On the uniform convergence of spectral expansions for a spectral problem with boundary conditions of the third kind one of which contains the spectral parameter

We study the uniform convergence, on a closed interval, of spectral expansions of Hölder functions in a given complete and minimal system of eigenfunctions corresponding to a spectral problem with spectral parameter in a boundary condition. We consider boundary conditions of the third kind and subject the function to be expanded to a condition of nonlocal type ensuring the uniform convergence. We prove a theorem stating that expansions in the entire system of eigenfunctions of the problem are possible without any additional conditions.     

On the convergence in W 2 m of spectral expansions for a spectral problem with boundary conditions of the third kind one of which contains the spectral parameter

We consider a spectral problem with boundary conditions of the third kind one of which contains the spectral parameter occurring linearly. We study the basis property of the system of eigenfunctions of this spectral problem in W ₂ ^m . We obtain conditions under which the system becomes a basis in this space after the deletion of any single eigenfunction.     

Some progress in spectral methods

In this paper,we review some results on the spectral methods.We frst consider the Jacobi spectral method and the generalized Jacobi spectral method for various problems,including degenerated and singular diferential equations.Then we present the generalized Jacobi quasi-orthogonal approximation and its applications to the spectral element methods for high order problems with mixed inhomogeneous boundary conditions.We also discuss the related spectral methods for non-rectangular domains and the irrational spectral methods for unbounded domains.Next,we consider the Hermite spectral method and the generalized Hermite spectral method with their applications.Finally,we consider the Laguerre spectral method and the generalized Laguerre spectral method for many problems defned on unbounded domains.We also present the generalized Laguerre quasi-orthogonal approximation and its applications to certain problems of non-standard type and exterior problems.     

Semi-classical behavior of the spectral function

We study the semi-classical behavior of the spectral function of the Schrödinger operator with short range potential. We prove that the spectral function is a semi-classical Fourier integral operator quantizing the forward and backward Hamiltonian flow relations of the system. Under a certain geometric condition we explicitly compute the phase in an oscillatory integral representation of the spectral function.

     

On a spectral problem in the theory of the heat operator

In the theory of the heat operator, we study a spectral problem with squared spectral parameter in the boundary condition. We construct the biorthogonal system and state a nonlocal spectral problem for a complete minimal eigenfunction system.     

Sharp upper and lower bounds for the Laplacian spectral radius and the spectral radius of graphs

In this paper, sharp upper bounds for the Laplacian spectral radius and the spectral radius of graphs are given, respectively. We show that some known bounds can be obtained from our bounds. For a bipartite graph G, we also present sharp lower bounds for the Laplacian spectral radius and the spectral radius, respectively.     

A note on the spectral theorem

We give necessary and sufficient conditions to prove a spectral theorem and a functional calculus for certain nonselfadjoint operators, H. Our method is non-perturbative: the conditions are given in terms of the resolvent (z-H)^–1. We give an example of an operator satisfying these conditions. This operator is not a spectral operator of scalar type. Its spectral projections are unbounded operators defined on a common dense domainD.This research was supported in part by Department of Energy Grant No. DE-AS05-80ER10711 and National Science Foundation Grant No. DMA-8312451.     

On the uniform convergence of the Fourier series for a spectral problem with squared spectral parameter in a boundary condition

We analyze the uniform convergence of the Fourier series expansions of Hölder functions in the system of eigenfunctions of a spectral problem with squared spectral parameter in a boundary condition. To this end, we first prove a theorem on the equiconvergence of such expansions with those in a well-known orthonormal basis.     

Filtering in Legendre spectral methods

We discuss the impact of modal filtering in Legendre spectral methods, both on accuracy and stability. For the former, we derive sufficient conditions on the filter to recover high order accuracy away from points of discontinuity. Computational results confirm that less strict necessary conditions appear to be adequate. We proceed to discuss a instability mechanism in polynomial spectral methods and prove that filtering suffices to ensure stability. The results are illustrated by computational experiments.

     

On the joint spectral radius of matrices of order 2 with equal spectral radius

We provide explicit formulas for the joint spectral radius of certain classes of pairs of real matrices of order 2 with equal spectral radius.     

On the unstable homotopy spectral sequences

We shall study the relation between the unstable homotopy spectral sequence associated with the cell decomposition of the source space and the one associated with the Postnikov decomposition of the target space. It is proved that the Maunder's result holds for these two types of spectral sequences. This result will be exploited to study the differentials of these spectral sequences and we obtain a generalization of a theorem due to Atiyah-Hirzebruch and Dold. Making use of these spectral sequences with the convergence lemma, some results on the phantom maps and the homotopy groups of map_*(Y, X) will be proved.     

