Harmonic Analysis on the Space of M-positive Vectors

Given a dilation matrix M, a so-called space of M-positive vectors in the Euclidean space is introduced and studied. An algebraic structure of this space is similar to the positive half-line equipped with the termwise addition modulo 2, which is used in the Walsh analysis. The role of harmonics is played by some analogues of the classical Walsh functions. The concept of Fourier transform is introduced, and the Poisson summation formula, Plancherel theorem, Vilenkin-Chrestenson formulas and so on are proved. A kind of analogue of the Schwartz class is studied. This class consists of functions such that both the function itself and its Fourier transform have compact support

Multivariate Anisotropic Interpolation on the Torus

We investigate the error of periodic interpolation, when sampling a function on an arbitrary pattern on the torus. We generalize the periodic Strang-Fix conditions to an anisotropic setting and provide an upper bound for the error of interpolation. These conditions and the investigation of the error especially take different levels of smoothness along certain directions into account