On the inverse diamond kernel of Marcel Riesz

In this paper, we define the diamond Marcel Riesz operator of order (α, β) on the function f by M (f) = Kα,β ∗ f, where Kα,β is diamond kernel of Marcel Riesz, α, β ∈ C, the symbol ∗ designates the convolution, and f ∈ S, S is the Schwartz space of functions. Our purpose of this paper is to obtain the operator N (α,β) = [ M (α,β) ]−1 such that if M (α,β)(f) = φ, then N (α,β)φ = f. Our results generalize formulae appearing in A. Kananthai [On the convolutions of the diamond kernel of Marcel Riesz, Applied Mathematics and Computation, 114(2000), 95 − 101]. Mathematics Subject Classification: 46F10, 46F12