Padé tables of entire functions of very slow and smooth growth,II

Letf(z):=Σ _j=0 ^∞ a _j z ^j , where a_j ≠ 0,j large enough, and for someq ε C such that ¦q¦ $$q_j : = a_{j - 1} a_{j + 1} /a_j^2 \to q,j \to \infty .$$ Define for m,n = 0,1,2,..., the Toeplitz determinant $$D(m/n): = \det (a_{m - j + k} )_{j,k = 1}^n .$$ Given ? > 0, we show that form large enough, and for everyn = 1,2,3,..., $$(1 - \varepsilon )^n \leqslant \left| {{{D(m/n)} \mathord{\left/ {\vphantom {{D(m/n)} {\left\{ {a_m^n \mathop \Pi \limits_{j - 1}^{n - 1} (1 - q_m^j )^{n - j} } \right\}}}} \right. \kern-\nulldelimiterspace} {\left\{ {a_m^n \mathop \Pi \limits_{j - 1}^{n - 1} (1 - q_m^j )^{n - j} } \right\}}}} \right| \leqslant (1 + \varepsilon )^n .$$ We apply this to show that any sequence of Padé approximants {[m _k /n _k ]} ₁ ^∞ tof, withm _k →∞ ask→ ∞, converges locally uniformly in C. In particular, the diagonal sequence {[n/n]} ₁ ^∞ converges throughout C. Further, under additional assumptions, we give sharper asymptotics forD(m/n).     

Padé approximants of special functions

Let $ {f_{\gamma }}(x) = \sum\nolimits_{{k = 0}}^{\infty } {{{{T_k (x)}} \left/ {{{{\left( \gamma \right)}_k}}} \right.}} $ , where (??) _k =??(??+1) ? (??+k?1) and T _k (x)=cos (k arccos x) are Padé?CChebyshev polynomials. For such functions and their Padé?CChebyshev approximations $ \pi_{n,m}^{ch}\left( {x;{f_{\gamma }}} \right) $ , we find the asymptotics of decreasing the difference $ {f_{\gamma }}(x) - \pi_{n,m}^{ch}\left( {x;{f_{\gamma }}} \right) $ in the case where 0 ? m ? m(n), m(n) = o (n), as n???? for all x ?? [?1, 1]. Particularly, we determine that, under the same assumption, the Padé?CChebyshev approximations converge to f _?? uniformly on the segment [?1, 1] with the asymptotically best rate.     

Padé Approximations of and preservation of quadratic Lyapunov functions

We investigate whether or not quadratic Lyapunov functions are preserved under Padé approximations. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Matrix Padé Approximation of rational functions

In many applications it is of major interest to decide whether a given formal power series with matrix-valued coefficients of arbitrary dimensions results from a matrix-valued rational function. As the main result of this paper we provide an answer to this question in terms of Matrix Padé Approximants of the given power series. Furthermore, given a matrix rational function, the smallest degrees of the matrix polynomials which represent it are not necessarily unique. Therefore we study a certain minimality-type, that is, minimum degrees. We aim to obtain all the minimum degrees for the polynomials which represent the function as equivalents. In addition, given that the rational representation of the function for the same pair of degrees need not be unique, we have obtained conditions to study the uniqueness of said representation. All the results obtained are presented graphically in tables setting out the above information. They lead to a number of properties concerning special structures, staired blocks, in the Padé Table.     

Uniform convergence of rows of the Padé table for functions with smooth Maclaurin series coefficients

Given a formal power seriesf(z)?∑ _j=0 ^∞ a _j z ^j for which the quantitya _j ?1a _j +1/a _j ² has a prescribed asymptotic behavior asj→∞, we obtain the asymptotic behavior of poles of rows of the Padé table, and the associated Toeplitz determinants. In particular, we can show for large classes of entire functions of zero, finite, and infinite order (including the Mittag-Leffler functions) and forn=1,2,3,..., that the poles of [m/n](z) approach ∞ with ratea _m /a _m+1 asm→∞.     

On Padé approximants of Markov-type meromorphic functions

The paper is devoted to the asymptotic properties of diagonal Padé approximants for Markov-type meromorphic functions. The main result is strong asymptotic formulas for the denominators of diagonal Padé approximants for Markov-type meromorphic functions f = \(\hat \sigma \) + r under additional constraints on the measure σ (r is a rational function). On the basis of these formulas, it is proved that, in a sufficiently small neighborhood of a pole of multiplicity m of such a meromorphic function f, all poles of the diagonal Padé approximants f _n are simple and asymptotically located at the vertices of a regular m-gon.     

