Generalized Jacobi Chebyshev Wavelet Approximation

General Background: Wavelet approximations are fundamental in numerical analysis and signal processing, with classical orthogonal polynomials like Jacobi and Chebyshev serving as key tools due to their strong approximation properties. Specific Background: The use of Chebyshev wavelets has been extended through generalized polynomial frameworks, such as Koornwinder’s generalization of Jacobi polynomials, offering more flexibility for function approximation on finite intervals. Knowledge Gap: Despite existing wavelet frameworks, the integration of generalized Jacobi and Chebyshev structures into a unified wavelet approximation scheme remains underexplored. Aims: This study introduces the Generalized Jacobi Chebyshev Wavelet (GJCW) approximation, establishing its theoretical foundations and demonstrating convergence and approximation capabilities. Results: It is shown that for a uniformly bounded function expanded in the GJCW basis, the partial sums yield both convergent and best uniform polynomial approximations. Novelty: The formulation of a new wavelet approximation based on a hybrid of generalized Jacobi and Chebyshev polynomials constitutes a novel contribution, supported by rigorous recurrence relations and multiresolution analysis. Implications: This work enhances the theoretical landscape of wavelet-based function approximation, with potential applications in computational mathematics, signal analysis, and numerical solutions of differential equations. Highlight : Wavelet Construction: The paper defines and constructs generalized Jacobi Chebyshev wavelets using orthogonal polynomials. Approximation Theory: It proves that if the wavelet series converges, then a uniform best polynomial approximation exists. Multiresolution Framework: The approach is grounded in Mallat’s multiresolution analysis, enabling efficient function approximation.