A note on the convolution theorem for the Fourier transform

Charles S. Kahane

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

Convolution in Colombeau's spaces of generalized functions. II: The convolution in $𝒢_{a}$ .

Nedeljkov, M., Pilipović, S. (1992)

Publications de l'Institut Mathématique. Nouvelle Série

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Convolution in Colombeau's Spaces of Generalized Functions; Part II: the Convolution in Ga

Marko Nedeljkov, Stevan Pilipović (1992)

Publications de l'Institut Mathématique

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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 $\tilde{f}$ , $\tilde{g}$ be ultradistributions in $𝒵^{'}$ and let ${\tilde{f}}_{n} = \tilde{f} * δ_{n}$ and ${\tilde{g}}_{n} = \tilde{g} * σ_{n}$ where ${δ_{n}}$ is a sequence in $𝒵$ which converges to the Dirac-delta function $δ$ . Then the neutrix product $\tilde{f} ⋄ \tilde{g}$ is defined on the space of ultradistributions $𝒵^{'}$ as the neutrix limit of the sequence ${\frac{1}{2} ({\tilde{f}}_{n} \tilde{g} + \tilde{f} {\tilde{g}}_{n})}$ provided the limit $\tilde{h}$ exist in the sense that $\underset{n \to \infty}{N - l i m} \frac{1}{2} 〈 {\tilde{f}}_{n} \tilde{g} + \tilde{f} {\tilde{g}}_{n}, ψ 〉 = 〈 \tilde{h}, ψ 〉$ 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 $\tilde{f} ⋄ \tilde{g}$ exist in $𝒵^{'}$ and the exchange formula $F (f ♦ * g) = \tilde{f} ⋄ \tilde{g}$ 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 $f (x) \to D \int_{ℝ_{+}^{2}} \frac{1}{u} (e^{- u cosh (x + v)} + e^{- u cosh (x - v)}) h (u) f (v) d u d v,$ 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...

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

Convolution in Colombeau's spaces of generalized functions. II: The convolution in 𝒢 a .

Convolution in Colombeau's Spaces of Generalized Functions; Part II: the Convolution in Ga

On the incomplete gamma function and the neutrix convolution

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

On the Fourier cosine—Kontorovich-Lebedev generalized convolution transforms

Convolution in Colombeau's spaces of generalized functions. II: The convolution in $𝒢_{a}$ .