Mathematics

In this article we first obtain an asymptotic expression of the N+1 eigenvalues of the Toeplitz matrix T_N(f) where f is a strictly increasing or strictly decreasing, differentiable and even function on [?�π,?] . In the particular case where the symbol only verifies these assumptions locally we specify the eigenvalues that belong to a certain interval. In the case where the symbol of the Toeplitz matrix is of the form h_α=(1?�cosθ ) α c with α> 1 2 and c a regular function we give the general expression of the eigenvalues and a second order of this asymptotic for a subset of the small eigenvalues as well as an approximation of the eigenvectors.

In this paper we consider Wronskian polynomials labeled by partitions that can be factorized via the combinatorial concepts of p -cores and p -quotients. We obtain the asymptotic behavior for these polynomials when the p -quotient is fixed while the size of the p -core grows to infinity. For this purpose, we associate the p -core with its characteristic vector and let all entries of this vector simultaneously tend to infinity. This result generalizes the Wronskian Hermite setting which is recovered when p=2 .

Our goal is to find an asymptotic behavior as n→∞ of the orthogonal polynomials P n (z) defined by Jacobi recurrence coefficients a n (off-diagonal terms) and b n (diagonal terms). We consider the case a n →∞ , b n →∞ in such a way that ∑ a −1 n <∞ ( that is, the Carleman condition is violated ) and γ n := 2 −1 b n ( a n a n−1 ) −1/2 →γ as n→∞ . In the case |γ|≠1 asymptotic formulas for P n (z) are known; they depend crucially on the sign of |γ|−1 . We study the critical case |γ|=1 . The formulas obtained are qualitatively different in the cases | γ n |→1−0 and | γ n |→1+0 . Another goal of the paper is to advocate an approach to a study of asymptotic behavior of P n (z) based on a close analogy of the Jacobi difference equations and differential equations of Schrödinger type.

In this paper, we use a Banach fixed point theorem to obtain suficient conditions satisfying the convergence and exponential convergence of solutions for the linear system of advanced differential equations. The considered system with multiple variable advanced arguments is discussed as well. The obtained theorems generalize previous results of Dung [8], from the one dimension to the n dimension.

We find two-sides estimates for the best uniform approximations of classes of convolutions of 2π -periodic functions from unit ball of the space L p ,1≤p<∞, with fixed kernels, modules of Fourier coefficients of which satisfy the condition ∑ k=n+1 ∞ ψ(k)<ψ(n). In the case of ∑ k=n+1 ∞ ψ(k)=o(1)ψ(n) the obtained estimates become the asymptotic equalities.

The Laplacian matrix is of fundamental importance in the study of graphs, networks, random walks on lattices, and arithmetic of curves. In certain cases, the trace of its pseudoinverse appears as the only non-trivial term in computing some of the intrinsic graph invariants. Here we study a double sum F n which is associated with the trace of the pseudo inverse of the Laplacian matrix for certain graphs. We investigate the asymptotic behavior of this sum as n→∞ . Our approach is based on classical analysis combined with asymptotic and numerical analysis, and utilizes special functions. We determine the leading order term, which is of size n 2 logn , and develop general methods to obtain the secondary main terms in the asymptotic expansion of F n up to errors of O(logn) and O(1) as n→∞ . We provide some examples to demonstrate our methods.

Asymptotic approximations of Jacobi polynomials are given in terms of elementary functions for large degree n and parameters α and β . From these new results, asymptotic expansions of the zeros are derived and methods are given to obtain the coefficients in the expansions. These approximations can be used as initial values in iterative methods for computing the nodes of Gauss--Jacobi quadrature for large degree and parameters. The performance of the asymptotic approximations for computing the nodes and weights of these Gaussian quadratures is illustrated with numerical examples.

We derive asymptotic expansions of the Kummer functions M(a,b,z) and U(a,b+1,z) for large positive values of a and b , with z fixed. For both functions we consider b/a≤1 and b/a≥1 , with special attention for the case a∼b . We use a uniform method to handle all cases of these parameters.

We study residual polynomials, R (e) x 0 ,n , e⊂R , x 0 ∈R∖e , which are the degree at most n polynomials with R( x 0 )=1 that minimize the sup norm on e . New are upper bounds on their norms (that are optimal in some cases) and Szegő--Widom asymptotics under fairly general circumstances. We also discuss several illuminating examples and some results in the complex case.

There is a vast theory of Chebyshev and residual polynomials and their asymptotic behavior. The former ones maximize the leading coefficient and the latter ones maximize the point evaluation with respect to an L ??norm. We study Chebyshev and residual extremal problems for rational functions with real poles with respect to subsets of R ¯ ¯ ¯ ¯ . We prove root asymptotics under fairly general assumptions on the sequence of poles. Moreover, we prove Szeg?--Widom asymptotics for sets which are regular for the Dirichlet problem and obey the Parreau--Widom and DCT conditions.