Extremal Polynomials with Preassigned Zeros and Rational Approximants

Let p _n be the n th orthonormal polynomial with respect to a positive finite measure μ supported by Δ=[-1,1] . It is well known that, uniformly on compact subsets of C/Δ , and, for a large class of measures μ , where g _Ω (z) is Green's function of with pole at infinity. It is also well known that these limit relations give convergence of the diagonal Padé approximants of the Markov function to f on Ω with a certain geometric speed measured by g _Ω (z) . We prove corresponding results when we restrict the freedom of p _n by preassigning some of the zeros. This means that the Padé approximants are replaced by Padé-type approximants where some of the poles are preassigned. We also replace Δ by general compact subsets of C. July 12, 1995. Date revised: October 1, 1996.     

On the Convergence in Capacity of Rational Approximants

Let {r _n } be a sequence of rational functions deg( r _n ≤ n) that converge rapidly in measure to an analytic function f on an open set in C ^N . We show that {r _n } converges rapidly in capacity to f on its natural domain of definition W _f (which, by a result of Goncar, is an open subset of C ^N ). In particular, for f meromorphic on C ^N and analytic near zero the sequence of Padé approximants {π _n (z, f, λ)} (as defined by Goncar) converges rapidly in capacity to f on C ^N . January 14, 1999. Date revised: October 7, 1999. Date accepted: November 1, 1999.     

The Calculation of Some Rational Approximants in Two Variables

Chisholm has shown how rational approximants may be definedfor functions of two variables, which reduce to diagonal Pad?Approximants when one variable is zero. We extend the idea toallow for different maximum powers in numerator and denominator,and state the relevant theorems. We discuss numerical difficultiesand possible singularities occurring in the construction ofthe approximants; approximants to the Beta function are calculatedand discussed in detail.     

General Order Pade-type Rational Approximants Defined from Double Power Series

Explicit determinant representations are given for general orderPadé-type rational approximants in two variables, andconvenient expressions are obtained for the Chisholm approximantsand "off-diagonal" Chisholm approximants, while the generalizationto three or more variables is obvious.     

Normal Elements of C*-Algebras of Real Rank Zero without Finite-Spectrum Approximants

We investigate inductive limits of Toeplitz-type C*-algebras.One example, which has real-rank zero, is the middle term ofan exact sequence where is a Bunce-Deddens algebra and I is AF. Using Berg's technique,we produce a normal element N that is not the limit of finite-spectrumnormals. Moreover, this is an example of a normal element inan inductive limit that is not the limit of normal elementsof the approximating subalgebras. A second example is an embedding of C() ( the closed disk) into , where is a simple AF algebra and is the Toeplitz algebra.Let _n, for n 2, be the CW complex obtained as the quotientof by an n-fold identification of the boundary. (So ₂ = RP².)Regarding C(_n) as a subalgebra of C(), we find nontrivial embeddingsof C(_n) into type I inductive limits. From this, we producea *-homomorphism, for n odd, C₀(_n\{pt}) _{n + 1}, that inducesan isomorphism on K-theory. More generally, for X a connectedCW complex minus a point, and for n odd, we show that the map is a split surjection.     

Branch Points, M Fractions and Rational Approximants Generated by Linear Equations

Rational approximants with error terms are obtained from continuedfractions generated by linear equations. A simple method forobtaining the branch points of a function with a quasi-periodiccontinued fraction is illustrated. Continued are derived whichmatch the derivatives of a function at two or more points inthe complex plane. Rational functions that match derivativesand also take on particular values are presented.     

The Generation of Chisholm Rational Polynomial Approximants to Power Series in Two Variables

The generation of successive Chisholm rational polynomial approximantsf_m/_m of f(x, y), a power series in two variables, is discussed. A necessary and sufficient condition for the non-degeneracyof f_m/_m is given. It is shown that the non-degeneracy of thediagonal Pad? approximants of order m in each variable separatelyis a necessary condition for the non-degeneracy of f_m/_m. In the case of a symmetric function, it is proved that the Chisholmapproximant f_m/_m is symmetric and non-degenerate if and onlyif all the diagonal Pad? approximants of order up to m in onevariable are non-degenerate. The generation of successive Chisholmapproximants to symmetric functions is also considered. The computational scheme, called the prong method, extends tocover the computation of Chisholm approximants in N-variables(Chisholm & McEwan, 1974).     

