Inversion formulas for the short-time Fourier transform

The inversion formula for the short-time Fourier transform is usually considered in the weak sense, or only for specific combinations of window functions and function spaces such as L₂. In the present article the so-called θ-summability (with a function parameter θ) is considered which induces norm convergence for a large class of function spaces. Under some conditions on θ we prove that the summation of the short-time Fourier transform of ƒ converges to ƒ in Wiener amalgam norms, hence also in the L_p sense for L_p functions, and pointwise almost everywhere.     

Inversion formula for the windowed Fourier transform, II

In this paper, we study the approximation of the inversion of windowed Fourier transforms using Riemannian sums. We show that for certain window functions, the Riemannian sums are well defined on L ^p (?), 1?<?p?<?∞, and tend to the function to be reconstructed as the sampling density tends to infinity.     

The Fourier Transform of Functions Satisfying the Lipschitz Condition on Rank 1 Symmetric Spaces

We prove an analog of the classical Titchmarsh theorem on the image under the Fourier transform of a set of functions satisfying the Lipschitz condition in L² for functions on noncompact rank 1 Riemannian symmetric spaces.     

Multiplier Theorems for the Short-Time Fourier Transform

So-called short-time Fourier transform multipliers (also called Anti-Wick operators in the literature) arise by applying a pointwise multiplication operator to the STFT before applying the inverse STFT. Boundedness results are investigated for such operators on modulation spaces and on L _p -spaces. Because the proofs apply naturally to Wiener amalgam spaces the results are formulated in this context. Furthermore, a version of the Hardy-Littlewood inequality for the STFT is derived. This paper was written while the author was researching at University of Vienna (NuHAG) supported by Lise Meitner fellowship No M733-N04. This research was also supported by the Hungarian Scientific Research Funds (OTKA) No K67642.     

Asymptotic properties of Gabor frame operators as sampling density tends to infinity

We study the asymptotic properties of Gabor frame operators defined by the Riemannian sums of inverse windowed Fourier transforms. When the analysis and the synthesis window functions are the same, we give necessary and sufficient conditions for the Riemannian sums to be convergent as the sampling density tends to infinity. Moreover, we show that Gabor frame operators converge to the identity operator in operator norm whenever they are generated with locally Riemann integrable window functions in the Wiener space.     

Vector-valued Modulation Spaces and Localization Operators with Operator-valued Symbols

We study the short-time Fourier transformation, modulation spaces, Gabor representations and time-frequency localization operators, for functions and tempered distributions that have as range space a Banach or a Hilbert space. In the Banach space case the theory of modulation spaces contains some modifications of the scalar-valued theory, depending on the Banach space. In the Hilbert space case the modulation spaces have properties similar to the scalar-valued case and the Gabor frame theory essentially works. For localization operators in this context symbols are operator-valued. We generalize two results from the scalar-valued theory on continuity on certain modulation spaces when the symbol belongs to an L^p,q space and M^∞, respectively. The first result is true for any Banach space as range space, and the second result is true for any Hilbert space as range space.     

Application of Interpolational Methods to the Study of Properties of Functions of Several Variables

In this paper, interpolation theorems for spaces of functions of several variables are used to generalize and refine Hörmander's theorem on the multipliers of the Fourier transform from L _p to L _q and the Hardy--Littlewood--Paley inequality for a class of multiple Fourier series in the multidimensional case.     

Convergence of Riemannian sums of inverse wavelet transforms

We study the approximation of the inverse wavelet transform using Riemannian sums.We show that when the Fourier transforms of wavelet functions satisfy some moderate decay condition,the Riemannian sums converge to the function to be reconstructed as the sampling density tends to infinity.We also study the convergence of the operators introduced by the Riemannian sums.Our result improves some known ones.     

Approximation of classes of analytic functions by Fourier sums in the metric of the space <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>

We establish asymptotic equalities for upper bounds of approximations by partial Fourier sums in the metrics of the spaces L _p , 1 ≤ p ≤ ∞, on classes of Poisson integrals of periodic functions belonging to the unit ball of the space L ₁. The results obtained are generalized to the classes of -differentiable (in the sense of Stepanets) functions that admit an analytic extension to a fixed strip of the complex plane. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 10, pp. 1395–1408, October, 2005.     

Inversion Formula for Shearlet Transform in Arbitrary Space Dimensions

In this article, we study the convergence of the inverse shearlet transform in arbitrary space dimensions. For every pair of admissible shearlets, we show that although the integral involved in the inversion formula from the continuous shearlet transform is convergent in the L² sense, it is not true in general whenever pointwise convergence is considered. We give some su?cient conditions for the pointwise convergence to hold. Moreover, for any pair of admissible shearlets we show that the Riemannian sums defined by the inverse shearlet transform are convergent to the original function as the sampling density tends to infinity.     

