Nonequispaced fast Fourier transforms for bandlimited functions
Melanie Kircheis, Daniel Potts
TL;DR
This work extends the nonequispaced FFT from trigonometric polynomials to bandlimited functions by developing an NFFT-like procedure based on regularized Shannon sampling. It introduces a framework with windowed sinc regularization $\psi$ and propagates Fourier-domain information $\hat f(\boldsymbol k)$ to spatial evaluations $f(\boldsymbol x_j)$ through a precomputation-and-iFFT pipeline, achieving $\mathcal{O}(|\mathcal I_{\boldsymbol M}| \log |\mathcal I_{\boldsymbol M}| + N)$ complexity. The method is expressed in matrix form as $\boldsymbol f = \boldsymbol \Psi \boldsymbol F \boldsymbol D_{\hat{\psi}} \hat{\boldsymbol f}$, paralleling the classical NFFT but tailored to bandlimited signals. A detailed comparison shows the NFFT-like approach yields improved accuracy over the classical NFFT, particularly for non-integer frequencies and larger bandwidths, underscoring its practical relevance for efficient and stable bandlimited function evaluation from irregular samples.
Abstract
In this paper we consider the problem of approximating function evaluations $f(\boldsymbol x_j)$ at given nonequispaced points $\boldsymbol x_j$, $j=1,\dots N$, of a bandlimited function from given values $\hat{f}(\boldsymbol k)$, $\boldsymbol k\in \mathcal I_{\boldsymbol M}$, of its Fourier transform. Note that if a trigonometric polynomial is given, it is already known that this problem can be solved by means of the nonequispaced fast Fourier transform (NFFT). In other words, we introduce a new NFFT-like procedure for bandlimited functions, which is based on regularized Shannon sampling formulas.