Poles Of Approximants Research Articles

Generic properties of Padé approximants and Padé universal series

We establish properties concerning the distribution of poles of Pade approximants, which are generic in Baire category sense. We also investigate Pade universal series, an analog of classical universal series, where Taylor partial sums are replaced with Pade approximants. In particular, we complement previous studies on this subject by exhibiting dense or closed infinite dimensional linear subspaces of analytic functions in a simply connected domain of the complex plane, containing the origin, whose all non zero elements are made of Pade universal series. We also show how Pade universal series can be built from classical universal series with large Ostrowski-gaps.

Trends in Classical Analysis, Geometric Function Theory, and Geometry of Conformal Invariants

The present special issue of the Journal of Abstract and Applied Analysis is devoted to trends in classical analysis, geometric function theory, and geometry of conformal mappings. In the sense of the title of this journal, we wanted to present a spectrum of research themes reaching from applied analysis to pure analysis and from applications of analysis in geometry to applications in differential equations and integral equations. Further, our aim was to find articles on classical function theory as well as on its generalizations in several directions. One additional aspect, that was paid attention to, is the tendency of mathematics to use computers to solve problems by new and effective algorithms. In detail, we addressed to the following themes. That we did not forget classical pure analysis is proved by an article that considers harmonic functions on Riemann manifolds and by an article on geometry and topology of Banach spaces. The classical geometric function theory is represented by an article on univalence criterions associated with the nth derivative. The relationships between conformal mappings and integral equations are in the scope of two articles, where algorithms are proved to compute the mappings of unbounded multiply connected as well as bounded multiply connected regions onto slit regions. Other old themes of classical function theory are entire functions for which we publish a paper on uniqueness theorems for monomials of entire functions. Concerning the generalizations of classical analysis, we incorporated a paper on the stability of solutions of fractional differential equations and two papers on harmonic mappings, specially one on the general theory of log-harmonic mappings and one on certain classes of harmonic mappings defined by convolutions. The papers on the generalizations of holomorphic functions are completed by a longer article on the distribution of zeros and poles of the rational approximants of a nonholomorphic function on an interval. The aspect of new algorithms is addressed in an article on Hermite interpolation using Mobius transformations of planar Pythagorean-hodograph cubics and in a paper where algorithms to calculate inverse Z-transforms on the unit disc by number-theoretical methods are proved. We hope that in this broad variety of analytic themes many researchers can find something interesting and new.

On the Distribution of Zeros and Poles of Rational Approximants on Intervals

The distribution of zeros and poles of best rational approximants is well understood for the functions f(x) = |x|α, α > 0. If f ∈ C[−1, 1] is not holomorphic on [−1, 1], the distribution of the zeros of best rational approximants is governed by the equilibrium measure of [−1, 1] under the additional assumption that the rational approximants are restricted to a bounded degree of the denominator. This phenomenon was discovered first for polynomial approximation. In this paper, we investigate the asymptotic distribution of zeros, respectively, a‐values, and poles of best real rational approximants of degree at most n to a function f ∈ C[−1, 1] that is real‐valued, but not holomorphic on [−1, 1]. Generalizations to the lower half of the Walsh table are indicated.

Asymptotics of Hermite-Pade Rational Approximants for Two Analytic Functions with Separated Pairs of Branch Points (Case of Genus 0)

We investigate the asymptotic behavior for type II Hermite-Pade approximation to two functions, where each function has two branch points and the pairs of branch points are separated. We give a classification of the cases such that the limiting counting measures for the poles of the Hermite-Pade approximants are described by an algebraic function of order 3 and genus 0. This situation gives rise to a vector-potential equilibrium problem for three measures and the poles of the common denominator are asymptotically distributed like one of these measures. We also work out the strong asymptotics for the corresponding Hermite-Pade approximants by using a 3x3 Riemann-Hilbert problem that characterizes this Hermite-Pade approximation problem.

