Construction of a Jacobi matrix from spectral data

A proof is given for the existence and uniqueness of a correspondence between two pairs of sequences {a},{b} and {ω},{μ}, satisfying b_i>0 for i=1,…,n?1 and ω₁<μ₁<?<μ_n?1<ω_n, under which the symmetric Jacobi matrices J(n,a,b) and J(n?1,a,b) have eigenvalues {ω} and {μ} respectively. Here J(m,a,b) is symmetric and tridiagonal with diagonal elements a_i (i=1,…,m) and off diagonal elements b_i (i=1,…,m?1). A new concise proof is given for the known uniqueness result. The existence result is new.     

The numerically stable reconstruction of a Jacobi matrix from spectral data

We show how to construct, from certain spectral data, a discrete inner product for which the associated sequence of monic orthogonal polynomials coincides with the sequence of appropriately normalized characteristic polynomials of the left principal submatrices of the Jacobi matrix. The generation of these orthogonal polynomials via their three-term recurrence relation, as popularized by Forsythe, then provides a stable means of computing the entries of the Jacobi matrix. Our construction provides, incidentally, very simple proofs of known results concerning the existence and uniqueness of a Jacobi matrix satisfying given spectral data and its continuous dependence on those data.     

On an inverse spectral problem for a quadratic Jacobi matrix pencil

Given two monic polynomials P2n and P2n−2 of degree 2n and 2n−2 (n?2) with complex coefficients and with disjoint zero sets. We give necessary and sufficient conditions on these polynomials such that there exist two n×n Jacobi matrices B and C for which

     

The Hilbert transform of a measure

Let e be a homogeneous subset of ℝ in the sense of Carleson. Let μ be a finite positive measure on ℝ and H _μ(x) its Hilbert transform. We prove that if lim _t→∞ t|e∩{x ‖H _μ(x)| > t}| = 0, then μ _s (e) = 0, where μ_s is the singular part of μ.     

The Delsarte extremal problem for the Jacobi transform

We give the solution of the Delsarte extremal problem for even entire functions of exponential type that are Jacobi transforms and prove the uniqueness of the extremal function. The quadrature Markov formula on the half-line with zeros of the modified Jacobi function are used.     

On the spectral radius of the Jacobi iteration matrix for a rectangular region with two different media

The spectral radius of the Jacobi iteration matrix plays an important role to estimate the optimum relaxation factor, when the successive overrelaxation (SOR) method is used for solving a linear system. The specific systems are finite difference forms of the Laplace equation satisfied on a rectanglar region with two different media. Though the potential function for the inhomogeneous closed region is continuous, the first order derivative is not continuous. So this requires internal boundary conditions or interface conditions. In this paper, the spectral radius of the Jacobi iteration matrix for the inhomogeneous rectangular region is formulated and the approximation for the explicit formula, suitable for the computation of the spectral radius, is deduced. It is also found by the proposed formula that the spectral radius and the optimum relaxation factor rigorously depend on the inhomogeneity or the internal boundary conditions in the closed region, and especially vary with the position of the internal boundary. These findings are also confirmed by the numerical results of the power method.The stationary iterative method using the proposed formula for calculating estimates of the spectral radius of the Jacobi iteration matrix is compared with Carré's method, Kulstrud's method and the stationary iterative method using Frankel's theoretical formula, all for the case of some numerical models with two different media. According to the results our stationary iterative method gives the best results ffor the estimate of the spectral radius of the Jacobi iteration matrix, for the required number of iterations to calculate solutions, and for the accuracy of the solutions.As a numerical example the microstrip transmission line is taken, the propating mode of which can be approximated by a TEM mode. The cross section includes inhomogeneous media and a strip conductor. Upper and lower bounds of the spectral radius of the Jacobi iteration matrix are estimated. Our method using these estimates is also compared with the other methods. The upper bound of the spectral radius of the Jacobi iteration matrix for more general closed regions with two different media might be given by the proposed formula.     

First order spectral perturbation theory of square singular matrix polynomials

We develop first order eigenvalue expansions of one-parametric perturbations of square singular matrix polynomials. Although the eigenvalues of a singular matrix polynomial P(λ) are not continuous functions of the entries of the coefficients of the polynomial, we show that for most perturbations they are indeed continuous. Given an eigenvalue λ₀ of P(λ) we prove that, for generic perturbations M(λ) of degree at most the degree of P(λ), the eigenvalues of P(λ)+?M(λ) admit covergent series expansions near λ₀ and we describe the first order term of these expansions in terms of M(λ₀) and certain particular bases of the left and right null spaces of P(λ₀). In the important case of λ₀ being a semisimple eigenvalue of P(λ) any bases of the left and right null spaces of P(λ₀) can be used, and the first order term of the eigenvalue expansions takes a simple form. In this situation we also obtain the limit vector of the associated eigenvector expansions.     

