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

Adem Kiliçman — 2001

Czechoslovak Mathematical Journal

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.

Some properties of geodesic semi E-b-vex functions

Adem Kiliçman; Wedad Saleh — 2015

Open Mathematics

In this study, we introduce a new class of function called geodesic semi E-b-vex functions and generalized geodesic semi E-b-vex functions and discuss some of their properties.

Some commutative neutrix convolution products of functions

Brian Fisher; Adem Kiliçman — 1995

Commentationes Mathematicae Universitatis Carolinae

The commutative neutrix convolution product of the locally summable functions ${cos}_{-} (λ x)$ and ${cos}_{+} (μ x)$ is evaluated. Further similar commutative neutrix convolution products are evaluated and deduced.

Commutative neutrix convolution products of functions

Brian Fisher; Adem Kiliçman — 1994

Commentationes Mathematicae Universitatis Carolinae

The commutative neutrix convolution product of the functions $x^{r} e_{-}^{λ x}$ and $x^{s} e_{+}^{μ x}$ is evaluated for $r, s = 0, 1, 2, ...$ and all $λ, μ$ . Further commutative neutrix convolution products are then deduced.

Some results on the non-commutative neutrix product of distributions and $Γ^{(r)} (x)$ .

Kilicman, Adem — 2000

Bulletin of the Malaysian Mathematical Sciences Society. Second Series

On the non-commutative neutrix product $ln x_{+} \circ x_{+}^{- s}$

Brian Fisher; Adem Kiliçman; Blagovest Damyanov; J. C. Ault — 1996

Commentationes Mathematicae Universitatis Carolinae

The non-commutative neutrix product of the distributions $ln x_{+}$ and $x_{+}^{- s}$ is proved to exist for $s = 1, 2, ...$ and is evaluated for $s = 1, 2$ . The existence of the non-commutative neutrix product of the distributions $x_{+}^{- r}$ and $x_{+}^{- s}$ is then deduced for $r, s = 1, 2, ...$ and evaluated for $r = s = 1$ .

Note on the numerical solutions of the general matrix convolution equations by using the iterative methods and box convolution product.

Kılıçman, Adem; Al Zhour, Zeyad — 2010

Abstract and Applied Analysis

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Currently displaying 1 – 20 of 27

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

Some properties of geodesic semi E-b-vex functions

Some commutative neutrix convolution products of functions

Commutative neutrix convolution products of functions

Some results on the non-commutative neutrix product of distributions and $Γ^{(r)} (x)$ .

On the non-commutative neutrix product $ln x_{+} \circ x_{+}^{- s}$

On the Fresnel integrals and the convolution.

On certain classes of $p$ -valent functions by using complex-order and differential subordination.

Product properties for pairwise Lindelöf spaces.

Some commutative neutrix convolution products of functions.

On the non-commutative neutrix product $(x_{+}^{r} ln x_{+}) \circ x_{-}^{- s}$ .

Note on the solution of transport equation by Tau method and Walsh functions.

On certain subclasses of meromorphically $p$ -valent functions associated by the linear operator $D_{λ}^{n}$ .

On the $S$ -invariance property for $S$ -flows.

Some results on ( $δ$ -pre, $s$ )-continuous functions.

A note on wave equation and convolutions.

Pairwise weakly regular-Lindelöf spaces.

Commutative neutrix convolution products of functions.

On the connection between Kronecker and Hadamard convolution products of matrices and some applications.

Note on the numerical solutions of the general matrix convolution equations by using the iterative methods and box convolution product.