An extension of the ‘ 1/9 ’-problem

We consider the rational approximation of a perturbed exponential function (1) f ( z ) = u 0 ( z ) + u 1 ( z ) e z on the negative real axis R − with u 0 and u 1 ≢ 0 being rational functions. Building upon results from the solution of the ‘ 1/9’-problem by Gonchar and Rakhmanov, it is shown that rational best approximants to f share the fast asymptotic convergence of rational approximants to the exponential function, despite the fact that the functions (1) may differ substantially from the exponential. Our main result is that for k ∈ Z fixed, we have (2) lim n → ∞ ‖ f − r n , n + k ∗ ‖ R − 1 / n = 1 9.28902549 … = H with H being the same Halphen constant as that already known from the solution of the ‘ 1/9’-problem and r n m ∗ = r n m ∗ ( f , R − ; ⋅ ) being a rational best approximant with denominator and numerator degrees at most n and m in the uniform norm on R − to the function f of type (1). Further, it is shown that the convergence on R − , which is characterized by (2), extends to compact subsets of the complex plane C and also to Hankel contours in C ∖ R − , and this extension holds with the same asymptotic rate of convergence H . Many of the results for rational best approximants extend also to rational close-to-best approximants, which by our definition are rational approximants with a convergence behavior on R − that is comparable in an n th root sense to that of the best approximants r n , n ∗ . The asymptotic distribution of the poles of approximants is studied and determined. In the present paper we present only results, complete proofs are not given, however, some of the main ideas of the proofs are sketched.

On the numerical condition of a generalized Hankel eigenvalue problem

The generalized eigenvalue problem $$\widetilde H y \,{=}\, \lambda H y$$ with H a Hankel matrix and $$\widetilde H$$ the corresponding shifted Hankel matrix occurs in number of applications such as the reconstruction of the shape of a polygon from its moments, the determination of abscissa of quadrature formulas, of poles of Pade approximants, or of the unknown powers of a sparse black box polynomial in computer algebra. In many of these applications, the entries of the Hankel matrix are only known up to a certain precision. We study the sensitivity of the nonlinear application mapping the vector of Hankel entries to its generalized eigenvalues. A basic tool in this study is a result on the condition number of Vandermonde matrices with not necessarily real abscissas which are possibly row-scaled.

Influence of poles on equioscillation in rational approximation

The error curve for the rational best approximation of ƒ ∈ C[−1, 1] is characterized by the well-known equioscillation property. Contrary to the polynomial case, the distribution of these alternations is not governed by the equilibrium distribution. It is known that these points need not be dense in [−1, 1]. The reason is the influence of the distribution of the poles of rational approximants. In this paper, we generalize the results known so far to situations where the requirements for the degrees of numerators and denominators are less restrictive.

The interaction of alternation points and poles in rational approximation

The interrelation of alternation points for the minimal error function and poles of best Chebyshev approximants is investigated if uniform approximation on the interval [ - 1 , 1 ] by rational functions of degree ( n ( s ) , m ( s ) ) is considered, s ∈ N . In general, the alternation points need not to be uniformly distributed with respect to the equilibrium measure on [ - 1 , 1 ] , even not to be dense on the interval. We show that, at least for a subsequence Λ ⊂ N , the asymptotic behaviour of the alternation points to the degrees ( n ( s ) , m ( s ) ) , s ∈ Λ , is completely determined by the location of the poles of the best approximants, and vice versa, if m ( s ) ⩽ n ( s ) or m ( s ) - n ( s ) = o ( s / log s ) as s → ∞ .

Poles and Alternation Points in Real Rational Chebyshev Approximation

The distribution of equi-oscillation points (alternation points) for the error in best Chebyshev approximation on [−1,1] by rational functions is investigated. In general, the alternation points need not be dense in [−1,1] when rational functions of degree (n, m) are considered and asymptotically n/m → κ with κ ≥ 1. We show that the asymptotic behavior of the alternation points is closely related to the behavior of the poles of the rational approximants. Hence, poles of the rational approximations are attracting points of alternations such that the well-known equi-distribution for the polynomial case can be heavily disturbed.

