A comparison on the commutative neutrix convolution of distributions and the exchange formula

Let f̃, g̃ be ultradistributions in ℒ' and let f̃n = f̃ * δn and g̃n = g̃ * σn where {δn} is a sequence in ℒ which converges to the Dirac-delta function δ. Then the neutrix product f̃ ◇ g̃ is defined on the space of ultradistributions ℒ' as the neutrix limit of the sequence {1/2(f̃ng̃ + f̃g̃n)} provided the limit h̃ exist in the sense that N-limn→∞1/2〈f̃ng̃ + f̃g̃n, ψ〉 = 〈h̃, ψ〉 for all ψ in ℒ. We also prove that the neutrix convolution product f g exist in' , if and only if the neutrix product f tild; ◇ g̃ exist in ℒ and the exchange formula F(f g) = f̃ ◇ g̃ is then satisfied.

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