On sequence spaces defined by a sequence of moduli and an extension of Kuttner’s theorem
Let A = (ank) be an infinite matrix with limn∑kank ≠ 0, and let p=(pk) be a sequence of positive real numbers. If 1≤pk<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<pk=˜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.  and ). 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.