Fast computation of spectral centroids

The spectral centroid of a signal is the curve whose value at any given time is the centroid of the corresponding constant-time cross section of the signal’s spectrogram. A spectral centroid provides a noise-robust estimate of how the dominant frequency of a signal changes over time. As such, spectral centroids are an increasingly popular tool in several signal processing applications, such as speech processing. We provide a new, fast and accurate algorithm for the real-time computation of the spectral centroid of a discrete-time signal. In particular, by exploiting discrete Fourier transforms, we show how one can compute the spectral centroid of a signal without ever needing to explicitly compute the signal’s spectrogram. We then apply spectral centroids to an emerging biometrics problem: to determine a person’s heart and breath rates by measuring the Doppler shifts their body movements induce in a continuous wave radar signal. We apply our algorithm to real-world radar data, obtaining heart- and breath-rate estimates that compare well against ground truth.     

On the Laplacian spectral ratio of connected graphs

The Laplacian spectrum of a graph is the eigenvalues of the associated Laplacian matrix. The quotient between the largest and second smallest Laplacian eigenvalues of a connected graph, is called the Laplacian spectral ratio. Some bounds on the Laplacian spectral ratio are considered. We improve a relation on the Laplacian spectral ratio of regular graphs. Especially, the first two smallest Laplacian spectral ratios of graphs with given order are determined. And some operations on Laplacian spectral ratio are presented.     

Every Banach algebra has the spectral radius property

Nylen and Rodman [NR] introduced the notion of spectral radius property in Banach algebras in order to generalize a classical theorem of Yamamoto on the asymptotic behaviour of the singular values of ann xn matrix. In this paper we prove a conjecture of theirs in the affirmative, namely that any unital Banach algebra has the spectral radius property. In fact a slightly more general spectral property holds. We show that for every element which has spectral points which are not of finite multiplicity, the essential spectral radius is the supremum of the set of absolute values of the spectral points that are not of finite multiplicity.     

On the power of combinatorial and spectral invariants

We extend the traditional spectral invariants (spectrum and angles) by a stronger polynomial time computable graph invariant based on the angles between projections of standard basis vectors into the eigenspaces (in addition to the usual angles between standard basis vectors and eigenspaces). The exact power of the new invariant is still an open problem. We also define combinatorial invariants based on standard graph isomorphism heuristics and compare their strengths with the spectral invariants. In particular, we show that a simple edge coloring invariant is at least as powerful as all these spectral invariants.     

On uniform continuity of the spectral radius in Banach algebras

We investigate relations between different forms of subadditivity and submultiplicativity of the spectral radius. In particular, we prove that if the spectral radius is uniformly continuous on a Banach algebra, then the algebra is commutative modulo the radical; this confirms a conjecture raised by the second author in [7].     

On matrices having equal spectral radius and spectral norm

In this paper we characterize all nxn matrices whose spectral radius equals their spectral norm. We show that for n?3 the class of these matrices contains the normal matrices as a subclass.     

An inverse spectral problem for the Laplacian

We propose an algorithm for the recovery of a potential from the knowledge of the eigenvalues of the Laplacian operator and the traces of its eigenfunctions. This inverse spectral problem is solved by recasting the operator as an infinite matrix and using transition matrices together with spectral projections on the boundary.     

On the eigenvalues of a nonlinear spectral problem

We consider a nonlinear eigenvalue problem of the Sturm–Liouville type with conditions of the third kind, which describes the propagation of polarized electromagnetic waves in a plane dielectric waveguide. The equation is nonlinear in the unknown function, and the boundary conditions depend on the spectral parameter nonlinearly. We obtain an equation for the spectral parameter and formulas for the zeros of the eigenfunctions and show that the problem has at most finitely many isolated eigenvalues.     

On the stability of the spectral Galerkin approximation

We study stability properties of the spectral Galerkin approximation of the solutions of semilinear problems. Assuming that the data of the problem are known within a certain error, we investigate when the solution of the Galerkin approximate equation provides a desired accuracy uniformly with respect to small perturbations of the data. We show that for certain classes of semilinear problems an additional compactness assumption is sufficient to assure that the spectral Galerkin method provides an accurate approximation to the exact solution uniformly with respect to small perturbations of the data. This revised version was published online in June 2006 with corrections to the Cover Date.