A Method for Constructing Wavelet Functions on the Real Number Field

TL;DR

This paper introduces a general method for constructing wavelet functions on the real number field using finite seed sequences, ensuring properties like continuity, finite support, and admissibility, with extensions to random wavelets.

Contribution

It presents a novel construction approach for wavelet functions based on finite seed sequences, including conditions for admissibility, higher order vanishing moments, and regularity of random wavelets.

Findings

Wavelet functions can be constructed from seed sequences with zero mean.

Admissibility of the wavelet function is guaranteed by the seed sequence's properties.

Conditions for higher order vanishing moments and regularity of random wavelets are established.

Abstract

A general method to construct wavelet function on real number ffeld is proposed in this article,which is based on finite length sequence.This finite length sequence is called the seed sequence, and the corresponding wavelet function is called the seed sequence wavelet function.The seed sequence wavelet function is continuous and energy concentrated in both time and frequency domains,That is, it has a finite support set in both time and frequency domains. It is proved that if and only if the seed sequence has 0 mean value, the interpolation function satisfy the admissible condition of wavelet function. The conditions corresponding to the higher order vanishing moment of the seed sequence wavelet function are also given in this article. On this basis, the concept of random wavelet function is proposed, and the condition of the regularity of random wavelet is discussed.