By Carothers N.L.

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**Extra resources for A short course on approximation theory (Math682)**

**Sample text**

Weierstrass gave a separate proof of this result in the same paper containing his theorem on approximation by algebraic polynomials, but it was later pointed out by Lebesgue (1898) that the two theorems are, in fact, equivalent. Lebesgue's proof is based on several elementary observations. We will outline these elementary facts as \exercises with hints," supplying a few proofs here and there, but leaving full details to the reader. We rst justify the use of the word \polynomial" in describing ( ).

The next several exercises concern the modulus of continuity. f ( ) ! 0 as ! 0. f ( ) ". f ( ) 1 for all > 0. 37. f ? g for g(x) = px. 38. f ( ) # 0 as # 0. f is continuous for 0. f . 39. f ( ;1 1 ] ). f ( ac+b ad+b ] a ). 40. Let f be continuously di erentiable on 0 1 ]. f 0 (1=(n + 1)). In order to see why this is of interest, nd a uniformly convergent sequence of polynomials whose derivatives fail to converge uniformly. ] Math 682 Trigonometric Polynomials 5/26/98 Introduction A (real) trigonometric polynomial, or trig polynomial for short, is a function of the form a0 + n ; X k=1 ak cos kx + bk sin kx () where a0 : : : an and b1 : : : bn are real numbers.

Let f be continuously di erentiable on 0 1 ]. f 0 (1=(n + 1)). In order to see why this is of interest, nd a uniformly convergent sequence of polynomials whose derivatives fail to converge uniformly. ] Math 682 Trigonometric Polynomials 5/26/98 Introduction A (real) trigonometric polynomial, or trig polynomial for short, is a function of the form a0 + n ; X k=1 ak cos kx + bk sin kx () where a0 : : : an and b1 : : : bn are real numbers. The degree of a trig polynomial is the highest frequency occurring in any representation of the form ( ) thus, ( ) has degree n provided that one of an or bn is nonzero.

### A short course on approximation theory (Math682) by Carothers N.L.

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