Efficient Methods for the Chebyshev-Type Prolate Spheroidal Wave Functions and Corresponding Eigenvalues
Abstract
:1. Introduction
- The CPSWFs are real and smooth on . They constitute a complete orthonormal system in withSince the normalized Chebyshev polynomials also constitute an orthonormal basis in , can be expanded by
- The eigenvalues of the Sturm–Liouville problem CPSWFs are distinct, real and positive, and can be ordered asFor any and , the following uniform bounds for were established by Wang and Zhang ([25], Lem. 3.1)
- is an even function when n is even, and an odd function when n is odd. There holds the parity
- has exactly n real distinct zeros in the interval . When , has real zeros in that interlace with the n zeros of . We define the roots of as the Chebyshev–Prolate–Lobatto points (CPL points). In computation, we use Newton’s iteration method with zeros of Chebyshev polynomial as the initial points [16].
2. The Eigenvalues of the Integral Operator
3. Evaluation of CPSWF and Related Quantities
4. Numerical Results
n | CPU Time (s) | Absolute Errors | |
---|---|---|---|
Fast Interpolation Method | Direct Method | ||
200 | 0.002 | 0.015 | |
400 | 0.004 | 0.036 | |
800 | 0.008 | 0.181 | |
1600 | 0.030 | 1.310 |
n | CPU Time (s) | Absolute Errors | |
---|---|---|---|
Fast Interpolation Method | Direct Method | ||
200 | 0.02 | 0.02 | |
400 | 0.03 | 0.06 | |
800 | 0.11 | 0.20 | |
1600 | 0.44 | 1.26 |
n | CPU Time (s) | Absolute Errors | |
---|---|---|---|
Fast Interpolation Method | Direct Method | ||
160 | 0.007 | 0.008 | |
320 | 0.017 | 0.021 | |
640 | 0.043 | 0.110 | |
1280 | 0.159 | 0.740 | |
2560 | 0.651 | 4.913 | |
5120 | 2.719 | 53.400 |
5. Applications
6. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
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Tian, Y.; Liu, G. Efficient Methods for the Chebyshev-Type Prolate Spheroidal Wave Functions and Corresponding Eigenvalues. Mathematics 2024, 12, 807. https://doi.org/10.3390/math12060807
Tian Y, Liu G. Efficient Methods for the Chebyshev-Type Prolate Spheroidal Wave Functions and Corresponding Eigenvalues. Mathematics. 2024; 12(6):807. https://doi.org/10.3390/math12060807
Chicago/Turabian StyleTian, Yan, and Guidong Liu. 2024. "Efficient Methods for the Chebyshev-Type Prolate Spheroidal Wave Functions and Corresponding Eigenvalues" Mathematics 12, no. 6: 807. https://doi.org/10.3390/math12060807
APA StyleTian, Y., & Liu, G. (2024). Efficient Methods for the Chebyshev-Type Prolate Spheroidal Wave Functions and Corresponding Eigenvalues. Mathematics, 12(6), 807. https://doi.org/10.3390/math12060807