Spectrum of certain tridiagonal matrices when their dimension goes to infinity

This paper deals with the spectra of matrices similar to infinite tridiagonal Toeplitz matrices with perturbations and with positive off-diagonal elements. We will discuss the asymptotic behavior of the spectrum of such matrices and we use them to determine the values of a matrix function, for an entire function. In particular we determine the matrix powers and matrix exponentials.     

The Hilbert Null-cone on Tuples of Matrices and Bilinear Forms

We describe the null-cone of the representation of G on M ^p , where either G = SL(W) × SL(V) and M = Hom(V,W) (linear maps), or G = SL(V) and M is one of the representations S ²(V ^*) (symmetric bilinear forms), Λ²(V ^*) (skew bilinear forms), or (arbitrary bilinear forms). Here V and W are vector spaces over an algebraically closed field K of characteristic zero and M ^p is the direct sum of p of copies of M. More specifically, we explicitly determine the irreducible components of the null-cone on M ^p . Results of Kraft and Wallach predict that their number stabilises at a certain value of p, and we determine this value. We also answer the question of when the null-cone in M ^p is defined by the polarisations of the invariants on M; typically, this is only the case if either dim V or p is small. A fundamental tool in our proofs is the Hilbert–Mumford criterion for nilpotency (also known as unstability).     

Cyclotomic matrices and hypergeometric functions over finite fields

With the help of hypergeometric functions over finite fields, we study some arithmetic properties of cyclotomic matrices involving characters and binary quadratic forms over finite fields. Also, we confirm some related conjectures posed by Zhi-Wei Sun.     

Asymptotic spectra of large matrices coming from the symmetrization of Toeplitz structure functions and applications to preconditioning

The singular value distribution of the matrix‐sequence {Y_nT_n[f]}_n , with T_n[f] generated by $f\in L^1([?\pi ,\pi ])$, was shown in [J. Pestana and A.J. Wathen, SIAM J Matrix Anal Appl. 2015;36(1):273‐288]. The results on the spectral distribution of {Y_nT_n[f]}_n were obtained independently in [M. Mazza and J. Pestana, BIT, 59(2):463‐482, 2019] and [P. Ferrari, I. Furci, S. Hon, M.A. Mursaleen, and S. Serra‐Capizzano, SIAM J. Matrix Anal. Appl., 40(3):1066‐1086, 2019]. In the latter reference, the authors prove that {Y_nT_n[f]}_n is distributed in the eigenvalue sense as $${?}_{|f|}(\theta )=\left\{\begin{array}{ll}|f(\theta )|,& \theta \in [0,2\pi ],\\ ?|f(?\theta )|,& \theta \in [?2\pi ,0),\end{array}\right.$$ under the assumptions that f belongs to $L^1([?\pi ,\pi ])$ and has real Fourier coefficients. The purpose of this paper is to extend the latter result to matrix‐sequences of the form {h(T_n[f])}_n , where h is an analytic function. In particular, we provide the singular value distribution of the sequence {h(T_n[f])}_n , the eigenvalue distribution of the sequence {Y_nh(T_n[f])}_n , and the conditions on f and h for these distributions to hold. Finally, the implications of our findings are discussed, in terms of preconditioning and of fast solution methods for the related linear systems.     

Convolution of Rayleigh functions with respect to the Bessel index

A convolution of Rayleigh functions with respect to the Bessel index can be treated as a special function in its own right. It appears in constructing global-in-time solutions for some semilinear evolution equations in circular domains and may control the smoothing effect due to nonlinearity. An explicit representation for it is derived which involves the special function ψ(x) (the logarithmic derivative of the Γ-function). The properties of the convolution in question are established. Asymptotic expansions for small and large values of the argument are obtained and the graph is presented.     

Interpolating functions of matrices on zeros of quasi‐kernel polynomials

The paper deals with Krylov methods for approximating functions of matrices via interpolation. In this frame residual smoothing techniques based on quasi‐kernel polynomials are considered. Theoretical results as well as numerical experiments illustrate the effectiveness of our approach. Copyright © 2004 John Wiley & Sons, Ltd.     

