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

Abstract

For q > 1 , the n th order q -advanced spherical Bessel functions of the first kind, j n ( q ; t ) , are introduced. Smooth perturbations, H q ( ω ) , of the Haar wavelet are derived. The inverse Fourier transforms F − 1 [ j n ( 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 ) j 0 ( q ; t ) ] ( ω ) is studied and shown to generate H q ( ω ) . For each n ≥ 1 , F − 1 [ j n ( q ; t ) ] ( ω ) is shown to be a Schwartz wavelet with vanishing j th moments for 0 ≤ j ≤ n − 1 and non-vanishing n th moment. Wavelet frame properties are developed. The family { 2 j / 2 H q ( 2 j ω − k ) | j , k ∈ Z } is seen to be a nearly orthonormal frame for L 2 ( 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 L 2 ( R ) .