An Lp Analog to AAK Theory for p⩾2

We develop an Lp analog to AAK theory on the unit circle that interpolates continuously between the case p=∞, which classically solves for best uniform meromorphic approximation, and the case p=2, which is equivalent to H2-best rational approximation. We apply the results to the uniqueness problem in rational approximation and to the asymptotic behaviour of poles of best meromorphic approximants to functions with two branch points. As pointed out by a referee, part of the theory extends to every p∈[1, ∞] when the definition of the Hankel operator is suitably generalized; this we discuss in connection with the recent manuscript by V. A. Prokhorov, submitted for publication.

Best Meromorphic Approximation of Markov Functions on the Unit Circle

Let E \subset(-1,1) be a compact set, let μ be a positive Borel measure with support \supp μ =E , and let H p (G),1≤ p ≤∈fty, be the Hardy space of analytic functions on the open unit disk G with circumference Γ={z \colon |z|=1} . Let Δ n,p be the error in best approximation of the Markov function \frac{1}{2π i} ∈t_E \frac{d μ(x)}{z-x} in the space L p (Γ) by meromorphic functions that can be represented in the form h=P/Q , where P ∈ H p (G),Q is a polynomial of degree at most n , Q\not \equiv 0 . We investigate the rate of decrease of Δ n,p ,1≤ p ≤∈fty , and its connection with n -widths. The convergence of the best meromorphic approximants and the limiting distribution of poles of the best approximants are described in the case when 1 - ∈fty.

A new estimation method in modal analysis

In this paper, the modal analysis model, which is made up of a linear combination of complex exponential functions, is considered. The problem of estimating the number of modes as well as the frequencies, decays, amplitudes, and phases and their variability is afforded by means of the condensed distribution of the poles of Pade approximants to the Z-transform of the data. This provides a unified framework in which all of these problems can be solved in an almost automatic way. It is experimentally proved that the results favorably compare with that provided by well-established methods.

Reconstruction of a Piecewise Constant Function from Noisy Fourier Coefficients by Padé Method

The problem of reconstructing a piecewise constant function from a finite number of its Fourier coefficients perturbed by noise is considered. A reconstruction method, based on the computation of the Pade approximants to the Z-transform of the sequence of the noisy Fourier coefficients is proposed. The method is based on the remark that the distribution of the poles of the Pade approximants shows, asymptotically, clusters in the complex plane which allow the identification of the discontinuities of the function. It turns out that the Z-transform is a multiple-valued function and the location of the clusters corresponds to the branch points of such a function. By using this property of the Pade poles, a very effective reconstruction method can be developed. Some numerical experiments are presented to show the feasibility of the method.

Continued Fractions and Szegö Polynomials in Frequency Analysis and Related Topics

This paper is an expository survey of recent research on the application of Szego polynomials and PPC-continuedfractions to the frequency analysis problem described as follows: We want to determine the unknown frequencies ω1, ω2, ..., ω I from a sample of N observed values x N (m), m = 0, 1, ..., N− 1, arising from a continuous waveform that is the superposition of a finite number of sinusoidal waves with frequencies ω1,ω2, ..., ω I . The method is based on the property that certain zerosof the Szego polynomials (and poles of the PPC-fraction approximants) converge (as N → ∞) to the frequency points ei ω j , j = ±1, ± 2, ..., ± I. The remaining zeros are bounded away from the unit circle |z|=1, asN → ∞. The Levinson algorithm is used to construct the Szego polynomials and PPC-fractions from the values x N (m). A discussion is given on connections between the topics: Caratheodory functions,the trigonometric moment problem, Szego polynomials and PPC-fractions. We also describe applications to Doppler radar, medicine, speech processing, speech therapy, meteorology and ocean tides.

