Classes of uniform convergence of spectral expansions for the one-dimensional Schrödinger operator with a distribution potential

For the self-adjoint Schrödinger operator ? defined on ? by the differential operation ?(d/dx)² + q(x) with a distribution potential q(x) uniformly locally belonging to the space W ₂ ^?1, we describe classes of functions whose spectral expansions corresponding to the operator ? absolutely and uniformly converge on the entire line ?. We characterize the sharp convergence rate of the spectral expansion of a function using a two-sided estimate obtained in the paper for its generalized Fourier transforms.     

A Wiener-Wintner theorem for 1/f power spectra

Wiener's generalized harmonic analysis (GHA) provides a theory of harmonic analysis for subspaces of tempered functions not accessible to the L¹,L², and Fourier series theories; and it does it in a way that is usually more quantitative than that provided by the theory of distributions. On the other hand, GHA does not yield an adequate spectral analysis of large classes of functions, including nonstationary processes and, in particular, 1/f noise. In this paper we adapt GHA to deal with 1/f noise by extending the Wiener-Wintner theorem to the case of 1/f power spectra.     

Mapping properties of generalized Fourier transforms and applications to Kirchhoff equations

We prove the differentiability of generalized Fourier transforms associated with a self-adjoint and strictly elliptic perturbation A of the Laplacian with variable coefficients in an exterior domain, using results on the spectral differentiability of the resolvent of A. Moreover we show that differentiable functions with bounded support and vanishing near the origin are mapped by the generalized Fourier transform into polynomially weighted L ²-spaces. As an application of the generalized Fourier transform and exploiting the previous results, we deal with equations of Kirchhoff type. We will not only show the global (in t) existence and uniqueness of solutions for a class of small data, but also an assertion on its time asymptotic behavior. In addition, we obtain amplified results for Schr?dinger operators . Received March 1999     

Generalized Pseudo-Butterworth Refinable Functions

In this paper we propose the generalized pseudo-Butterworth refinable functions which involve pseudo-splines of type I and II, Butterworth refinable functions, pseudo-Butterworth refinable functions, and almost all symmetric and causal fractional B-splines. Furthermore, the convergence of cascade algorithms associated with the new masks is proved, and Riesz wavelet bases in L ₂(?) corresponding to the parameters are constructed. The regularity of the generalized pseudo-Butterworth refinable functions is also analyzed by Fourier analysis.     

A Gel’fand-type spectral radius formula and stability of linear constrained switching systems

Using ergodic theory, in this paper we present a Gel’fand-type spectral radius formula which states that the joint spectral radius is equal to the generalized spectral radius for a matrix multiplicative semigroup S⁺ restricted to a subset that need not carry the algebraic structure of S⁺ This generalizes the Berger–Wang formula. Using it as a tool, we study the absolute exponential stability of a linear switched system driven by a compact subshift of the one-sided Markov shift associated to S.     

Probability and Fourier Duality for Affine Iterated Function Systems

Let d be a positive integer, and let μ be a finite measure on ? ^d . In this paper we ask when it is possible to find a subset Λ in ? ^d such that the corresponding complex exponential functions e _λ indexed by Λ are orthogonal and total in L ²(μ). If this happens, we say that (μ,Λ) is a spectral pair. This is a Fourier duality, and the x-variable for the L ²(μ)-functions is one side in the duality, while the points in Λ is the other. Stated this way, the framework is too wide, and we shall restrict attention to measures μ which come with an intrinsic scaling symmetry built in and specified by a finite and prescribed system of contractive affine mappings in ? ^d ; an affine iterated function system (IFS). This setting allows us to generate candidates for spectral pairs in such a way that the sets on both sides of the Fourier duality are generated by suitably chosen affine IFSs. For a given affine setup, we spell out the appropriate duality conditions that the two dual IFS-systems must have. Our condition is stated in terms of certain complex Hadamard matrices. Our main results give two ways of building higher dimensional spectral pairs from combinatorial algebra and spectral theory applied to lower dimensional systems.     

The generalized analytic signal

This paper is an explication of the analytic signal in the generalized case, i.e., the analytic signal of a generalized function and of a generalized stochastic process. The contributions of the author are: (1) an L²-theory of distributions which, in the study of the analytic signal, has an advantage over the usual Schwartz-Itô-Gel'fand theory because the Cauchy representation is defined; (2) a proof (Theorem 2.5) that the Schwartz distributions δ, δ⁺, δ^? and ? may be extended to the L₂-case, expressions (Theorems 2.6 and 2.7) for their Hilbert and Fourier transforms in the L₂-case, and expressions (Section 2.1) for their analytic signals; (3) a proof (Theorem 3.3) that an orthogonal L₂-process, and therefore the Fourier transform of a second-order stationary stochastic process (Theorem 3.4), is strictly generalized; (4) a representation theorem (Theorem 3.5) which extends the Itô spectral representation theorem for stationary random distributions to the nonspectral, nonstationary, L₂-case; (5) expressions for the Cauchy representation (Theorem 3.6) and the analytic signal (Theorem 3.7) of an L₂-process; (6) an expression for and the covariance kernel of the analytic signal of white noise (Section 3.4). The word application in the text refers to the application of previously developed concepts.     