Determination of an infinite non-self-adjoint Jacobi matrix from its generalized spectral function

Restoration from the generalized spectral function of the equations $$b_0 y_0 + a_0 y_1 = \lambda y_{0,} a_{n - 1} y_{n - 1} + b_n y_n + a_n y_{n + 1} = \lambda y_{n,} n = 1,2,3,...,$$ wherea _n and b_n are arbitrary complex numbers,a _n≠ 0 (n = o, i, 2,...), λ is a complex parameter, and {P _n (λ)} ₀ ^∞ is the required solution, is investigated. Necessary and sufficient conditions for solvability of the inverse problem are obtained, and the restoration procedure is described.     

On a spectral property of Jacobi matrices

Let be a Jacobi matrix with elements on the main diagonal and elements on the auxiliary ones. We suppose that is a compact perturbation of the free Jacobi matrix. In this case the essential spectrum of coincides with , and its discrete spectrum is a union of two sequences 2, x^-_j<-2$">, tending to . We denote sequences and by and , respectively.

The main result of the note is the following theorem.

Theorem.     Let be a Jacobi matrix described above and be its spectral measure. Then if and only if

-\infty,\qquad {ii)} \sum_j(x^\pm_j\mp2)^{7/2}<\infty. \end{displaymath}">

     

Computing the Hilbert transform of a Jacobi weight function

We discuss the evaluation of the Hilbert transformf _–1 ¹ (t-)^–1 w(, )(t)dt,–1<<1, of the Jacobi weight functionw(, )(t)=(1–t))(1+t) by analytic and numerical means and also comment on the recursive computation of the quantitiesf _–1 ¹ )(t–)^–1 _n (t;w ^{(, ))} w ^{(, )}(t)dt,n=0, 1, 2, ..., where _n (·;w ^{(, )}) is the Jacobi polynomial of degreen.The work of the first author was supported in part by the National Science Foundation under grant DCR-8320561. The work of the second author was supported by the National Science Foundation under grant DMS-8419086.     

The differential Fourier transform method

We suggest the two new discrete differential sine and cosine Fourier transforms of a complex vector which are based on solving by a finite difference scheme the inhomogeneous harmonic differential equations of the first order with complex coefficients and of the second order with real coefficients, respectively. In the basic version, the differential Fourier transforms require by several times less arithmetic operations as compared to the basic classicalmethod of discrete Fourier transform. In the differential sine Fourier transform, the matrix of the transformation is complex,with the real and imaginary entries being alternated, whereas in the cosine transform, the matrix is purely real. As in the classical case, both matrices can be converted into the matrices of cyclic convolution; thus all fast convolution algorithms including the Winograd and Rader algorithms can be applied to them. The differential Fourier transform method is compatible with the Good–Thomas algorithm of the fast Fourier transform and can potentially outperform all available methods of acceleration of the fast Fourier transform when combined with the fast convolution algorithms.     

The quadratic-phase Fourier wavelet transform

In this paper, we define the quadratic-phase Fourier wavelet transform (QPFWT) and discuss its basic properties including convolution for QPFWT. Further, inversion formula and the Parseval relation of QPFWT are also discussed. Continuity of QPFWT on some function spaces are studied. Moreover, some applications of quadratic-phase Fourier transform (QPFT) to solve the boundary value problems of generalized partial differential equations.     

Spectrum of a Jacobi matrix with exponentially growing matrix elements

A Jacobi matrix with an exponential growth of its elements and the corresponding symmetric operator are considered. It is proved that the eigenvalue problem for some self-adjoint extension of this operator in some Hilbert space is equivalent to the eigenvalue problem of the Sturm-Liouville operator with a discrete self-similar weight. An asymptotic formula for the distribution of eigenvalues is obtained.     

How well does the finite Fourier transform approximate the Fourier transform?

We show that the answer to the question in the title is “very well indeed.” In particular, we prove that, throughout the maximum possible range, the finite Fourier coefficients provide a good approximation to the Fourier coefficients of a piecewise continuous function. For a continuous periodic function, the size of the error is estimated in terms of the modulus of continuity of the function. The estimates improve commensurately as the functions become smoother. We also show that the partial sums of the finite Fourier transform provide essentially as good an approximation to the function and its derivatives as the partial sums of the ordinary Fourier series. Along the way we establish analogues of the Riemann‐Lebesgue lemma and the localization principle. © 2004 Wiley Periodicals, Inc.