On \(q\)-advanced spherical Bessel functions of the first kind and perturbations of the Haar wavelet
The authors study in great detail a number of so called \(q\)-advanced special functions that converge to their classical counterparts as \(q\to 1+\) and the objective is to obtain functions with desirable wavelet properties and study their properties. The \(q\)-advanced spherical Bessel functions \(j_n(q;t)\) are defined by \[ j_n(q;t)=(-1)^n\left (\frac 1t \frac {d}{dt}\right )^n \frac {{}_qSin(t)}{t}, \] where \(n\in \mathbb N_0\), \(q>1\), and \[ {}_qSin(t)=\text{sign}(t)\frac{\sum_{M=-\infty}^\infty (-1)^Me^{-q^M|t|}q^{-M(M-1)}}{\sum_{M=-\infty}^\infty (-1)^Mq^{-M^2}}, \] (with a similar definition for \({}_qCos(t)\)). The functions \(j_n(q;t)\) are shown to belong to the Schwartz class and satisfy the multiplicative advanced differential equation \[ j_n''(q;t)+ \frac 2t j_n'(q;t)- \frac {n(n+1)}{t^2} j_n(q;t)= -q^{n+3}j_n(q;qt), \] and each of the odd-order functions \(j_{2k+1}(q;t)\) is a mother wavelet generating a frame for \(\mathcal L^2(\mathbb R)\). Other special functions studied are the inverse Fourier transforms \(\mathcal F^{-1}[j_n(q;t)](\omega)\) and \(\mathcal F^{-1}[\sin(t)j_n(q;t)](\omega)\). The former leads to \(q\)-advanced Legendre polynomials and from the latter one gets a \(\mathcal C^\infty\)-perturbation of the Haar wavelet that gives an almost orthogonal frame for \(\mathcal L^2(\mathbb R)\).
\(q\)-advanced spherical Bessel functions
smooth approximations to the Haar wavelet
scientific article; zbMATH DE number 6825639
scientific article