Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
8898236 | Applied and Computational Harmonic Analysis | 2018 | 64 Pages |
Abstract
For q>1, the nth order q-advanced spherical Bessel functions of the first kind, jn(q;t), are introduced. Smooth perturbations, Hq(Ï), of the Haar wavelet are derived. The inverse Fourier transforms Fâ1[jn(q;t)](Ï) are expressed in terms of the Jacobi theta function and are shown to give genesis to the q-advanced Legendre polynomials PËn(q;Ï). The wavelet Fâ1[sinâ¡(t)j0(q;t)](Ï) is studied and shown to generate Hq(Ï). For each nâ¥1, Fâ1[jn(q;t)](Ï) is shown to be a Schwartz wavelet with vanishing jth moments for 0â¤jâ¤nâ1 and non-vanishing nth moment. Wavelet frame properties are developed. The family {2j/2Hq(2jÏâk)|j,kâZ} is seen to be a nearly orthonormal frame for L2(R) and a perturbation of the Haar basis. The corresponding multiplicatively advanced differential equations (MADEs) satisfied by these new functions are presented. As the parameter qâ1+, convergence of the q-advanced functions to their classical counterparts is shown. A q-Wallis formula is given. Symmetry of the Jacobi theta function is shown to preclude Gibb's type phenomena. A Schwartz function with lower moments vanishing is shown to be a mother wavelet for a frame generating L2(R).
Keywords
Related Topics
Physical Sciences and Engineering
Mathematics
Analysis
Authors
D.W. Pravica, N. Randriampiry, M.J. Spurr,