Abstract
We consider a singular differential oper- ator ∆ on the half line which generalizes the Bessel operator. Using harmonic analysis tools corresponding to ∆, we construct and investigate a new continuous wavelet transform on (0, ∞( tied to ∆. We apply this wavelet transform to invert an intertwining operator between ∆ and the second derivative operator d 2 /dx 2 .
Highlights
Consider the second-order singular differential operator on the half line d2f 2α + 1 df 4n(α + n) ∆f (x) = dx2 + x− dx x2 f (x), where α > −1/2 and n = 0, 1
A well known harmonic analysis on the half line generated by the Bessel operator Lα, is amply and brilliantly exposed by Trimeche in [14]
∫1 × f(1 − t2)α+2n−1/2 dt is a topological isomorphism between two suitable functional spaces, satisfying the intertwining relation d2 X ◦ dx2 = ∆ ◦ X, Through the intertwining operator X, a completely new commutative harmonic analysis on the half line related to the differential operator ∆, was initiated
Summary
Consider the second-order singular differential operator on the half line d2f 2α + 1 df 4n(α + n). A well known harmonic analysis on the half line generated by the Bessel operator Lα, is amply and brilliantly exposed by Trimeche in [14]. ∫1 × f (tx)(1 − t2)α+2n−1/2 dt is a topological isomorphism between two suitable functional spaces, satisfying the intertwining relation d2 X ◦ dx2 = ∆ ◦ X , Through the intertwining operator X , a completely new commutative harmonic analysis on the half line related to the differential operator ∆, was initiated. Classical wavelets have wide applications, ranging from signal analysis in geophysics and acoustics to quantum theory and pure mathematics (see [3, 4, 7] and the references therein)
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