On the Representation of Functions as Fourier Transforms of Unbounded Measures

Abstract

exists and is finite. The motivation for the present work stems from Cooper's paper ([3]), where he generalized the notion of a positive definite function, which owes its inception to 1\iathias ([7]) and its importa,nce to Bochner ([1]). It will turn out, that every locally summable function that is posit,ive definite in Cooper's sense belongs to R(<I»; such a function admits a representation of type (I) in terms of a posit?:ve measure. The symbol (C, I) prefixing the integral appearing in (I) means that the integral is summable by Cesaro's method of order 1 to the value ~(27T)f(x), and ,ve adhere to this notation throughout the paper. Other methods of summability could also he used ,: in particular, Cesaro's method of order ex, for every positive ex, and Abel's method; and it can be shown that the integral (1) is summable by any of these methods to ~(27T)f(x). ahnost eveEY~Yl1(jre; moreover, if f is locally in LZJ for some p, I ::s; p ::s; 00, then