Approximation of Derivatives of Functions Belonging to Lip (𝜶, 𝒑) Class by Legendre Wavelet Method
Anjalee S. Srivastava
Dr. Anjalee S. Srivastava, Assistant Professor, Department of Mathematics, Tolani College of Arts and Science, Adipur, Kachchh, Affiliated to KSAKV Kachchh University, Bhuj, Kachchh (Gujarat), India.
Manuscript received on 06 August 2024 | Revised Manuscript received on 15 September 2024 | Manuscript Accepted on 15 October 2024 | Manuscript published on 30 October 2024 | PP: 31-34 | Volume-4 Issue-2, October 2024 | Retrieval Number: 100.1/ijam.A123806010424 | DOI: 10.54105/ijam.A1238.04021024
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Abstract: 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 ≤ 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.
Keywords: Lipschitz Class, Legendre Wavelets, Haar Wavelets, Fourier Series, Degree of Approximation.
Scope of the Article: Applied Mathematics