The continuous wavelet derived by smoothing function and its application in cosmology

TL;DR

This paper introduces a new method for constructing continuous wavelets using derivatives of smoothing functions, simplifying inverse transforms and demonstrating its effectiveness in cosmological data analysis.

Contribution

A novel wavelet construction scheme based on derivatives of smoothing functions, enhancing ease of use and computational efficiency in cosmological applications.

Findings

Wavelet transforms are easier to compute with the new scheme.

The method effectively analyzes power spectra in cosmology.

Transforms demonstrate practical convenience and strength.

Abstract

The wavelet analysis technique is a powerful tool and is widely used in broad disciplines of engineering, technology, and sciences. In this work, we present a novel scheme of constructing continuous wavelet functions, in which the wavelet functions are obtained by taking the first derivative of smoothing functions with respect to the scale parameter. Due to this wavelet constructing scheme, the inverse transforms are only one-dimensional integrations with respect to the scale parameter, and hence the continuous wavelet transforms constructed in this way are more ready to use than the usual scheme. We then apply the Gaussian-derived wavelet constructed by our scheme to computations of the density power spectrum for dark matter, the velocity power spectrum and the kinetic energy spectrum for baryonic fluid. These computations exhibit the convenience and strength of the continuous wavelet transforms. The transforms are very easy to perform, and we believe that the simplicity of our wavelet scheme will make continuous wavelet transforms very useful in practice.