A Note on Fractional Gabor Systems in L2(ℝ)

In this paper, we introduce the notion of Walsh shift-invariant space and present a unified approach to the study of shift-invariant systems to be frames in L2(ℝ+). We obtain a necessary condition and three sufficient conditions under which the Walsh shift-invariant systems constitute frames for L2(ℝ+). Furthermore, we discuss applications of our main results to obtain some known conclusions about the Gabor frames and wavelet frames on positive half line.

Gabor Frames in L2(R)

The purpose of this text is to present some new results in the spectral theory of orthogonal polynomials and Schrodinger operators. These results concern perturbations of the free Schrodinger operator and of the free case for orthogonal polynomials on the unit circle (which corresponds to Verblunsky coefficients equal to 0) and the real line (which corresponds to off-diagonal Jacobi coefficients equal to 1 and diagonal Jacobi coefficients equal to 0). The condition central to our results is that of generalized bounded variation. This class consists of finite linear combinations of sequences of rotated bounded variation with an L^1 perturbation. This generalizes both usual bounded variation and expressions of the form lambda(x) cos(phi x + alpha) with lambda(x) of bounded variation (and, in particular, with lambda(x) = x^gamma, Wigner-von Neumann potentials) as well as their finite linear combinations. Assuming generalized bounded variation and an L^p condition (with any finite p) on the perturbation, our results show preservation of absolutely continuous spectrum, absence of singular continuous spectrum, and that embedded pure points in the continuous spectrum can only occur in an explicit finite set.

It is shown that the Gabor frame operator for frames with lattice such thatab=1/N for some integerN≥1 is continuous on the modulation spaces ([9], [20]) whenever the basic functiong is a Schwartz function and (g, a, b) generates a frame forL2 (R). From this a sufficient condition on the lattice is given which guarantees that one has a Banach frame for the modulation spaces of Feichtinger.

We provide explicit constructions of particularly convenient dual pairs of Gabor frames. We prove that arbitrary polynomials restricted to sufficiently large intervals will generate Gabor frames, at least for small modulation parameters. Unfortunately, no similar function can generate a dual Gabor frame, but we prove that almost any such frame has a dual generated by a B-spline. Finally, for frames generated by any compactly supported function φ whose integer-translates form a partition of unity, e.g., a B-spline, we construct a class of dual frame generators, formed by linear combinations of translates of φ. This allows us to chose a dual generator with special properties, for example, the one with shortest support, or a symmetric one in case the frame itself is generated by a symmetric function. One of these dual generators has the property of being constant on the support of the frame generator.

The far field (FF) of a radiating or scattering object immersed in an acoustic fluid can be represented in the frequency domain as a finite dimension linear combination of basis functions, which are determined by the shape and the size of the object in acoustic wavelengths. The coefficients of the linear combination are obtained as the outputs of a bank of spatial filters whose inputs are either the boundary normal velocity—for the radiation problem—or the boundary pressure and the normal velocity—for the scattering problem—measured by a dense set of sensors. The filter outputs provide the information sufficient to reconstruct the radiated or scattered far field, amplitude and phase. The structure of the filters is identified from the singular valve decomposition of the appropriate radiation or scattering operator, mapping the boundary normal velocity, or the combination of normal velocity and pressure, into the FF. Also, from the filter outputs, the efficiently radiating component of the boundary field can be extracted. For example the boundary total normal velocity may have a localized high peak, which however may be completely or almost completely absent in the radiating component. Selected results of an extensive simulation are presented illustrating the accuracy of the reconstruction of the FF from the spatial filtering of the boundary field.

Now we reach the centerpiece of this volume, which is the theory of bases in Banach spaces. Since every Banach space is a vector space, it has a basis in the ordinary vector space sense, i.e., a set that spans and is linearly independent.However, this definition of basis restricts us to using only finite linear combinations of vectors, while in any normed space it makes sense to deal with infinite series. Restricting to finite linear combinations when working in an infinite-dimensional space is simply too restrictive for most purposes. Moreover, the proof that a vector space basis exists is nonconstructive in general, as it relies on the Axiom of Choice. Hence we need a new notion of basis that is appropriate for infinite-dimensional Banach spaces, and that is the main topic of this chapter.

Signal processing by alternate dual Gabor frames

Oscillations of finite liquid columns

This paper provides and efficient for the derivation of the delay distribution for a TDMA system with a finite buffer. Assuming successive intervals of length equal to the duration of a slot, the density is expressed as (finite) linear combinations of gamma densities and beta densities with positive coefficients. The results are verified by simulations. Comparison with a previously proposed solution demonstrates that the solution presented here is more efficient.

We discuss properties of a class of real-valued functions on a set X2 constructed as finite (real) linear combinations of functions denoted as [(X, ∗); μ], where (X, ∗) is a groupoid (binary system) and μ is a fuzzy subset of X and where [(X, ∗); μ](x, y): = μ(x∗y) − min⁡{μ(x), μ(y)}. Many properties, for example, μ being a fuzzy subgroupoid of X, ∗), can be restated as some properties of [(X, ∗); μ]. Thus, the context provided opens up ways to consider well-known concepts in a new light, with new ways to prove known results as well as to provide new questions and new results. Among these are identifications of many subsemigroups and left ideals of (Bin(X), □) for example.

On Quasi-interpolation with Radial Basis Functions

Direct and Inverse Approximation Theorems for Local Trigonometric Bases

The properties of Fourier transforms and single-center expansions of Slater-type orbitals are used to obtain formulas for general Coulomb integrals as finite linear combinations of overlap and nuclear attraction integrals, thereby permitting the explicit expression of Coulomb integrals in terms of the auxiliary functions normally used in one-electron integral calculations. The formulas are relatively simple and have a form which indicates the avoidance of numerical difficulties found in other evaluation methods. Working formulas are given for all integrals arising from orbitals of principal quantum numbers 1 and 2.

Poincaré and Eisenstein series for Jacobi forms of lattice index