Strong Weighted Mean Summability and Kuttner's Theorem

In this paper we study sequence spaces that arise from the concept of strong weighted mean summability. Let q = ( q n ) be a sequence of positive terms and set Q n = ∑ n k =1 q k . Then the weighted mean matrix M q = ( a nk ) is defined bya n k =q kQ nif k ⩽ n , a nk =0 if k > n . It is well known that M q defines a regular summability method if and only if Q n →∞. Passing to strong summability, we let 0< p <∞. Then[ M q , p ] 0 = { X : 1 Q n∑ k = 1 nq k| x k | p → 0         a s         n → ∞ } ,[ M q , p ] = { X : 1 Q n∑ k = 1 nq k| x k ‐ l | p → 0         a s         n → ∞ ,   f o r   s o m e   n u m b e r   l } ,[ M q , p ] ∞ = { X : sup n 1 Q n∑ k = 1 nq k| x k | p < ∞ } ,are the spaces of all sequences that are strongly M q ‐ summable with index p to 0, strongly M q ‐ summable with index p and strongly M q ‐ bounded with index p , respectively. The most important special case is obtained by taking M q = C 1 , the Cesàro matrix, which leads to the familiar sequence spaces w 0 ( p ), w ( p ) and w ∞ ( p ), respectively, see [ 4 , 21 ]. We remark that strong summability was first studied by Hardy and Littlewood [ 8 ] in 1913 when they applied strong Cesàro summability of index 1 and 2 to Fourier series; orthogonal series have remained the main area of application for strong summability. See [ 32 , §6] for further references. When we abstract from the needs of summability theory certain features of the above sequence spaces become irrelevant; for instance, the q k simply constitute a diagonal transform. Hence, from a sequence space theoretic point of view we are led to study the spaces