Study of a class of wavelet and their applications

Abstract

From the past four decades, the subject of wavelet analysis has drawn much attention from the mathematician, engineers and researchers from different fields. Wavelets are oscillatory basis functions that have been cleverly constructed to have several appealing properties not found in big waves (sines and cosines), such as multiscale structure, the ability to represent a variety of functions in a sparse manner, and simultaneous localization in time and frequency. Wavelets are generated from different function which are called scaling function. Depending on the application and required features, a researcher chooses the scaling function. newlineHere Cardinal B-splines is chosen as scaling function. B-splines are very simple and impressive function because it is a polynomial function and have specific formulae in both time and frequency domain and easy to handle from both computational and implementation purposes. Hence they are more attractive for analyzing and constructing wavelets. We go through a rigorous study of B-splines and its properties. newlineThere is the great interest in the investigation of compactly supported wavelets. The study of compactly supported wavelets has sparked a lot of attention. The computing capabilities of such wavelets, as well as the breadth of their applications, have piqued curiosity. The compactly supported B-spline wavelets have shown to be a useful tool in a variety of scientific and practical applications, such as the finite element technique and image processing. In compared to other known wavelets, they are also utilised and produce extremely good results in numerous areas of applied sciences due to some of their outstanding qualities and mathematical simplicity. Here, Daubechies approach are applied to construct compactly supported B-spline wavelets with orthonormal scaling function. And examples for B-spline wavelets of order 2,3 and 4 are illustrated. Then evaluate the vanishing moment of the same wavelet. newlineNext frames were studied. newline