Abstract
We consider a singular differential-difference operator Λ on R which includes as a particular case the one-dimensional Dunkl operator. By using harmonic analysis tools corresponding to Λ, we introduce and study a new continuous wavelet transform on R tied to Λ. Such a wavelet transform is exploited to invert an intertwining operator between Λ and the first derivative operator d/dx.
Highlights
In this paper we consider the first-order singular differential-difference operator on R fx df dx x f x x q x where 1 2 and q is a C real-valued odd function on R
We consider a singular differential-difference operator Λ on R which includes as a particular case the one-dimensional Dunkl operator
By using harmonic analysis tools corresponding to Λ, we introduce and study a new continuous wavelet transform on R tied to Λ
Summary
In this paper we consider the first-order singular differential-difference operator on R f. For q = 0, we regain the differential-difference operator f x f x , x which is referred to as the Dunkl operator with parameter 1 2 associated with the reflection group Z2 on R Those operators were introduced and studied by Dunkl [1,2,3] in connection with a generalization of the classical theory of spherical harmonics. The intertwining operator X has been exploited to initiate a quite new commutative harmonic analysis on the real line related to the differential-difference operator Λ in which several analytic structures on R were generalized. The classical theory of wavelets on R is extended to the differential-difference operator Λ. Classical wavelets have wide applications, ranging from signal analysis in geophysics and acoustics to quantum theory and pure mathematics (see [12,13,14] and the references therein)
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