Power series equivalent to rational functions: A shifting-origin kronecker type theorem,and normality of Padé tables

Letf(z) be a function analytic in a neighbourhood of zero. For each pair of non-negative integers (m, n), form then byn Toeplitz determinantD(m/n) whose entries are the Maclaurin series coefficients off, namely,

Convergence of Padé approximants of partial theta functions and the Rogers-Szegö polynomials

We investigate the convergence of sequences of Padé approximants for the partial theta function $$h_q (z): = \sum\limits_{j = 0}^\infty { q^{j(j - 1)/2_{Z^j } } } , q = e^{i\theta } , \theta \in [0,2\pi ).$$ Whenθ/(2π) is irrational, this function has the unit circle as its natural boundary. We determine subrogions of ¦z¦ < 1 in which sequences of Padé approximants converge uniformly, and subrogions in which they converge in capacity, but not uniformly. In particular, we show that only a proper subsequence of the diagonal sequence {[n/n]} _n=1 ^∞ converges locally uniformly in all of ¦z¦< l; in contrast, no subsequence of any Padé row {[m/n]} _m=1 ^∞ (withn ≥ 2 fixed) can converge locally uniformly in all of ¦z¦ < 1. Further, we obtain the zero and pole distributions of sequences of Padé approximants by analyzing the zero distribution of the Rogers-Szegö polynomials $$G_n (z): = \sum\limits_{j = 0}^n {\left[ {\begin{array}{*{20}c} n \\ j \\ \end{array} } \right]} z^j , n = 0,1,2,....$$      

The convergence of diagonal Padé approximants and the Padé Conjecture

Questions related to the convergence problem of diagonal Padé approximants are discussed. A central place is taken by the Padé Conjecture (also known as the Baker-Gammel-Wills Conjecture). Partial results concerning this conjecture are reviewed and weaker and more special versions of the conjecture are formulated and their plausibility is investigated. Great emphasis is given to the role of spurious poles of the approximants. A conjecture by Nuttall (1970) about the number and distribution of such poles is stated and its importance for the Padé Conjecture is analyzed.     

Padé approximants to certain elliptic-type functions

Among all continua joining non-collinear points a ₁, a ₂, a ₃ ∈ ?, there exists a unique compact Δ ? ? that has minimal logarithmic capacity. For a complex-valued non-vanishing Dini-continuous function h on Δ, we define $${f_h}(z): = \frac{1}{{\pi i}}\int_\Delta {\frac{{h(t)}}{{t - z}}\frac{{dt}}{{{w^ + }(t)}}} $$ , where $w(z): = \sqrt {\prod\nolimits_{k = 0}^3 {(z - {a_k})} } $ and w ⁺ is the one-sided value according to some orientation of 1. In this work, we present strong asymptotics of diagonal Padé approximants to f _h and describe the behavior of the spurious pole and the regions of locally uniform convergence from a generic perspective.     

The A-Stability of Methods with Padé and Generalized Padé Stability Functions

This paper surveys some stability results and suggests the use of order arrows as an alternative to order stars in studying questions about the possible A-stability of a numerical method. A discussion of the so-called Butcher–Chipman conjecture includes a proof of a partial result.     

On the growth and approximation of entire functions represented by Laplace–Stieltjes’ transformation

In the present paper, we obtain the characterization of various growth parameters of an entire function F(s) represented by Laplace–Stieltjes transformation in terms of the rate of decrease of \(E_n ( {F,\beta } ),\)where \(E_n ( {F,\beta } )\) represents the error in approximating the function F(s) by exponential polynomials.     

Approximation of partially smooth functions

In this paper we discuss approximation of partially smooth functions by smooth functions. This problem arises naturally in the study of laminated currents.     

Defects and the Convergence of Padé Approximants

A selective survey is given of convergence results for sequences of Padé approximants. Various approaches for dealing with the convergence problems due to `defects" are discussed. Attention is drawn to the close relationship between analyticity properties of a function and the `smoothness" of its Taylor series coefficients. A new theorem on the convergence of horizontal sequences of Padé approximants to functions in the Baker–Gammel–Wills conjecture function class is presented.     

Existence and Uniqueness of Matrix Padé Approximants

For the problems of the left and right matrix Padé approximations, we give the necessary and sufficient conditions for the existence of their solutions. If the left Padé approximant exists, then we prove that its uniqueness is equivalent to the existence of right Padé approximants, and we further give the exact results about the dimension of the linear space $^LR^{*}(m,n)$ formed from the left Padé approximants.