High relative accuracy with some special matrices related to Γ and β functions

For some families of totally positive matrices using $\mathrm{\Gamma }$ and $\beta$ functions, we provide their bidiagonal factorization. Moreover, when these functions are defined over integers, we prove that the bidiagonal factorization can be computed with high relative accuracy and so we can compute with high relative accuracy their eigenvalues, singular values, inverses and the solutions of some associated linear systems. We provide numerical examples illustrating this high relative accuracy.     

Spectra of infinite-dimensional sample covariance matrices

We study spectral functions of infinite-dimensional random Gram matrices of the form RR^T, where R is a rectangular matrix with an infinite number of rows and with the number of columns N → ∞, and the spectral functions of infinite sample covariance matrices calculated for samples of volume N → ∞ under conditions analogous to the Kolmogorov asymptotic conditions. We assume that the traces d of the expectations of these matrices increase with the number N such that the ratio d/N tends to a constant. We find the limiting nonlinear equations relating the spectral functions of random and nonrandom matrices and establish the asymptotic expression for the resolvent of random matrices. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 148, No. 2, pp. 309–322, August, 2006.     

Jacobi matrices generated by ratios of hypergeometric functions

A problem of determining zeroes of the Gauss hypergeometric function goes back to Klein, Hurwitz, and Van Vleck. In this very short note we show how ratios of hypergeometric functions arise as m-functions of Jacobi matrices and we then revisit the problem based on the recent developments of the spectral theory of non-Hermitian Jacobi matrices.     

Generalized boundary value problems for analytic functions with convolutions and its applications

In this paper, we first establish a locality theory for the Noethericity of generalized boundary value problems on the spaces . By means of this theory, of the classical boundary value theory, and of the theory of Fourier analysis, we discuss the necessary and sufficient conditions of the solvability and obtain the general solutions and the Noether conditions for one class of generalized boundary value problems. All cases as regards the index of the coefficients in the equations are considered in detail. Moreover, we apply our theoretical results to the solvability of singular integral equations with variable coefficients. Thus, this paper will be of great significance for the study of improving and developing complex analysis, integral equation, and boundary value theory.     

Approximation of semigroups and cosine functions in spaces of periodic functions

In this article we furnish a representation of the solutions of some classes of first-order and second-order evolution problems as limit of iterates of classical sequences of approximating operators. The method is based on Trotter's theorem on the approximation of semigroups which is applied here also for the approximation of groups and cosine functions. We apply this method in spaces of continuous periodic functions and using some classical sequences of trigonometric polynomials.     

Distribution of Eigenvalues of Real Symmetric Palindromic Toeplitz Matrices and Circulant Matrices

Consider the ensemble of real symmetric Toeplitz matrices, each independent entry an i.i.d. random variable chosen from a fixed probability distribution p of mean 0, variance 1, and finite higher moments. Previous investigations showed that the limiting spectral measure (the density of normalized eigenvalues) converges weakly and almost surely, independent of p, to a distribution which is almost the standard Gaussian. The deviations from Gaussian behavior can be interpreted as arising from obstructions to solutions of Diophantine equations. We show that these obstructions vanish if instead one considers real symmetric palindromic Toeplitz matrices, matrices where the first row is a palindrome. A similar result was previously proved for a related circulant ensemble through an analysis of the explicit formulas for eigenvalues. By Cauchy’s interlacing property and the rank inequality, this ensemble has the same limiting spectral distribution as the palindromic Toeplitz matrices; a consequence of combining the two approaches is a version of the almost sure Central Limit Theorem. Thus our analysis of these Diophantine equations provides an alternate technique for proving limiting spectral measures for certain ensembles of circulant matrices. A. Massey’s current address: Department of Mathematics, UCLA, Los Angeles, CA 90095, USA. e-mail: amassey3102@math.ucla.edu.     

