On sequence spaces defined by a sequence of moduli and an extension of Kuttner’s theorem

Abstract

Let  A = (a_nk) be an infinite matrix with  lim_n∑_ka_nk ≠ 0, and let p=(p_k) be a sequence of positive real numbers. If 1≤p_k<H<∞, then the strong A-summability field (with the exponent p) is included in the summability field of A. But the result known as Kuttner’s theorem asserts that if 0<p_k=˜p<1 and A is regular, then there is a sequence which is strongly (C,1)-summable but which is not A-summable. This result was extended by Thorpe (cf. Theorem 6) and Maddox (cf. [8] and [9]). The purpose of the present paper is to extend the results of Thorpe and Maddox to a lacunary strong summability with respect to a sequence of modulus functions.