Piecewise Polynomial Methods of Haar Transform in Digital Signal Processing

This paper analyzes the mathematical foundations and practical significance of piecewise polynomial methods based on the Haar orthogonal basis in the process of digital signal processing. Algorithms for calculating spectral coefficients in Haar, Schauder, and spline bases are compared, and their structural and computational efficiency is presented through graphs and formulas. In particular, the advantages of fast transform algorithms adapted for piecewise-constant, piecewise-linear, and piecewise-quadratic bases are demonstrated, along with the challenges encountered during their implementation and possible solutions. It is shown that piecewise polynomial methods of the Haar transform can be effectively applied in signal processing systems that require high accuracy and speed.