Irregular sampling, Toeplitz matrices, and the approximation of entire functions of exponential type

In many applications one seeks to recover an entire function of exponential type from its non-uniformly spaced samples. Whereas the mathematical theory usually addresses the question of when such a function in can be recovered, numerical methods operate with a finite-dimensional model. The numerical reconstruction or approximation of the original function amounts to the solution of a large linear system. We show that the solutions of a particularly efficient discrete model in which the data are fit by trigonometric polynomials converge to the solution of the original infinite-dimensional reconstruction problem. This legitimatizes the numerical computations and explains why the algorithms employed produce reasonable results. The main mathematical result is a new type of approximation theorem for entire functions of exponential type from a finite number of values. From another point of view our approach provides a new method for proving sampling theorems.

     

Convolution operators and zeros of entire functions

Let be a real entire function of order less than with only real zeros. Then we classify certain distribution functions such that the convolution has only real zeros.

     

Distribution of Eigenvalues for the Ensemble of Real Symmetric Toeplitz Matrices

Consider the ensemble of real symmetric Toeplitz matrices whose entries are i.i.d. random variable from a fixed probability distributionpof mean 0,variance 1, and finite moments of all order. The limiting spectral measure (the density of normalized eigenvalues) converges weakly to a new universal distribution with unbounded support, independent of pThis distribution’s moments are almost those of the Gaussian’s, and the deficit may be interpreted in terms of obstructions to Diophantine equations; the unbounded support follows from a nice application of the Central Limit Theorem. With a little more work, we obtain almost sure convergence. An investigation of spacings between adjacent normalized eigenvalues looks Poissonian, and not GOE. A related ensemble (real symmetric palindromic Toeplitz matrices) appears to have no Diophantine obstructions, and the limiting spectral measure’s first nine moments can be shown to agree with those of the Gaussian; this will be considered in greater detail in a future paper.     

On classes of analytic functions defined by convolution with incomplete beta functions

Carlson and Shaffer [SIAM J. Math. Anal. 15 (1984) 737-745] defined a convolution operator L(a,c) on the class A of analytic functions involving an incomplete beta function ?(a,c;z) as L(a,c)f=?(a,c)?f. We use this operator to introduce certain classes of analytic functions in the unit disk and study their properties including some inclusion results, coefficient and radius problems. It is shown that these classes are closed under convolution with convex functions.     

On inverses of GCD matrices associated with multiplicative functions and a proof of the Hong-Loewy conjecture

Let C={1,2,…,m} and f be a multiplicative function such that (f∗μ)(k)>0 for every positive integer k and the Euler product converges. Let (C_f)=(f(i,j)) be the m×m matrix defined on the set C having f evaluated at the greatest common divisor (i,j) of i and j as its ij-entry. In the present paper, we first obtain the least upper bounds for the ij-entry and the absolute row sum of any row of (C_f)^-1, the inverse of (C_f), in terms of ζ_f. Specializing these bounds for the arithmetical functions f=N^ε,J_ε and σ_ε we examine the asymptotic behavior the smallest eigenvalue of each of matrices (C_N^ε),(C_Jε) and (C_σε) depending on ε when m tends to infinity. We conclude our paper with a proof of a conjecture posed by Hong and Loewy [S. Hong, R. Loewy, Asymptotic behavior of eigenvalues of greatest common divisor matrices, Glasg. Math. J. 46 (2004) 551-569].     

Representation of measurable functions in Franklin general system

The paper proves that every almost everywhere finite measurable function is representable by an absolute convergent series in the Franklin systems generated by quasi-dyadic, weakly regular partitions.     

Jordan canonical form of Pascal-type matrices via sequences of binomial type

In this paper, we study the Jordan canonical form of the generalized Pascal functional matrix associated with a sequence of binomial type, and demonstrate that the transition matrix between the generalized Pascal functional matrix and its Jordan canonical form is the iteration matrix associated with the binomial sequence. In addition, some combinatorial identities are derived from the corresponding matrix factorization.