Spectral Approximation of Convolution Operators

Fast Convolution Algorithm

The paper A fast algorithm for the convolution of functions with compact support using Fourier extensions by Xu, Austin, and Wei presents a fast algorithm for the convolution of functions with compact support using Fourier extensions. The authors propose a method that leverages the properties of Fourier transforms to efficiently compute convolutions, which is particularly useful in applications where computational speed is critical.

This algorithm can be summarized as follows:

1. Choose an extension parameter on which to base the approximations; taking is standard and should suffice in general.
2. Compute Fourier extension approximations to and as prescribed in the paper by the method given in [5].
3. Compute Fourier extension approximations to , , and by approximately by using the FFT to compute the coefficients of the approximations.
4. Undo the changes of variables to obtain Fourier extension approximations to , , and .

Spectral Approximation of Convolution Operators

In the paper Spectral approximation of convolution operators by Xu and Loureiro, the authors study the spectral approximation of convolution operators. They provide a comprehensive analysis of the convergence properties of these approximations and their applications in various fields, including numerical analysis and applied mathematics.

Volterra-type Convolution of Classical Polynomials

The paper Volterra-type convolution of classical polynomials by Loureiro and Xu investigates the properties of Volterra-type convolutions of classical polynomials. The authors explore the implications of these convolutions in the context of polynomial approximation and provide insights into their computational efficiency.

Spectral Approximation of Convolution Operators of Fredholm Type

Paper Spectral approximation of convolution operators of Fredholm type by Liu, Deng, and Xu focuses on the spectral approximation of convolution operators of Fredholm type. The authors analyze the convergence properties of these approximations and their applications in various fields, including numerical analysis and applied mathematics.

References

1. Xu, K., Austin, A. P., & Wei, K. (2017). A fast algorithm for the convolution of functions with compact support using Fourier extensions. SIAM Journal on Scientific Computing, 39(4), A1552-A1575.
2. Xu, K., & Loureiro, A. F. (2018). Spectral approximation of convolution operators. SIAM Journal on Scientific Computing, 40(4), A2331-A2352.
3. Loureiro, A. F., & Xu, K. (2019). Volterra-type convolution of classical polynomials. Mathematics of Computation, 88(316), 1931-1950.
4. Liu, X., Deng, K., & Xu, K. (2024). Spectral approximation of convolution operators of Fredholm type. SIAM Journal on Scientific Computing, 46(1), A1-A22.
5. Matthysen, R., & Huybrechs, D. (2016). Fast algorithms for the computation of Fourier extensions of arbitrary length. SIAM Journal on Scientific Computing, 38(2), A899-A922.