Function Representation in Hilbert Spaces Using Haar Wavelet Series

Abstract

This work explores the application of integral transforms using Scale and Haar wavelet functions to numerically represent a function \( f(t) \). It is based on defining a vector space where any function can be represented as a linear combination of orthogonal basis functions. In this case, the Haar wavelet transform is used, employing Haar functions generated from Scale functions. First, the fundamental mathematical concepts such as Hilbert spaces and orthogonality, necessary for understanding the Haar wavelet transform, are presented. Then, the construction of the Scale and Haar wavelet functions and the process for determining the coefficients for function representation are detailed. The methodology is applied to the function \( f(t) = t^2 \) over the interval \( t \in [-3, 3] \), showing how to calculate the series coefficients for different resolution levels. As the resolution level increases, the approximation of \( f(t) \) improves significantly. Furthermore, the representation of the function \( f(t) = \sin(t) \) over the interval \( t \in [-6, 6] \) using the Haar wavelet series is presented.