Estimates for coefficients in Jacobi series for functions with limited regularity by fractional calculus

• Published in: Advances in computational mathematics Vol. 50; no. 4
• Main Authors: Liu, Guidong, Liu, Wenjie, Duan, Beiping
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.08.2024; Springer Nature B.V
• Subjects: Computational Mathematics and Numerical Analysis; Computational Science and Engineering; Estimates; Fractional calculus; Mathematical and Computational Biology; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Singularity (mathematics); Sobolev space; Visualization; Jacobi polynomials; Projection error; 41A25; 65N35; Jacobi expansion coefficients; 65M70; 41A10; Generalized Jacobi functions; 41A50; Fractional calculus
• ISSN: 1019-7168; 1572-9044
• DOI: 10.1007/s10444-024-10159-y

In this paper, optimal estimates on the decaying rates of Jacobi expansion coefficients are obtained by fractional calculus for functions with algebraic and logarithmic singularities. This is inspired by the fact that integer-order derivatives fail to deal with singularity of fractional-type, while fractional calculus can. To this end, we first introduce new fractional Sobolev spaces defined as the range of the L p -space under the Riemann-Liouville fractional integral. The connection between these new spaces and classical fractional-order Sobolev spaces is then elucidated. Under this framework, the optimal decaying rate of Jacobi expansion coefficients is obtained, based on which the projection errors under different norms are given. This work is expected to introduce fractional calculus into traditional fields in approximation theory and to explore the possibility in solving classical problems by this ‘new’ tool.