMiZkaad6gadaWgaaWcbaGaaGimaaqabaaaaa@3A7E@ , extend the estimate [2] to

|f(x) B n f(x)|< ε 2 + m 2n δ 2 ε 2 + ε 2 =ε MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadAgacaGGOaGaamiEaiaacMcacqGHsislcaWGcbWaaSbaaSqaaiaad6gaaeqaaOGaamOzaiaacIcacaWG4bGaaiykaiaacYhacqGH8aapdaWcaaqaaiabew7aLbqaaiaaikdaaaGaey4kaSYaaSaaaeaacaWGTbaabaGaaGOmaiaad6gacqaH0oazdaahaaWcbeqaaiaaikdaaaaaaOGaeyizIm6aaSaaaeaacqaH1oqzaeaacaaIYaaaaiabgUcaRmaalaaabaGaeqyTdugabaGaaGOmaaaacqGH9aqpcqaH1oqzaaa@5517@

Thus the Weierstrass theorem is valid for the interval [0,1] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4waiaaicdacaGGSaGaaGymaiaac2faaaa@39D1@ . The general case is easily reduced to this special one:

For an arbitrary interval [a,b] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4waiaadggacaGGSaGaamOyaiaac2faaaa@3A29@ we consider the linear (and thus continuous) function g(ba)X+a MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zaiabg2da9iaacIcacaWGIbGaeyOeI0IaamyyaiaacMcacaWGybGaey4kaSIaamyyaaaa@3E96@ . It is a bijection from [0,1] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4waiaaicdacaGGSaGaaGymaiaac2faaaa@39D1@ to [a,b] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4waiaadggacaGGSaGaamOyaiaac2faaaa@3A29@ with  f C 0 ([a,b]) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiabgIGiolaadoeadaahaaWcbeqaaiaaicdaaaGccaGGOaGaai4waiaadggacaGGSaGaamOyaiaac2facaGGPaaaaa@3FAA@ being equivalent to  fg C 0 ([0,1]) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiablIHiVjaadEgacqGHiiIZcaWGdbWaaWbaaSqabeaacaaIWaaaaOGaaiikaiaacUfacaaIWaGaaiilaiaaigdacaGGDbGaaiykaaaa@4178@ . Thus for any ε>0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTduMaeyOpa4JaaGimaaaa@3955@ there is a polynomial p such that

|fg(x)p(x)|<ε  for all  x[0,1] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadAgacqWIyiYBcaWGNbGaaiikaiaadIhacaGGPaGaeyOeI0IaamiCaiaacIcacaWG4bGaaiykaiaacYhacqGH8aapcqaH1oqzcaqGMbGaaei=aiaabkhacaqGGaGaaeyyaiaabYgacaqGSbGaaeyzaiaadIhacqGHiiIZcaGGBbGaaGimaiaacYcacaaIXaGaaiyxaaaa@5246@ [5]

As the inverse of g is linear the function p g 1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCaiablIHiVjaadEgadaahaaWcbeqaaiabgkHiTiaaigdaaaaaaa@3ADC@ is a polynomial as well. The equivalence of x[a,b] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgIGiolaacUfacaWGHbGaaiilaiaadkgacaGGDbaaaa@3CAA@ and g 1 (x)[0,1] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zamaaCaaaleqabaGaeyOeI0IaaGymaaaakiaacIcacaWG4bGaaiykaiabgIGiolaacUfacaaIWaGaaiilaiaaigdacaGGDbaaaa@4076@ now allows to restate [5] as follows:

|f(x)p g 1 (x)|=|fg( g 1 (x))p( g 1 (x))|<ε  for all  x[a,b] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadAgacaGGOaGaamiEaiaacMcacqGHsislcaWGWbGaeSigI8Maam4zamaaCaaaleqabaGaeyOeI0IaaGymaaaakiaacIcacaWG4bGaaiykaiaacYhacqGH9aqpcaGG8bGaamOzaiablIHiVjaadEgacaGGOaGaam4zamaaCaaaleqabaGaeyOeI0IaaGymaaaakiaacIcacaWG4bGaaiykaiaacMcacqGHsislcaWGWbGaaiikaiaadEgadaahaaWcbeqaaiabgkHiTiaaigdaaaGccaGGOaGaamiEaiaacMcacaGGPaGaaiiFaiabgYda8iabew7aLjaabAgacaqG8dGaaeOCaiaabccacaqGHbGaaeiBaiaabYgacaqGLbGaamiEaiabgIGiolaacUfacaWGHbGaaiilaiaadkgacaGGDbaaaa@696A@
 

The applet below creates and illustrates the Bernstein polynomials for three selected functions.


6.6. 6.8.