On perfect methods of summability

On Matrices of Mazur Type on Rate Spaces

“…Mazur [6] showed that Cesaro metrices of all positive orders are of type M. Later Hill [5] established that under certain conditions the Norlund, Hausdorff and Weighted Mean are of type M. Chandrasekhara Rao proved that the Abel and the Borel methods are also of type M. General properties of methods of type M are discussed in Wilansky's book [8]. The present paper is devoted to matrix methods of type M in respect of rate spaces and related topics.…”

Commutants for Some Classes of Hausdorff Matrices

“…Multiply both sides of (1) by (1 -t)k, use the representation for H, and sum over k to get For each fixed u , 0 < u < 1, Bl-)-,,^¿H|Vt'"""' Suppose that 77 is such that the solution of (4) for each n = 0, I, ... , m-I implies that ank = 0 for k>n,0<n<m, i.e., from (3), that gno(t) is a polynomial of degree < n . Then, for n = m , (4) becomes (5) / gmo(tu)dq(u) = pmgmo(t) + Pm-l(t),…”

Some Hausdorff matrices not of type M

“…Matrices of type M were first introduced by Mazur [3] and sonamed by Hill [2]. In [2], Hill developed several sufficient conditions for a Hausdorff matrix to be of type M. He showed that there exists a regular Hausdorff matrix not of type M. The particular matrix used contained a zero on the main diagonal.…”

“…In [2], Hill developed several sufficient conditions for a Hausdorff matrix to be of type M. He showed that there exists a regular Hausdorff matrix not of type M. The particular matrix used contained a zero on the main diagonal. He also posed the following question: Does there exist a regular Hausdorff matrix which has no zero elements on the main diagonal and which is not of type M?…”

“…Let ß= {ßk} be a sequence, A a forward difference operator defined by Auk = Hk is commonly referred to as the mass function for p. In [2] it has been shown that, for regular Hausdorff matrices, (1) can be written in the form…”