$Q$-FOURIER LIPSCHITZ FUNCTIONS FOR THE GENERALIZED FOURIER TRANSFORM IN SPACE $L^{2}_{Q}(\mathbb{R})$

Year 2017, Volume: 5 Issue: 1, 99 - 105, 01.04.2017

S. El Ouadıh , R. Daher

https://izlik.org/JA32UK22PB

Abstract

In this paper, we prove the generalization of Titchmarsh's theorem for the generalized Fourier transform for functions satisfying the $Q$-Fourier Lipschitz condition in the space $L^{2}_{Q}(\mathbb{R})$.

Keywords

Generalized Fourier transform; Differential-difference operator; Generalized dual translation operator.

References

• [1] E. A. Al Zahran, H. El Mir and M. A. Mourou: Intertwining operators associated with a Dunkl type operator on the real line and applications. Far East J. Appl. Math. Appl. 64(2), 129-144 (2012).
• [2] E. A. Al Zahrani and M. A. Mourou: The continuous wavelet transform associated with a Dunkl type operator on the real line. Adv. Pure Math. 3(5), 443-450 (2013).
• [3] E. C. Titchmarsh: Introduction to the theory of Fourier integrals. Claredon, oxford, (1948), Komkniga. Moxow. (2005).
• [4] E. S. Belkina and S. S. Platonov: Equivalence of K-functionals and modulus of smoothness constructed by generalized Dunkl translations. Izv. Vyssh. Uchebn. Zaved. Mat 8, 315 (2008).
• [5] R. F. Al Subaie and M. A. Mourou: Equivalence of K-functionals and modulus of smoothness generated by a generalized Dunkl operator on the real line. Adv. Pure Math. 5(6), 367-376 (2015).
• [6] R. F. Al Subaie and M. A. Mourou: The equivalence theorem for a K-functional and a modulus of smoothness constructed by a singular differential-difference operator on R. BJMCS, 10(4), 1-14 (2015). DOI: 10.9734/BJMCS/2015/19652.