The nonequispaced fast Fourier transform (NFFT) allows the fast approximate evaluation of trigonometric polynomials with frequencies supported on full box-shaped grids at arbitrary sampling nodes. Due to the curse of dimensionality, the total number of frequencies and thus, the total arithmetic complexity can already be very large for small refinements at medium dimensions. In this paper, we present an approach for the fast approximate evaluation of trigonometric polynomials with frequencies supported on an arbitrary subset of the full grid at arbitrary sampling nodes, which is based on Taylor expansion and rank-1 lattice methods. For the special case of symmetric hyperbolic cross index sets in frequency domain, we present error estimates and numerical results.

We consider multivariate trigonometric polynomials with frequencies supported on a fixed but arbitrary frequency index set I, which is a finite set of integer vectors of length d. Naturally, one is interested in spatial discretizations in the d-dimensional torus such that - the sampling values of the trigonometric polynomial at the nodes of this spatial discretization uniquely determines the trigonometric polynomial, - the corresponding discrete Fourier transform is fast realizable, and - the corresponding fast Fourier transform is stable. An algorithm that computes the discrete Fourier transform and that needs a computational complexity that is bounded from above by terms that are linear in the maximum of the number of input and output data up to some logarithmic factors is called fast Fourier transform. We call the fast Fourier transform stable if the Fourier matrix of the discrete Fourier transform has a condition number near one and the fast algorithm does not corrupt this theoretical stability. We suggest to use rank-1 lattices and a generalization as spatial discretizations in order to sample multivariate trigonometric polynomials and we develop construction methods in order to determine reconstructing sampling sets, i.e., sets of sampling nodes that allow for the unique, fast, and stable reconstruction of trigonometric polynomials. The methods for determining reconstructing rank-1 lattices are component{by{component constructions, similar to the seminal methods that are developed in the field of numerical integration. During this thesis we identify a component{by{component construction of reconstructing rank-1 lattices that allows for an estimate of the number of sampling nodes M |I|\le M\le \max\left(\frac{2}{3}|I|^2,\max\{3\|\mathbf{k}\|_\infty\colon\mathbf{k}\in I\}\right) that is sufficient in order to uniquely reconstruct each multivariate trigonometric polynomial with frequencies supported on the frequency index set I. We observe that the bounds on the number M only depends on the number of frequency indices contained in I and the expansion of I, but not on the spatial dimension d. Hence, rank-1 lattices are suitable spatial discretizations in arbitrarily high dimensional problems. Furthermore, we consider a generalization of the concept of rank-1 lattices, which we call generated sets. We use a quite different approach in order to determine suitable reconstructing generated sets. The corresponding construction method is based on a continuous optimization method. Besides the theoretical considerations, we focus on the practicability of the presented algorithms and illustrate the theoretical findings by means of several examples. In addition, we investigate the approximation properties of the considered sampling schemes. We apply the results to the most important structures of frequency indices in higher dimensions, so-called hyperbolic crosses and demonstrate the approximation properties by the means of several examples that include the solution of Poisson's equation as one representative of partial differential equations.

We consider multivariate trigonometric polynomials with frequencies supported on a fixed but arbitrary frequency index set I, which is a finite set of integer vectors of length d. Naturally, one is interested in spatial discretizations in the d-dimensional torus such that - the sampling values of the trigonometric polynomial at the nodes of this spatial discretization uniquely determines the trigonometric polynomial, - the corresponding discrete Fourier transform is fast realizable, and - the corresponding fast Fourier transform is stable. An algorithm that computes the discrete Fourier transform and that needs a computational complexity that is bounded from above by terms that are linear in the maximum of the number of input and output data up to some logarithmic factors is called fast Fourier transform. We call the fast Fourier transform stable if the Fourier matrix of the discrete Fourier transform has a condition number near one and the fast algorithm does not corrupt this theoretical stability. We suggest to use rank-1 lattices and a generalization as spatial discretizations in order to sample multivariate trigonometric polynomials and we develop construction methods in order to determine reconstructing sampling sets, i.e., sets of sampling nodes that allow for the unique, fast, and stable reconstruction of trigonometric polynomials. The methods for determining reconstructing rank-1 lattices are component{by{component constructions, similar to the seminal methods that are developed in the field of numerical integration. During this thesis we identify a component{by{component construction of reconstructing rank-1 lattices that allows for an estimate of the number of sampling nodes M |I|\le M\le \max\left(\frac{2}{3}|I|^2,\max\{3\|\mathbf{k}\|_\infty\colon\mathbf{k}\in I\}\right) that is sufficient in order to uniquely reconstruct each multivariate trigonometric polynomial with frequencies supported on the frequency index set I. We observe that the bounds on the number M only depends on the number of frequency indices contained in I and the expansion of I, but not on the spatial dimension d. Hence, rank-1 lattices are suitable spatial discretizations in arbitrarily high dimensional problems. Furthermore, we consider a generalization of the concept of rank-1 lattices, which we call generated sets. We use a quite different approach in order to determine suitable reconstructing generated sets. The corresponding construction method is based on a continuous optimization method. Besides the theoretical considerations, we focus on the practicability of the presented algorithms and illustrate the theoretical findings by means of several examples. In addition, we investigate the approximation properties of the considered sampling schemes. We apply the results to the most important structures of frequency indices in higher dimensions, so-called hyperbolic crosses and demonstrate the approximation properties by the means of several examples that include the solution of Poisson's equation as one representative of partial differential equations.

The nonequispaced fast Fourier transform (NFFT) allows the fast approximate evaluation of trigonometric polynomials with frequencies supported on full box-shaped grids at arbitrary sampling nodes. Due to the curse of dimensionality, the total number of frequencies and thus, the total arithmetic complexity can already be very large for small refinements at medium dimensions. In this paper, we present an approach for the fast approximate evaluation of trigonometric polynomials with frequencies supported on an arbitrary subset of the full grid at arbitrary sampling nodes, which is based on Taylor expansion and rank-1 lattice methods. For the special case of symmetric hyperbolic cross index sets in frequency domain, we present error estimates and numerical results.