Wavelets generated by Riesz potentials of KdV solitons

Abstract

In recent years the study of wavelets gained much popularity among mathematicians and applied scientists. It firmly established itself in the field of series and integral expansions and signal processing and led to new and interesting applications. Our interest in this article stems from finding a completely new representative of wavelets, the one coming from the fractional derivative of Korteweg–de Vries solitons. More precisely, we mean by this Riesz fractional derivatives of the well-known KdV solitons. The Riesz fractional derivative and its conjugate are given via the Hilbert transform.

Appendix

In order to complement the study of [23] we present here

Theorem 8

(Poisson’s Summation Formula) For $\alpha >0$ and $x\in \mathbb R$

$$\begin{aligned} \sum _{k=-\infty }^{\infty }u_{\alpha }(x+2\pi k)&= \sum _{k=-\infty }^{\infty }\frac{k|k|^{\alpha }}{\sinh (\pi k/2)}e^{ikx},\\ \sum _{k=-\infty }^{\infty }v_{\alpha }(x+2\pi k)&= -i\sum _{k=-\infty }^{\infty }\text{ sgn}(k)\frac{k|k|^{\alpha }}{\sinh (\pi k/2)}e^{ikx}. \end{aligned}$$

Proof

It follows from the asymptotics of the functions $u_{\alpha }(x)$ and $v_{\alpha }(x)$ (see [23]) that for $\alpha >0$ and $|x|\rightarrow \infty$

$$\begin{aligned} u_{\alpha }(x),\; v_{\alpha }(x)=O\left(\frac{1}{\left(1+|x|^{2}\right){}^{(1+\alpha )/2}}\right). \end{aligned}$$

(3.2) and (3.4) show that similar estimates hold for their Fourier transforms. Therefore, by Corollary 2.26 of [3], p. 46, the Poisson Summation formula holds true for these functions. The theorem is proved.$\square$