Abstract We prove that if F is a nonzero (possibly noncuspidal) vector-valued Siegel modular form of any degree, then it has infinitely many Fourier coefficients which are indexed by half-integral matrices having odd, square-free (and thus fundamental) discriminant. The proof uses an induction argument in the setting forms. Further, as application variant our result and complementing work A. Po...

Hypercomplex or quaternions numbers have been used recently for both greyscale and colour image processing. Fast, numerous hypercomplex 2D Fourier transforms were presented as a generalization of the complex 2D Fourier transform to this new hypercomplex space. Thus, the major problem was to put an interpretation of what information the Fourier coefficients could provide. In this paper, we first...

In this article we prove that if the coefficients of a Fourier–Legendre expansion satisfy a suitable Hausdorff–type condition, then the series converges to a function which admits a holomorphic extension to a cut–plane. Furthermore, we prove that a Laplace–type (Laplace composed with Radon) transform of the function describing the jump across the cut is the unique Carlsonian interpolation of th...

A straightforward discretisation of problems in d spatial dimensions often leads to an exponential growth in the number of degrees of freedom. Thus, even efficient algorithms like the fast Fourier transform (FFT) have high computational costs. Hyperbolic cross approximations allow for a severe decrease in the number of used Fourier coefficients to represent functions with bounded mixed derivati...

The probability density function of single-point velocity fluctuations in turbulence is studied systematically using Fourier coefficients in the energy-containing range. In ideal turbulence where energy-containing motions are random and independent, the Fourier coefficients tend to Gaussian and independent of each other. Velocity fluctuations accordingly tend to Gaussian. However, if energy-con...

This paper considers the use of the anisotropic total variation seminorm to recover a two dimensional vector x ∈ CN×N from its partial Fourier coefficients, sampled along Cartesian lines. We prove that if (xk,j − xk−1,j)k,j has at most s1 nonzero coefficients in each column and (xk,j − xk,j−1)k,j has at most s2 nonzero coefficients in each row, then, up to multiplication by log factors, one can...