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Displaying similar documents to “Linear transforms supporting circular convolution over a commutative ring with identity”

On the incomplete gamma function and the neutrix convolution

Brian Fisher, Biljana Jolevska-Tuneska, Arpad Takači (2003)

Mathematica Bohemica

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The incomplete Gamma function γ ( α , x ) and its associated functions γ ( α , x + ) and γ ( α , x - ) are defined as locally summable functions on the real line and some convolutions and neutrix convolutions of these functions and the functions x r and x - r are then found.

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

Adem Kiliçman (2001)

Czechoslovak Mathematical Journal

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Let f ˜ , g ˜ be ultradistributions in 𝒵 ' and let f ˜ n = f ˜ * δ n and g ˜ n = g ˜ * σ n where { δ n } is a sequence in 𝒵 which converges to the Dirac-delta function δ . Then the neutrix product f ˜ g ˜ is defined on the space of ultradistributions 𝒵 ' as the neutrix limit of the sequence { 1 2 ( f ˜ n g ˜ + f ˜ g ˜ n ) } provided the limit h ˜ exist in the sense that N - l i m n 1 2 f ˜ n g ˜ + f ˜ g ˜ n , ψ = h ˜ , ψ for all ψ in 𝒵 . We also prove that the neutrix convolution product f * g exist in 𝒟 ' , if and only if the neutrix product f ˜ g ˜ exist in 𝒵 ' and the exchange formula F ( f * g ) = f ˜ g ˜ is then satisfied.