Weinstein Gabor Transform and Applications

Affiliation(s)

Department of Mathematics, College of Sciences, King Faisal University, Ahsaa, Kingdom of Saudi Arabia.

Department of Mathematics, College of Education, King Khalid University, Mohayil, Kingdom of Saudi Arabia.

Department of Mathematics, College of Sciences, King Faisal University, Ahsaa, Kingdom of Saudi Arabia.

Department of Mathematics, College of Education, King Khalid University, Mohayil, Kingdom of Saudi Arabia.

ABSTRACT

In this paper we consider Weinstein operator. We define and study the continuous Gabor transform associated with this operator. We prove a Plancherel formula, an inversion formula and a weak uncertainty principle for it. As applications, we obtain analogous of Heisenberg’s inequality for the generalized continuous Gabor transform. At the end we give the practical real inversion formula for the generalized continuous Gabor transform.

In this paper we consider Weinstein operator. We define and study the continuous Gabor transform associated with this operator. We prove a Plancherel formula, an inversion formula and a weak uncertainty principle for it. As applications, we obtain analogous of Heisenberg’s inequality for the generalized continuous Gabor transform. At the end we give the practical real inversion formula for the generalized continuous Gabor transform.

Cite this paper

H. Mejjaoli and A. Salem, "Weinstein Gabor Transform and Applications,"*Advances in Pure Mathematics*, Vol. 2 No. 3, 2012, pp. 203-210. doi: 10.4236/apm.2012.23029.

H. Mejjaoli and A. Salem, "Weinstein Gabor Transform and Applications,"

References

[1] Z. B. Nahia and N. B. Salem, “Spherical Harmonics and Appli-cations Associated with the Weinstein Operator,” Potential The-ory-ICPT 94, 1996, pp. 235-241.

[2] Z. B. Nahia and N. B. Salem, “On a Mean Value Property Associated with the Weinstein Operator,” Potential Theory-ICPT 94, 1996, pp. 243-253.

[3] M. Brelot, “Equation de Weinstein et Potentiels de Marcel Riesz,” Lecture Notes in Mathematics 681, Séminaire de Théorie de Potentiel Paris, No. 3, 1978, pp. 18-38.

[4] D. Gabor, “Theory of Communication,” Proceedings of the Institute of Electrical Engineers, Vol. 93, No. 26, 1946, pp. 429-457.

[5] S. Saitoh, “Theory of Reproducing Kernels and Its Ap- plications,” Longman Scientific Technical, Harlow, 1988.

[6] H. Mejjaoli and N. Sraieb, “Gabor Transform in Quantum Calculus and Applications,” Fractional Calculus and Applied Analysis, Vol. 12, No. 3, 2009, pp. 320-336.

[1] Z. B. Nahia and N. B. Salem, “Spherical Harmonics and Appli-cations Associated with the Weinstein Operator,” Potential The-ory-ICPT 94, 1996, pp. 235-241.

[2] Z. B. Nahia and N. B. Salem, “On a Mean Value Property Associated with the Weinstein Operator,” Potential Theory-ICPT 94, 1996, pp. 243-253.

[3] M. Brelot, “Equation de Weinstein et Potentiels de Marcel Riesz,” Lecture Notes in Mathematics 681, Séminaire de Théorie de Potentiel Paris, No. 3, 1978, pp. 18-38.

[4] D. Gabor, “Theory of Communication,” Proceedings of the Institute of Electrical Engineers, Vol. 93, No. 26, 1946, pp. 429-457.

[5] S. Saitoh, “Theory of Reproducing Kernels and Its Ap- plications,” Longman Scientific Technical, Harlow, 1988.

[6] H. Mejjaoli and N. Sraieb, “Gabor Transform in Quantum Calculus and Applications,” Fractional Calculus and Applied Analysis, Vol. 12, No. 3, 2009, pp. 320-336.