Approximation of Derivatives of Functions Belonging to Lip (𝛼, 𝑝) Class by Legendre Wavelet Method

This paper investigates the approximation of functions by Legendre wavelet expansions when their first and second derivatives belong to the generalized Lipschitz class Lip(𝜶, 𝒑), 0 < 𝛼 less then 1. Explicit error bounds are obtained in the L 2 -norm, showing that the rate of convergence depends on both the resolution level and the polynomial degree of the wavelet basis. The analysis reveals that Legendre wavelet estimators achieve sharper approximation orders than classical Fourier series and Haar wavelet methods under comparable smoothness assumptions. These results extend earlier studies on Lipschitztype approximation and highlight the effectiveness of Legendre wavelets for functions with higher-order regularity.