Discrete wavelets and the Vilenkin-Chrestenson transform

In the spaces of complex periodic sequences, we use the Vilenkin-Chrestenson transforms to construct new orthogonal wavelet bases defined by finite collections of parameters. Earlier similar bases were defined for the Cantor and Vilenkin groups by means of generalized Walsh functions. It is noted that similar constructions can be realized for biorthogonal wavelets as well as for the space ? ²(?₊).     

Integrability andL 1-convergence of sine series with generalized quasi-convex coefficients

In this paper we obtain a necessary and sufficient condition for the sine series with generalized quasi-convex coefficients to be a Fourier series. Also we studyL ¹-convergence of this series under the said condition on the coefficients.     

A calculus of Fourier integral operators with inhomogeneous phase functions on R<Superscript> <Emphasis Type="Italic">d</Emphasis> </Superscript>

We construct a calculus for generalized SG Fourier integral operators, extending known results to a broader class of symbols of SG type. In particular, we do not require that the phase functions are homogeneous. An essential ingredient in the proofs is a general criterion for asymptotic expansions within the Weyl-Hörmander calculus. We also prove the L²(R^d)-boundedness of the generalized SG Fourier integral operators having regular phase functions and amplitudes uniformly bounded on R^2d.     

Optimal decay rates for the system of elastic waves in R n with structural damping

We consider the Cauchy problem in R ⁿ for the system of elastic waves with structural damping. We derive (almost) optimal decay rates in time for the L ²-norm and the total energy which improves previous results for this system. To derive the estimates for elastic waves, we employ an improvement in a method in the Fourier space, which was developed in our previous works. Our estimates came from those for a generalized energy of α-order in the Fourier space.     

Exactness of homotopy functors of spaces

We will provide an analysis of the generalized Atiyah-Hirzebruch spectral sequence (GAHSS), which was introduced by Hakim-Hashemi and Kahn. To do so, we introduce a new class of functors, called n-exact functors, which are analogous to Goodwillie’s n-excisive functors. In the study of these functors, we introduce a new spectral sequence, the homological Barratt-Goerss spectral sequence (HBGSS), which has properties similar to those of the classical Barratt-Goerss Spectral Sequence on homotopy. We close by giving an identification of the E² term of the GAHSS in the case of 2-exact functors on Moore spaces.     

Generalized Walsh Bases and Applications

We investigate convergence properties of generalized Walsh series associated with signals f∈L ¹[0,1]. We also show how the dependence of the generalized Walsh bases on N×N unitary matrices allows for applications in signal encoding and encryption, provided the signals are piece-wise constant on N-adic subintervals of [0,1].     

A spectral Paley-Wiener theorem

The Fourier inversion formula in polar form is \(f(x) = \int_0^\infty {P_\lambda } f(x)d\lambda \) for suitable functionsf on ? ⁿ , whereP _λ f(x) is given by convolution off with a multiple of the usual spherical function associated with the Euclidean motion group. In this form, Fourier inversion is essentially a statement of the spectral theorem for the Laplacian and the key question is: how are the properties off andP _λ f related? This paper provides a Paley-Wiener theorem within this avenue of thought generalizing a result due to Strichartz and provides a spectral reformulation of a Paley-Wiener theorem for the Fourier transform due to Helgason. As an application we prove support theorems for certain functions of the Laplacian.     

Weyl type theorems for operators satisfying the single-valued extension property

Let T be a bounded linear operator acting on a Banach space X such that T or its adjoint T^∗ has the single-valued extension property. We prove that the spectral mapping theorem holds for the B-Weyl spectrum, and we show that generalized Browder's theorem holds for f(T) for every analytic function f defined on an open neighborhood U of σ(T). Moreover, we give necessary and sufficient conditions for such T to satisfy generalized Weyl's theorem. Some applications are also given.     

Fourier frequencies in affine iterated function systems

We examine two questions regarding Fourier frequencies for a class of iterated function systems (IFS). These are iteration limits arising from a fixed finite families of affine and contractive mappings in Rd, and the “IFS” refers to such a finite system of transformations, or functions. The iteration limits are pairs (X,μ) where X is a compact subset of Rd (the support of μ), and the measure μ is a probability measure determined uniquely by the initial IFS mappings, and a certain strong invariance axiom. The two questions we study are: (1) existence of an orthogonal Fourier basis in the Hilbert space L²(X,μ); and (2) explicit constructions of Fourier bases from the given data defining the IFS.     

Wavelet bases in the Lebesgue spaces on the field of p-adic numbers

In this paper we prove that p-adic wavelets form an unconditional basis in the space L ^r (? _p ⁿ ) and give the characterization of the space L ^r (? _p ⁿ ) in terms of Fourier coefficients of p-adic wavelets.Moreover, the Greedy bases in the Lebesgue spaces on the field of p-adic numbers are also established.     

Some remarks concerning the Fourier transform in the space L 2(ℝ n )

For the Fourier transform in the space L ₂(?²) of square integrable multivariable functions, two practically useful estimates are proved in certain classes of functions characterized by a generalized continuity modulus.     

On the uniform convergence in <Emphasis Type="Italic">C</Emphasis><Superscript>1</Superscript> of Fourier series for a spectral problem with squared spectral parameter in a boundary condition

We study the uniform convergence in C ¹ of the Fourier series of a Hölder function in a system of eigenfunctions corresponding to a spectral problem with squared spectral parameter in a boundary condition. We preliminarily study one more spectral problem with a spectral parameter in a boundary condition.