Boundedness of Multilinear Pseudo-differential Operators on Modulation Spaces

Boundedness results for multilinear pseudodifferential operators on products of modulation spaces are derived based on ordered integrability conditions on the short-time Fourier transform of the operators’ symbols. The flexibility and strength of the introduced methods is demonstrated by their application to the bilinear and trilinear Hilbert transform.     

Annihilating sets for the short time Fourier transform

We obtain a class of subsets of R2d such that the support of the short time Fourier transform (STFT) of a signal f∈L²(Rd) with respect to a window g∈L²(Rd) cannot belong to this class unless f or g is identically zero. Moreover we prove that the L²-norm of the STFT is essentially concentrated in the complement of such a set. A generalization to other Hilbert spaces of functions or distributions is also provided. To this aim we obtain some results on compactness of localization operators acting on weighted modulation Hilbert spaces.     

Frames and Coorbit Theory on Homogeneous Spaces with a Special Guidance on the Sphere

The topic of this article is a generalization of the theory of coorbit spaces and related frame constructions to Banach spaces of functions or distributions over domains and manifolds. As a special case one obtains modulation spaces and Gabor frames on spheres. Group theoretical considerations allow first to introduce generalized wavelet transforms. These are then used to define coorbit spaces on homogeneous spaces, which consist of functions having their generalized wavelet transform in some weighted L_p space. We also describe natural ways of discretizing those wavelet transforms, or equivalently to obtain atomic decompositions and Banach frames for the corresponding coorbit spaces. Based on these facts we treat aspects of nonlinear approximation and show how the new theory can be applied to the Gabor transform on spheres. For the S¹ we exhibit concrete examples of admissible Gabor atoms which are very closely related to uncertainty minimizing states.     

Approximation of Classes of Analytic Functions by Fourier Sums in Uniform Metric

We find asymptotic equalities for upper bounds of approximations by partial Fourier sums in the uniform metric on classes of Poisson integrals of periodic functions belonging to the unit balls in the spaces L _p , 1 ≤ p ≤ ∞. We generalize the results obtained to the classes of (ψ, )-differentiable (in the sense of Stepanets) functions that admit an analytic extension to a fixed strip of the complex plane. __________ Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 8, pp. 1079 – 1096, August, 2005.     

Tight Gabor Frames Associated with Non-separable Lattices and the Hyperbolic Secant

We consider Gabor systems generated by a window given by the hyperbolic secant function. We show that such a system forms a Parseval frame for L ²(?) when the translations and modulations of the window are associated with certain non-separable lattices in ?² which we explicitly describe. We also study the more general problem of characterizing the positive Borel measures μ on ?²ⁿ with the property that the short-time Fourier transform defines an isometric embedding from L ²(? ⁿ ) to L _μ ² (?²ⁿ ) when the window belongs to the Schwartz class and, in particular, we characterize the extreme points of this set. In the case where the window is the hyperbolic secant function, we consider the situation where the measure is discrete with constant weights and supported on a non-separable lattice yielding a Parseval frame. We provide arithmetic conditions on the parameters defining the lattice characterizing when the associated measure is an extreme point.     

Beurling's theorem for Riemannian symmetric spaces II

We prove two versions of Beurling's theorem for Riemannian symmetric spaces of arbitrary rank. One of them uses the group Fourier transform and the other uses the Helgason Fourier transform. This is the master theorem in the quantitative uncertainty principle.

     

On the kolmogorov theorems on Fourier series and conjugate functions

We consider Fourier series of summable functions from spaces ??wider?? than L ₁. We describe classes ??(L) which contain conjugate functions, where their conjugate Fourier series converge. The obtained results are more general than A. N. Kolmogorov theorems on the convergence of Fourier series in metrics weaker than that of L ₁.     

Shift-invariant spaces of tempered distributions and Lp-functions

This paper studies the structure of shift-invariant spaces. A characterization for the univariate shift-invariant spaces of tempered distributions is given. In L_p case, an inclusive relation in terms of Fourier transform is established.     

Asymptotic expansions of Eisenstein integrals and Fourier transform on symmetric spaces

This paper studies the asymptotic expansions of spherical functions on symmetric spaces and Fourier transform of rapidly decreasing functions of L^p type (0 < p ? 2) on these spaces.