# A Fourier transform and convolution of Diamond operator

Volume 12, Issue 10, pp 656--666
Publication Date: June 14, 2019 Submission Date: July 20, 2017 Revision Date: February 27, 2019 Accteptance Date: April 30, 2019
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### Authors

Wanchak Satsanit - Department of Mathematics, Faculty of Science, Maejo University, Chiang Mai, 50290, Thailand.

### Abstract

In this paper, we define a new operator and give a sense of distribution theory to find the Fourier transform of new operator. It was found that the Fourier transform of new operator related to the Fourier transform of ultrahyperbolic operator and Diamond operator. And we also study the convolution products $\Box^{k}\delta \ast \Box^{l}$ and $\diamondsuit^{k}\delta\ast \diamondsuit^{l}.$

### Share and Cite

##### ISRP Style

Wanchak Satsanit, A Fourier transform and convolution of Diamond operator, Journal of Nonlinear Sciences and Applications, 12 (2019), no. 10, 656--666

##### AMA Style

Satsanit Wanchak, A Fourier transform and convolution of Diamond operator. J. Nonlinear Sci. Appl. (2019); 12(10):656--666

##### Chicago/Turabian Style

Satsanit, Wanchak. "A Fourier transform and convolution of Diamond operator." Journal of Nonlinear Sciences and Applications, 12, no. 10 (2019): 656--666

### Keywords

• Diamond operator
• Fourier transform
• hypersurface

•  46F10
•  46F12

### References

• [1] M. A. Aguirre Tellez, A. Kananthai, On the convolution product of the Distributional Families related to the Diamond operator, Matematiche (Catania), 57 (2002), 39--48

• [2] M. A. Aguirre Tellez, S. E. Trione, The distribution Multiplicative product $P^{-\frac{s}{2}}_{\pm}\delta(x)$, Rev. Colombiana Mat., 27 (1993), 1--7

• [3] I. M. Gel'fand, G. E. Shilov, Generalized Function , (Translated from the Russian by Eugene Saletan), Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London (1964)

• [4] A. González Domínguez, S. E. Trione, On the Laplace transform of retarded Lorentz invariant functions, Adv. in Math., 31 (1979), 51--62

• [5] A. Kananthai, On the Solution of the $n$-Dimensional Diamond Operator, Appl. Math. Comput., 88 (1997), 27--37

• [6] A. Kananthai, On the Fourier transform of the Diamond kernel of Marcel Riesz, Appl. Math. Comput., 101 (1999), 151--158

• [7] A. Kananthai, K. Nonlaopon, On the Residue of Generalized Function $P^{\lambda}$, Thai J. Math., 1 (2003), 49--57

• [8] K. Nonlaopon, M. Aguirre Tellez, The Residue of the Generalized function $P^{\lambda}_{+}$ in hypercone $P=(ar)^{2}-(br)^{2}$, to appear, (),

• [9] Y. Nozaki, On Riemann-Liouville integral of ultra--hyperbolic type, Kodai Math. Sem. Rep., 16 (1964), 69--87

• [10] S. E. Trione, M. A. Aguirre Tellez, The distribution convolution products of Marcel Riesz's ultra-hyperbolic kernel, Ravista de la Union Mathematica Argentina, 39 (1995), 115--124