Modification of a theorem of A. A. Gonchar on the rate of rational approximation

In this paper results of A. A. Gonchar [1] on uniform approximation by rational functions are extended to certain classes of unbounded functions.     

Analytic continuation of functions of several variables

Hartogs' theorem concerning analytic continuation of functions of several complex variables is generalized to the case in which part of the variables are real and the continuation is carried out over the complex variables.Translated from Matematicheskie Zametki, Vol. 14, No. 1, pp. 49–54, July, 1973.The author wishes to thank B. V. Shabat, A. A. Gonchar, and A. G. Vitushkin for entering into discussions of the work presented here.     

Approximation to the function z α by rational fractions in a domain with zero external angle

Rational approximations to the function z ^α , α ∈ ? \ ?, were studied by Newman, Gonchar, Bulanov, Vyacheslavov, Andersson, Stahl, and others. The present paper deals with the order of best rational approximations to this function in a domain with zero external angle and vertex at the point z = 0. In particular, the obtained results show that the conditions imposed on the boundary of the domain in the Jackson-type inequality proved by the author in 2001 for the best rational approximations in Smirnov spaces cannot be weakened significantly.     

Approximation of the function sign x in the uniform and integral metrics by means of rational functions

Estimates are obtained for the nonsymmetric deviations R_n [sign x] and R_n [sign x]_L of the function sign x from rational functions of degree ≤n, respectively, in the metric $$c([ - 1, - \delta ] \cup [\delta ,1]), 0< \delta< exp( - \alpha \surd \overline n ), \alpha > 0,$$ and in the metric L[?1, 1]: $$\begin{gathered} R_n [sign x] _{\frown }^\smile exp \{ - \pi ^2 n/(2 ln 1/\delta )\} , n \to \infty , \hfill \\ 10^{ - 3} n^{ - 2} \exp ( - 2\pi \surd \overline n )< R_n [sign x_{|L}< \exp ( - \pi \surd \overline {n/2} + 150). \hfill \\ \end{gathered} $$ Let 0 < δ < 1, Δ (δ)=[?1, ? δ] ∪ [δ, 1]; $$\begin{gathered} R_n [f;\Delta (\delta )] = R_n [f] = inf max |f(x) - R(x)|, \hfill \\ R_n [f;[ - 1,1] ]_L = R_n [f]_L = \mathop {inf}\limits_{R(x)} \smallint _{ - 1}^1 |f(x) - R(x)|dx, \hfill \\ \end{gathered} $$ where R(x) is a rational function of order at most n. Bulanov [1] proved that for δ ε [e^?n, e^?1] the inequality $$\exp \left( {\frac{{\pi ^2 n}}{{2\ln (1/\delta }}} \right) \leqslant R_n [sign x] \leqslant 30 exp\left( {\frac{{\pi ^2 n}}{{2\ln (1/\delta + 4 ln ln (e/\delta ) + 4}}} \right)$$ is valid. The lower estimate in this inequality was previously obtained by Gonchar ([2], cf. also [1]).     

Chebyshev expansions and rational approximations

Let A(z) = A_m(z) + a_mz^mB(z,m) where A_m(z) is a polynomial in z of degree m-1. Suppose A(z) and B(z,m) are approximated by main diagonal Padé approximations of order n and r respectively. Suppose that the number of operations needed to evaluate both sides of the above equations by means of the Padé approximations and polynomial noted are the same. Thus 4n = 3m + 4r. We address ourselves to the question of which procedure is more efficient? That is, which procedure produces the smallest error? A variant of this problem is the situation where A(z) and B(z,m) are approximated by their representations in infinite series of Chebyshev polynomials of the first kind truncated after n and r terms respectively. Here n = m + r.Let F(z) have two different series type representations in overlapping or completely disjoint regions of the complex z-plane. Suppose that for each representation there is a sequence of rational approximations of the same type, say of the Padé class, which converge for |arg z| < π except possibly for some finite set of points. Assume that the number of machine operations required to make evaluations using the noted approximations are the same. Again, we ask which procedure is best? Other variants are studied.General answers to the above questions are not known. Instead, we illustrate the ideas for a number of the rather common special functions.     