The Solution of Volterra Integral Equations of the Convolution Type Using Two-Point Rational Approximants

This paper applies the method of Laplace transform inversionusing two-point rational approximants, introduced by Grundy(1977), to the solution of Volterra integral equations of theconvolution type. The transforms which occur in solving theseequations using Laplace transform methods are, in many cases,of the type which can be inverted by the method introduced inGrundy (1977). There are however additional difficulties whichhave to be met. Examples are given to illustrate their resolutionand at the same time indicate the scope of the method.     

Discontinuity of Best Harmonic Approximants

LetDR²be the open unit disk. We consider best harmonic approximation to functions continuous onD. In a basic paper, Haymanet al.characterized best harmonic approximants which are themselves continuous onD. In this paper we give sufficient conditions and many simple examples of functions continuous onDwhich have no best harmonic approximants which are continuous onD.     

Some Properties of Chisholm Approximants

The rational fraction approximants to functions of two variablesdefined by Chisholm are shown (a) to factorize into Pad? approximantswhen the given function is a product function and (b) to beunitary when the given function is unitary.     

Asymptotic Behavior of Logarithmic Potential of Zero Kind

Under a fairly general condition on the behavior of a Borel measure,we obtain unimprovable asymptotic formulas for its logarithmic potential.     

Asymptotic Uniqueness of Best Rational Approximants of Given Degree to Markov Functions in L 2 of the Circle

We improve over a sufficient condition given in [8] for uniqueness of a nondegenerate critical point in best rational approximation of prescribed degree over the conjugate-symmetric Hardy space of the complement of the disk. The improved condition connects to error estimates in AAK approximation, and is necessary and sufficient when the function to be approximated is of Markov type. For Markov functions whose defining measure satisfies the Szego condition, we combine what precedes with sharp asymptotics in multipoint Padé approximation from [43], [40] in order to prove uniqueness of a critical point when the degree of the approximant goes large. This lends perspective to the uniqueness issue for more general classes of functions defined through Cauchy integrals.     

全平面上零级Dirichlet级数的增长性

在较一般的指数条件下,研究了全平面上零级Dirichlet级数的增长性,得到了一些新的结果.     

On the Bernstein Inequality for Rational Functions with a Prescribed Zero

We extend some results of Giroux and Rahman (Trans. Amer. Math. Soc.193(1974), 67–98) for Bernstein-type inequalities on the unit circle for polynomials with a prescribed zero atz=1 to those for rational functions. These results improve the Bernstein-type inequalities for rational functions. The sharpness of these inequalities is also established. Our approach makes use of the Malmquist–Walsh system of orthogonal rational functions on the unit circle associated with the Lebesgue measure.     

Asymptotic Zero Behavior of Laguerre Polynomials with Negative Parameter

We consider Laguerre polynomials $L_n^{(\alpha_n)}(nz)$ with varying negative parameters $\alpha_n$, such that the limit $A = -\!\lim_n \alpha_n/n$ exists and belongs to $(0,1)$. For $A > 1$, it is known that the zeros accumulate along an open contour in the complex plane. For every $A \in (0,1)$, we describe a one-parameter family of possible limit sets of the zeros. Under the condition that the limit $r= - \!\lim_n {1}/{n} \log [\dist\!(\alpha_n, \mbox{\footnotesize{\bf Z}})]$ exists, we show that the zeros accumulate on $\Gamma_r \cup [\beta_1,\beta_2]$ with $\beta_1$ and $\beta_2$ only depending on $A$. For $r \in [0,\infty)$, $\Gamma_r$ is a closed loop encircling the origin, which for $r = +\infty$, reduces to the origin. This shows a great sensitivity of the zeros to $\alpha_n$s proximity to the integers. We use a Riemann–Hilbert formulation for the Laguerre polynomials, together with the steepest descent method of Deift and Zhou to obtain asymptotics for the polynomials, from which the zero behavior follows.     

Zero Distribution of Composite Polynomials and Polynomials Biorthogonal to Exponentials

We analyze polynomials P _n that are biorthogonal to exponentials , in the sense that

Padé-Type Approximants of Markov and Meromorphic Functions

We study the question of convergence of Padé and Padé-type approximants to functions meromorphic in a domain. As an example we investigate in detail the case of functions of the formwhereμ(z) is a Markov function.