The Convolution and Fractional Derivative of Distributions

Let {τn} be a certain sequence of functions in D converging to 1 in D .The commutative neutrix convolution f ♦ * g of two distributions f and g in D is defined to be the neutrix limit of the sequence 1 2 {(f τn) * g + f * (gτn)} , provided the limit exists.We present relations between this new convolution and other existing distributional convolutions, and demonstrate its strong computational power in evaluating convolutions as well as applications to defining new fractional derivatives and integrals of generalized functions in the new space H which contains D (R + ).The neutrix convolutions x λ -♦ * x µ + for λ, µ, λ + µ = 0, ±1, ±2, • • • and x λ -♦ * x s + for λ = 0, ±1, ±2, • • • and s = 0, 1, 2, • • • are evaluated, from which other neutrix convolutions are deduced.