Related papers: On the convergence of complex Jacobi methods
We study Jacobi matrices on trees whose coefficients are generated by multiple orthogonal polynomials. Hilbert space decomposition into an orthogonal sum of cyclic subspaces is obtained. For each subspace, we find generators and the…
We obtain bounds for the spectrum and for the total width of the spectral gaps for Jacobi matrices on $\ell^2(\Z)$ of the form $(H\psi)_n= a_{n-1}\psi_{n-1}+b_n\psi_n+a_n\psi_{n+1}$, where $a_n=a_{n+q}$ and $b_n=b_{n+q}$ are periodic…
We consider the 1D periodic Jacobi matrices. The spectrum of this operator is purely absolutely continuous and consists of intervals separated by gaps. We solve the inverse problem (including characterization) in terms of vertical slits on…
We obtain various versions of classical Lieb--Thirring bounds for one- and multi-dimensional complex Jacobi matrices. Our method is based on Fan-Mirski Lemma and seems to be fairly general.
The Jacobi polynomial has been advocated by many authors as a useful tool to evolve non-singlet structure functions to higher $Q^2$. In this work, it is found that the convergence of the polynomial sum is not absolute, as there is always a…
The generalized coherent states attached to the Jacobi group realize the squeezed states. Imposing hermitian conjugacy to the generators of the Jacobi algebra, we find out the form of the weight function appearing in the scalar product. We…
We discuss the proof of and systematic application of Case's sum rules for Jacobi matrices. Of special interest is a linear combination of two of his sum rules which has strictly positive terms. Among our results are a complete…
In this paper, we study the Jacobi frame approximation with equispaced samples and derive an error estimation. We observe numerically that the approximation accuracy gradually decreases as the extended domain parameter $\gamma$ increases in…
A modification of the well-known step-by-step process for solving Nevanlinna-Pick problems in the class of $\bR_0$-functions gives rise to a linear pencil $H-\lambda J$, where $H$ and $J$ are Hermitian tridiagonal matrices. First, we show…
In this paper, we establish a "global" Morse index theorem. Given a hypersurface $M^{n}$ of constant mean curvature, immersed in $\mathbb{R}^{n+1}$. Consider a continuous deformation of "generalized" Lipschitz domain $D(t)$ enlarging in…
We analyze several versions of Jacobi's method for the symmetric eigenvalue problem. Our goal is to reduce the asymptotic cost of the algorithm as much as possible, as measured by the number of arithmetic operations performed and associated…
We study the perturbative power-series expansions of the eigenvalues and eigenvectors of a general tridiagonal (Jacobi) matrix of dimension d. The(small) expansion parameters are being the entries of the two diagonals of length d-1…
We consider a generalization of Jacobi theta series and show that every such function is a quasi-Jacobi form. Under certain conditions we establish transformation laws for these functions with respect to the Jacobi group and prove such…
Persymmetric Jacobi matrices are invariant under reflection with respect to the anti-diagonal. The associated orthogonal polynomials have distinctive properties that are discussed. They are found in particular to be also orthogonal on the…
We equip the complex polynomial algebra C[t] with the involution which is the identity on C and sends t to -t. Answering a question raised by V.G. Kac, we show that every hermitian or skew-hermitian matrix over this algebra is congruent to…
In this paper, we construct Hamilton-Jacobi equations for a great variety of mechanical systems (nonholonomic systems subjected to linear or affine constraints, dissipative systems subjected to external forces, time-dependent mechanical…
We present and analyze a simple numerical method that diagonalizes a complex normal matrix A by diagonalizing the Hermitian matrix obtained from a random linear combination of the Hermitian and skew-Hermitian parts of A.
We propose new iterative methods for computing nontrivial extremal generalized singular values and vectors. The first method is a generalized Davidson-type algorithm and the second method employs a multidirectional subspace expansion…
A special class of generalized Jacobi operators which are self-adjoint in Krein spaces is presented. A description of the resolvent set of such operators in terms of solutions of the corresponding recurrence relations is given. In…
We develop relative oscillation theory for Jacobi matrices which, rather than counting the number of eigenvalues of one single matrix, counts the difference between the number of eigenvalues of two different matrices. This is done by…