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The aim of this book is introduce the distributional family M (x) and give a sense the Fourier transfor of M (x), the convolution product M (x) Mbeta(x), the elemental solution of the hyperbolic operator, the inverse of M (x), the Fourier transform of (k-1) derivative of Dirac s delta in the quadratic form defined by the formula(148);our results generalized properties of Marcel Riesz ultrahyperbolic kernel.