Fourier Analysis and Approximation by Paul Leo Butzer

By Paul Leo Butzer

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The name approximate identity is justified by the fact that forfe C,, the sequence {f*x,,} tends uniformly toffor n --+ m. Indeed, in view of the properties of {x,,(x)}, the magnitude of the convolution integral f * xn for large n essentially depends upon the value of its integrand near u = 0 (fbeing bounded). )(x) is roughly equal tof(x)(l/27r) x,,(u) du, which tends tof(x) for large n. One of the important features of the convolution integral f* x,, is that the 'best' properties of each of its factors are inherited by the product itself.

Therefore as a counterpart to Prop. 5. Let f E X(R), p E BV. 9) * dp a9 defined through * dp E X(R), and 5 IlfIIxta,IlPllsv. Proof. The case X(R) = C being obvious, let X(R) = Lp, 1 I p < Fu bini's theorem that This implies that the assertion forp 03. It follows by If(x - u)Ipldp(u)I exists for almost all x and belongs to L', proving If 1 < p < 03, by Holder's inequality = 1. since the p-measure of R is finite. , and the proof may be completed as for Prop. 2. In the literature, Lebesgue-Stieltjes integrals are usually considered with respect to arbitrary Borel measures rather than for functions of bounded variation as is the case in this volume.

H/a]*]‘. [ Show that A K f ; x ) E Can for every h E (0, 2 4 and limh,o IIAicf; every f E Xzn (and r E N). 0) - f(o)llxan =0 for 38 APPROXIMATION BY SINGULAR INTEGRALS 6. Let f be defined in a neighbourhood of a point x E R. For (sufficiently small)! &-'f(x). Show (by induction) that + 7. (i) Let f~ Xan. Show that limh,o Aicf; x) = f ( x ) almost everywhere, in particular at all points of continuity off(for r = 1 compare with Prop. 1). (ii) Let f E Xzn. ). 1. 1. ). [l, p. 2541, TIMAN [2, p. 163 ff]) (Hint: GRAVES 8.

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