Some properties of the asymptotic location of poles of Pade approximants to noisy rational functions, relevant for modal analysis

The modal analysis model, made up by a linear combination of complex exponential functions, is considered. Pade approximants to the Z-transform of a noisy sample are then considered, and the asymptotic locus of their poles is studied. It turns out that this locus is strongly related to the complex exponentials of the model. By exploiting these properties, powerful methods for estimating the model parameters can be devised, which have both denoising and super-resolution capabilities.

Incomplete rational approximation in the complex plane

We consider rational approximations of the form $$\left\{ {(1 + z)^{\alpha n + 1} \frac{{p_{cn} (z)}}{{q_n (z)}}} \right\}$$ in certain natural regions in the complex plane wherep cn andq n are polynomials of degreecn andn, respectively. In particular we construct natural maximal regions (as a function of α andc) where the collection of such rational functions is dense in the analytical functions. So from this point of view we have rather complete analog theorems to the results concerning incomplete polynomials on an interval. The analysis depends on an examination of the zeros and poles of the Pade approximants to (1+z)αn+1. This is effected by an asymptotic analysis of certain integrals. In this sense it mirrors the well-known results of Saff and Varga on the zeros and poles of the Pade approximant to exp. Results that, in large measure, we recover as a limiting case. In order to make the asymptotic analysis as painless as possible we prove a fairly general result on the behavior, inn, of integrals of the form $$\int_0^1 {[t(1 - t)f_z (t)]^n {\text{ }}dt,}$$ wheref z (t) is analytic inz and a polynomial int. From this we can and do analyze automatically (by computer) the limit curves and regions that we need.

Distribution of poles of diagonal rational approximants to functions of fast rational approximability

We show that the most of the time, most poles of diagonal multipoint Pade or best rational approximants to functions admitting fast rational approximation, leave the region of meromorphy. Following is a typical result: Letf be single-valued and analytic in CS, where cap(S)=0. Let {n j } =1 ∞ be an increasing sequence of positive integers withn j+1/n j →1 asj→∞. Then there exists an infinite sequenceL of positive integers such that asj→∞,j∈L the total multiplicity of poles of any sequence of type (n j ,n j ) multipoint Pade or best rational approximants tof, iso(n j ) in any compactK in whichf is meromorphic. The sequenceL is independent of the particular sequence of multipoint Pade or best approximants, and yields the same behavior for “near-best” approximants. If the errors of best approximation on some compact set satisfy a weak regularity condition, then we may takeL={1,2,3,⋯}.

Generalized moment representations and Pad�-Chebyshev approximations

An approach to the application of Dzyadyk's generalized moment representations in problems of construction and investigation of the Pade-Chebyshev approximants is developed. With its help, certain properties of the Pade-Chebyshev approximants of a class of functions that is a natural analog of the class of Markov functions are studied. In particular, it is proved that the poles of the Pade-Chebyshev approximants of these functions lie outside their domain of analyticity.

Steep standing waves at a fluid interface

An algorithm is formulated for computing perturbation-series solutions for standing waves on the interface between two semi-infinite fluids of different but uniform densities. Using a comppter, the series solutions are computed to fifth order for a general value of r, the ratio of the density of the upper fluid to that of the lower fluid (0 ≤ r ≤ l), and to 21st order for five specific values of this ratio: r = 0, 10−3, 0·1, 5·0, 1·0. The series for the period, the energy, and the interface profile of the waves are summed using Pade approximants. The maximum wave height for each of the above five density ratios is estimated from the locations of the poles of the Pade approximants for the wave period and the wave energy. At maximum height the interface appears to be vertical at a point on the interface that is very near the crest for r = 10−3 and approaches the midpoint between the crest and the trough as r approaches 1·0.

Padé approximants and variational methods for operator series

A variational method for evaluating matrix elements and poles of Pade approximants to operator series is discussed. Error bounds and convergence properties are considered for the approximants of arbitrary order. Some applications to potential models are presented.