Wall Rational Functions and Khrushchev’s Formula for Orthogonal Rational Functions

The Nevalinna–Pick algorithm yields a continued fraction expansion of every Schur function, whose approximants are identified. These approximants are quotients of rational functions which can be understood as the rational analogs of the Wall polynomials. The properties of these Wall rational functions and the corresponding approximants permit us to obtain a Khrushchev’s formula for orthogonal rational functions. An introduction to the convergence of the Wall approximants in the indeterminate case is presented. This work was partially realized during two stays of the second author at the Norwegian University of Science and Technology (NTNU) financed respectively by Secretaría de Estado de Universidades e Investigación from the Ministry of Education and Science of Spain and by the Department of Mathematical Sciences of NTNU. The work of the second author was also partially supported by the Spanish grants from the Ministry of Education and Science, project code MTM2005-08648-C02-01, and the Ministry of Science and Innovation, project code MTM2008-06689-C02-01, and by Project E-64 of Diputación General de Aragón (Spain).     

A fascinating polynomial sequence arising from an electrostatics problem on the sphere

A positive unit point charge approaching from infinity a perfectly spherical isolated conductor carrying a total charge of +1 will eventually cause a negatively charged spherical cap to appear. The determination of the smallest distance ρ(d) (d is the dimension of the unit sphere) from the point charge to the sphere where still all of the sphere is positively charged is known as Gonchar’s problem. Using classical potential theory for the harmonic case, we show that 1+ρ(d) is equal to the largest positive zero of a certain sequence of monic polynomials of degree 2d?1 with integer coefficients which we call Gonchar polynomials. Rather surprisingly, ρ(2)?is the Golden ratio and ρ(4) the lesser known Plastic number. But Gonchar polynomials have other interesting properties. We discuss their factorizations, investigate their zeros and present some challenging conjectures.     

The computation of orthogonal rational functions and their interpolating properties

We shall consider nested spacesl _n ,n = 0, 1, 21... of rational functions withn prescribed poles outside the unit disk of the complex plane. We study orthogonal basis functions of these spaces for a general positive measure on the unit circle. In the special case where all poles are placed at infinity,l _n = _n , the polynomials of degree at mostn. Thus the present paper is a study of orthogonal rational functions, which generalize the orthogonal Szegö polynomials. In this paper we shall concentrate on the functions of the second kind which are natural generalizations of the corresponding polynomials.The work of the first author is partially supported by a research grant from the Belgian National Fund for Scientific Research     

Uniform and tangential harmonic approximation of continuous functions on arbitrary sets

Necessary and sufficient conditions which must be imposed on a set E are derived, such that functions continuous on E G can be approximated by functions harmonic in a region G (Rⁿ.Translated from Matematicheskie Zametki, Vol. 9, No. 2, pp. 131–142, February, 1971.In conclusion I wish to thank my scientific director S. N. Mergelyan and also A. A. Gonchar for their valuable advice.     

Wachstumsordnung und Index der Lösungen linearer Differentialgleichungen mit rationalen Koeffizienten

Let z=∞ be an irregular singular point of the differential equation wⁿ+p_n?1(z)w^(n?1)+...+p₀(z)w=0 with rational coefficients. The functions of the canonical set of solutions relative to z=∞ are of the form $$w(z) = z^\rho \cdot \sum { d_m (z) (\log z)^m , } \rho \varepsilon \mathbb{C}$$ with univalent functions d_m(z) in a neighbourhood of z=∞. Let λ(w)=max {λ(d_m)} denote the maximal order of growth of an irregular solution relative to z=∞, then it is shown that there exists a branch of w in the plane cut along a half ray, which attains the maximal order λ(w). An important tool for the proof is the index of the branches of w.     

Approximation by rational functions in integral metrics and differentiability in the mean

The paper deals with approximations of a functionf of space L_p[0, 1] by rational functions in the metric of this same space (0n(f, p) of functionf of rational functions of degree no higher than n is evidence of the presence inf of derivatives and differentials of a definite order if differentiation is understood as differentiation in the metric of space L_q[0, 1], with 0相似文献   

A method of approximation by rational functions on the real line

For a given system of numbers {z _k } _k=1 ⁿ , IMz _k > 0, rational functions of order 4n — 2 are constructed which effect for a functionf(x∈C _∞) an approximation of the same order as the best approximation by proper rational functions having poles at the points {z _{k k=1} ⁿ and . Translated from Matematicheskie Zametki, Vol. 22, No. 3, pp. 375–380, September, 1977. In conclusion the author thanks E. P. Dolzhenko and S. B. Stechkin, whose discussions contributed to improvements in this work.     

