An extension of Mercer’s theorem

Abstract

Some of the known generalisations [4, 7, 8, 9] replace oa by other kinds of summation means. We propose to show that Oa may be replaced by the means of any regular Toeplitz method of summation; and even by those of any convergence-preserving method (if the limits (1+q)s and s are left unspecified). Some reduction in the domain of values of q is not unexpected. No other restriction is required, however, than that Sn should be bounded; and this is, in the generality of Toeplitz methods, necessary for the existence of the means. Mercer's theorem takes the form summability implies convergence (without Tauberian condition) when expressed in terms of the summation means rn= (sn+qun)/(A +q); this is the character of the very general Mercerian theorems of Pitt [8]. They refer to certain integral means, and to Hausdorff means not usually restricted to have the form of rn; so our theorem may not be closely related to them. Agnew [6] gave three theorems of the same type, referring to Toeplitz means. We show that two of these together compose exactly a special case of our Corollary 1 on positive regular summation matrices, apart from our hypothesis of boundedness which Agnew shows to be unnecessary in his more special context. His other theorem of this kind is slightly more general, but only in that it permits the matrix to differ negligibly from one which is positive; it is included in our Corollary 2. Our restrictions on q may perhaps be heavier than necessary, at any rate in the contexts of particular methods of summation. For instance, Hardy [2] showed that Mercer's original theorem with arithmetic means a-n holds throughout the half-plane '(q > -1 (and nowhere