Margit Pap

Hyperbolic Wavelets and Multiresolution in the Hardy Space of the Upper Half Plane

A multiresolution analysis in the Hardy space of the unit disc was introduced recently (see Pap in J. Fourier Anal. Appl. 17(5):755–776, 2011). In this paper we will introduce an analogous construction in the Hardy space of the upper half plane. The levels of the multiresolution are generated by localized Cauchy kernels on a special hyperbolic lattice in the upper half plane. This multiresolution has the following new aspects: the lattice which generates the multiresolution is connected to the Blaschke group, the Cayley transform and the hyperbolic metric. The second: the nth level of the multiresolution has finite dimension (in classical affine multiresolution this is not the case) and still we have the density property, i.e. the closure in norm of the reunion of the multiresolution levels is equal to the Hardy space of the upper half plane. The projection operator to the nth resolution level is a rational interpolation operator on a finite subset of the lattice points. If we can measure the values of the function on the points of the lattice the discrete wavelet coefficients can be computed exactly. This makes our multiresolution approximation very useful from the point of view of the computational aspects.

Discrete Orthogonality of Zernike Functions and Its Application to Corneal Measurements

The optical aberrations of human eyes – as well as, those of other optical systems – are often characterized with the Zernike coefficients. Although, these coefficients are normally obtained from discrete measurement data via discrete computations, the developers and programmers of these computer programs could not rely on the discrete orthogonality of the Zernike functions despite the orthogonality of the continuous Zernike functions. Recently, meshes of points over the unit disk were found and reported that ensure this property. In the present paper, such meshes of points are used for computing Zernike coefficients in respect of cornea-like test surfaces. Test results are presented concerning the precision of the surface reconstruction from the aforementioned coefficients. The meshes proposed, however, are not exactly like what the developers hoped for. Further work is necessary in two respects, firstly, how to tune the conventional measurements so that the advantages of the proposed meshes can be exploited, secondly, how to design optical sensors that are based on such meshes.

Coorbit Theory and Bergman Spaces

Coorbit theory arose as an attempt to describe in a unified fashion the properties of the continuous wavelet transform and the STFT (Short-time Fourier transform) by taking a group theoretical viewpoint. As a consequence H.G. Feichtinger and K.H. Grochenig have established a rather general approach to atomic decomposition for families of Banach spaces (of functions, distributions, analytic functions, etc.) through integrable group representations (see Feichtinger and Grochenig (Lect. Notes Math. 1302:52–73, 1988; J. Funct. Anal. 86(2):307–340, 1989; Monatsh. Math. 108(2–3):129–148, 1989), Grochenig (Monatsh. Math. 112(3):1–41, 1991)), now known as coorbit theory.They gave also examples for the abstract theory and until now this approach gives new insights on atomic decompositions, even for cases where concrete examples can be obtained by other methods. Due to the flexibility of this theory the class of possible atoms is much larger than it was supposed to be in concrete cases. It is a remarkable fact that almost all classical function spaces in real and complex variable theory occur naturally as coorbit spaces related to certain integrable representations.In the present paper we present an overview of the general theory and applications for the case of the weighted Bergman spaces over the unit disc, indicating the benefits of the group theoretic perspective (more flexibility, at least at a qualitative level, more general atoms).

Sampling and Rational Interpolation for Non-band-limited Signals

This paper concentrates on the frequency domain representation of non-band-limited continuous-time signals. Many LTI systems of practical interest can be represented using an Nth-order linear differential equation with constant coefficients. The frequency response of these systems is a rational function. Hence our aim is to give sampling and interpolation algorithms with good convergence properties for rational functions. A generalization of the Fourier-type representation is analyzed using special rational orthogonal bases: the Malmquist–Takenaka system for the upper and lower half plane. This representation is more efficient in particular classes of signals characterized with a priori fixed properties. Based on the discrete orthogonality of the Malmquist–Takenaka system we introduce new rational interpolation operators for the upper and lower half plane as well. Combining these two interpolations we can give exact interpolation for a large class of rational functions among them for the Runge test function. We study the properties of these rational interpolation operators.

Spectral Properties of Toeplitz Operators Acting on Gabor Type Reproducing Kernel Hilbert Spaces

This is a survey presenting an overview of the results describing the behavior of the eigenvalues of compact Gabor–Toeplitz operators and Gabor multipliers. We introduce Gabor–Toeplitz operators and Gabor multipliers as Toeplitz operators defined in the context of general reproducing kernel Hilbert spaces. In the first case the reproducing kernel Hilbert space is derived from the continuous Gabor reproducing formula, and in the second case, out of the discrete Gabor reproducing formula, based on tight Gabor frames. The extended metaplectic representation provides all affine transformations of the phase-space. Both classes of operators satisfy natural transformation properties with respect to this group, and both have natural interpretations from the point of view of phase space geometry. Toeplitz operators defined on the Fock space of several complex variables are at the background of the topic. The Berezin transform of general reproducing kernel Hilbert spaces applied to both kinds of Toeplitz operators shares in both cases the same natural phase-space interpretation of the Fock space model. In the first part of the survey we discuss the dependence of the eigenvalues on symbols and generating functions. Then we concentrate on Szego type asymptotic formulae in order to analyze the dependence on the symbol and on Schatten class cutoff phenomena in dependence on the generating function. In the second part we restrict attention to symbols which are characteristic functions of phase space domains, called localization domains in the current context. The corresponding Toeplitz operators are called localization operators. We present results expressing mutual interactions between localization domains and generating functions from the point of view of the eigenvalues of the localization operators. In particular, we discuss asymptotic boundary forms quantifying these interactions locally at the boundary points of localization domains. Our approach to localization operators is motivated by the principles of the semiclassical limit. We finish the survey with a list of open problems and possible future research directions.

Lebesgue functions of rational interpolations of non-band-limited functions

This paper concentrates on the Lebesgue functions of rational interpolation of non-band-limited continuous time signals. Approximation based on sampling and interpolation are cornerstones of applied mathematics. In the last years rational interpolations has been in the focus of the investigations, because they have better approximation properties then the polynomial interpolations. The Whittaker-Kotelnikov-Shannon sampling theorem is for band-limited signals and requires the a priori knowledge of the band-width. In [1], [2] new rational interpolation operators were developed for the transfer function of non-band-limited signals, which can be used also in cases when the band-width is not known a priori. The construction of these operators is based on the discrete orthogonality of the Malmquist-Takenaka systems. Combining these interpolations one can give exact interpolation on the real line for a large class of rational functions among them for the Runge test function. Our aim is to study the properties of the Lebesgue function of these rational interpolation operators.