Displaying similar documents to “A note on the convolution theorem for the Fourier transform”

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.

On the Fourier cosine—Kontorovich-Lebedev generalized convolution transforms

Nguyen Thanh Hong, Trinh Tuan, Nguyen Xuan Thao (2013)

Applications of Mathematics

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We deal with several classes of integral transformations of the form f ( x ) → D ∫ ℝ + 2 1 u ( e - u cosh ( x + v ) + e - u cosh ( x - v ) ) h ( u ) f ( v ) d u d v , where D is an operator. In case D is the identity operator, we obtain several operator properties on L p ( ℝ + ) with weights for a generalized operator related to the Fourier cosine and the Kontorovich-Lebedev integral transforms. For a class of differential operators of infinite order, we prove the unitary property of these transforms on L 2 ( ℝ + ) and define the inversion formula. Further, for an other class of differential operators...

On tempered convolution operators

Saleh Abdullah (1994)

Commentationes Mathematicae Universitatis Carolinae

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In this paper we show that if S is a convolution operator in S ' , and S * S ' = S ' , then the zeros of the Fourier transform of S are of bounded order. Then we discuss relations between the topologies of the space O c ' of convolution operators on S ' . Finally, we give sufficient conditions for convergence in the space of convolution operators in S ' and in its dual.