Rational quadrature formulae on the unit circle with arbitrary poles

Interpolatory quadrature rules exactly integrating rational functions on the unit circle are considered. The poles are prescribed under the only restriction of not lying on the unit circle. A computable upper bound of the error is obtained which is valid for any choice of poles, arbitrary weight functions and any degree of exactness provided that the integrand is analytic on a neighborhood of the unit circle. A number of numerical examples are given which show the advantages of using such rules as well as the sharpness of the error bound. Also, a comparison is made with other error bounds appearing in the literature. The work of the first author was supported by the Dirección General de Investigación, Ministerio de Educación y Ciencia, under grants MTM2006-13000-C03-02 and MTM2006-07186 and by UPM and Comunidad de Madrid under grant CCG06-UPM/MTM-539. The work of the second author was partially supported by the Dirección General de Investigación, Ministerio de Educación y Ciencia, under grant MTM2005-08571.     

Fast decreasing rational functions

Necessary and sufficient conditions are obtained for the existence of sequences of rational functions of the formr _n(x) =p _n(x)/p_n(−x), withp _n a polynomial of degreen, that decrease geometrically on (0, 1] in accordance with a specified rate function. The technique of proof involves minimum energy problems for Green potentials in the presence of an external field. Applications are given for the construction of rational approximations of |x| and sgn(x) on [−1, 1] having geometric rates of convergence forx ≠ 0. The research of this author was supported, in part, by National Science Foundation grant DMS-9501130.     

Rates of Best Uniform Rational Approximation of Analytic Functions by Ray Sequences of Rational Functions

In this paper, problems related to the approximation of a holomorphic function f on a compact subset E of the complex plane C by rational functions from the class of all rational functions of order (n,m) are considered. Let ρ _n,m = ρ _n,m (f;E) be the distance of f in the uniform metric on E from the class . We obtain results characterizing the rate of convergence to zero of the sequence of the best rational approximation { ρ _n,m(n) } _n=0 ^∞ , m(n)/n → θ ∈ (0,1] as n → ∞ . In particular, we give an upper estimate for the liminf _{n →∞} ρ _n,m(n) ^1/(n+m(n)) in terms of the solution to a certain minimum energy problem with respect to the logarithmic potential. The proofs of the results obtained are based on the methods of the theory of Hankel operators. June 16, 1997. Date revised: December 1, 1997. Date accepted: December 1, 1997. Communicated by Ronald A. DeVore.     

Approximation by Faber-Laurent rational functions in the mean of functions of callsL p(Γ) with 1

Çavu§  A.  Israfilov  D. M. 《分析论及其应用》1995,11(1):105-118

In what sense is the rational interpolation problem well posed

It is well known that the nonlinear problem of interpolatingm+n+1 data by a rational function of type (m, n) may have no solution, but that the corresponding linearized problem (obtained by multiplying through by the denominator) always leads to a unique rational function, which is often still called the rational interpolant. For fixedm andn, and fixed (possibly multiple) interpolation points, the dependence of this interpolant on the prescribed function values is studied here. For ten notions of convergence in the space _{m, n} the question of the continuity of this interpolation operator is investigated.Communicated by William B. Gragg.AMS classification: 41A24, 30E05, 41A20, 65D05.     

Solomon's zeta functions over algebraic function fields

L. Solomon recently introduced a wide-ranging but concrete generalization of the Riemann and Dedkind zeta functions, as well as of Hey's zeta function for a simple algebra over the rationals. The coefficients of Solomon's zeta function give the numbers of certain types of sublattices in a given lattice over an order in a semisimple rational algebra. This paper studies the analogous zeta function and coefficients which arise for an order in a semi-simpleF _q (X) -algebra, whereF _q (X) is a field of rational functions over a finite fieldF _q . Use is made of the analogues for function fields of results on his zeta functions which were first conjectured by Solomon, and later established by C J Bushnell and l Reiner.     

Error bounds for rational quadrature formulae of analytic functions

We study the error of rational quadrature rules when functions which are analytic on a neighborhood of the set of integration are considered. A computable upper bound of the error is presented which is valid for a broad range of rational quadrature formulae and a comparison is made with the exact error for a number of numerical examples.This work was supported by the Dirección General de Investigación (DGI), Ministerio de Ciencia y Tecnología, under grants BFM2003-06335-C03-02 and BFM2002-04315- C02-01.