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The product of distributions on R m

Cheng Lin-ZhiBrian Fisher — 1992

Commentationes Mathematicae Universitatis Carolinae

The fixed infinitely differentiable function ρ ( x ) is such that { n ρ ( n x ) } is a regular sequence converging to the Dirac delta function δ . The function δ 𝐧 ( 𝐱 ) , with 𝐱 = ( x 1 , , x m ) is defined by δ 𝐧 ( 𝐱 ) = n 1 ρ ( n 1 x 1 ) n m ρ ( n m x m ) . The product f g of two distributions f and g in 𝒟 m ' is the distribution h defined by error n 1 error n m f 𝐧 g 𝐧 , φ = h , φ , provided this neutrix limit exists for all φ ( 𝐱 ) = φ 1 ( x 1 ) φ m ( x m ) , where f 𝐧 = f * δ 𝐧 and g 𝐧 = g * δ 𝐧 .

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