# Aliasing error of the exp$(\beta \sqrt{1-z^2})$ kernel in the nonuniform fast Fourier transform

@article{Barnett2020AliasingEO, title={Aliasing error of the exp\$(\beta \sqrt\{1-z^2\})\$ kernel in the nonuniform fast Fourier transform}, author={Alex H. Barnett}, journal={arXiv: Numerical Analysis}, year={2020} }

The most popular algorithm for the nonuniform fast Fourier transform (NUFFT) uses the dilation of a kernel $\phi$ to spread (or interpolate) between given nonuniform points and a uniform upsampled grid, combined with an FFT and diagonal scaling (deconvolution) in frequency space. The high performance of the recent FINUFFT library is in part due to its use of a new ``exponential of semicircle'' kernel $\phi(z)=e^{\beta \sqrt{1-z^2}}$, for $z\in[-1,1]$, zero otherwise, whose Fourier transform…

## 2 Citations

How exponentially ill-conditioned are contiguous submatrices of the Fourier matrix?

- Mathematics, Computer ScienceArXiv
- 2020

The proof uses the Kaiser-Bessel transform pair, and estimates on sums over distorted sinc functions, to construct a localized trial vector whose DFT is also localized, and proves a lower bound on the condition number of any cyclically contiguous submatrix of the discrete Fourier transform (DFT) matrix.

Continuous window functions for NFFT

- Computer Science, MathematicsAdv. Comput. Math.
- 2021

This paper considers the continuous/discontinuous Kaiser--Bessel, continuous $\exp$- type, and continuous $\cosh$-type window functions and presents novel explicit error estimates for NFFT with such a window function and derive rules for the optimal choice from the parameters involved in N FFT.

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