On q-advanced spherical Bessel functions of the first kind and perturbations of the Haar wavelet

• Published in: Applied and computational harmonic analysis 2018-03, Vol.44 (2), p.350-413
• Main Authors: Pravica, D.W., Randriampiry, N., Spurr, M.J.
• Format: Article
• Language: English
• Subjects: Advanced differential equation; Jacobi theta function; q-Advanced cosine; q-Advanced Haar wavelet; q-Advanced sine; q-Advanced spherical Bessel functions; q-Legendre polynomials; Smooth approximations to the Haar wavelet; Wavelet frame
• ISSN: 1063-5203; 1096-603X
• DOI: 10.1016/j.acha.2016.05.003

For q>1, the nth order q-advanced spherical Bessel functions of the first kind, jn(q;t), are introduced. Smooth perturbations, Hq(ω), of the Haar wavelet are derived. The inverse Fourier transforms F−1[jn(q;t)](ω) are expressed in terms of the Jacobi theta function and are shown to give genesis to the q-advanced Legendre polynomials P˜n(q;ω). The wavelet F−1[sin⁡(t)j0(q;t)](ω) is studied and shown to generate Hq(ω). For each n≥1, F−1[jn(q;t)](ω) is shown to be a Schwartz wavelet with vanishing jth moments for 0≤j≤n−1 and non-vanishing nth moment. Wavelet frame properties are developed. The family {2j/2Hq(2jω−k)|j,k∈Z} 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 q→1+, 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).