State and prove Theorem 28 6 and Corollary 28 8 for an arbitrary compact interval ( a, b ) Data from theorem 28 6 Data from corollary 28 8 Theorem 28 6 (Weierstra) Polynomials are dense in C 0, 1 w r t uniform convergence Proof (S N Bernstein) Take sequence (Si)ieN of independent2 measur able funct...

Theorem 286 The polynomials are dense in Ca b with respect to uniform convergence ...

State and prove Theorem 28.6 and Corollary 28.8 for an arbitrary compact interval ([a, b]). Data from

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State and prove Theorem 28.6 and Corollary 28.8 for an arbitrary compact interval $[a, b]$.

Data from theorem 28.6

Data from corollary 28.8

This does not quite work for the Hermite and Laguerre polynomials, which are defined on infinite intervals.

The Trigonometric System and Fourier Series We consider now L = L((-, ), B(-, ), =|(-)). As before we use dx

(28.2) follows easily from the classical result that for j, k = No and l, mN 0, if kj, I cos(jx) cos(kx) dx =

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Measures Integrals And Martingales

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State and prove Theorem 28.6 and Corollary 28.8 for an arbitrary compact interval ([a, b]). Data from

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Theorem 28.6 (Weierstra) Polynomials are dense in C[0, 1] w.r.t. uniform convergence. Proof (S. N. Bernstein) Take sequence (Si)ieN of independent2 measur- able functions on ([0, 1], [0, 1], dx) which are Bernoulli (p, 1 - p)-distributed, 0

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Measures Integrals And Martingales

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