Summability Kernels for Lp Multipliers

In this article we study the problem of extending FourierMultipliers on L ^p (T) to those on L ^p (R)by taking convolution with a kernel, called a summabilitykernel. We characterize the space of such kernels for the cases p = 1 and p = 2. For other values of p we give anecessary condition for a function to be asummability kernel. For the case p = 1, we presentproperties of measures which are transferred from M(T) toM(R) by summability kernels. Furthermore it isshown that every l _p sequence can be extended to someL ^q (R) multipliers for certain values of p and q.