A note on summability methods and spectral analysis

Abstract

We are concerned with analyzing the spectrum of a bounded measurable function on the real line by means of certain summability methods. If fEL', the Fourier transform is 7(t) --fexp (itx)f(x)dx and Z(f) denotes the set of zeros of 7. Given 4 ELI* we may form the convolution fo4 (x) ff(y)o (x-y)dy. The spectrum of 4 is defined by A())= nZ(f) where the intersection is taken over all fELL such that fo4) O. The underlying heuristic principle of this investigation is that t*A(0) if and only if the trigonometric integral fexp (-isx)0(x)dx is summable, in some suitable sense, to 0 in a neighborhood of t. Since we do not assume that fx+l I 4(y) I dy = o(l) as I x| oo , ordinary convergence is not suitable. Beurling(') has treated Abel summability and Pollard, [4], the (R, 2) method. However their results may be extended to a large class of summability methods which are quite easy to describe. (Some of our theorems are new even for the Abel and (R, 2) cases.) DEFINITION. A function kGL' is a spectral kernel if k(O) = 1 and k(x) =f-Zjk'(y)dy where k' L'. (We shall consider k' extended to negative arguments as an odd function.) Set kh(t) =fexp (-itx)k(hx)q5(x)dx. Taking the limit as h-*O gives a regular summability method.