On Kernel polynomials and self-perturbation of orthogonal polynomials

Given an orthogonal polynomial system {Q _n (x)} _n=0 ^∞, define another polynomial system by where α _n are complex numbers and t is a positive integer. We find conditions for {P _n (x)} _n=0 ^∞ to be an orthogonal polynomial system. When t=1 and α₁≠0, it turns out that {Q _n (x)} _n=0 ^∞ must be kernel polynomials for {P _n (x)} _n=0 ^∞ for which we study, in detail, the location of zeros and semi-classical character. Received: November 25, 1999; in final form: April 6, 2000?Published online: June 22, 2001     

Strong Approximation of Eigenvalues of Large Dimensional Wishart Matrices by Roots of Generalized Laguerre Polynomials

The purpose of this note is to establish a link between recent results on asymptotics for classical orthogonal polynomials and random matrix theory. Roughly speaking it is demonstrated that the ith eigenvalue of a Wishart matrix W(I_n,s) is close to the ith zero of an appropriately scaled Laguerre polynomial, when . As a by-product we obtain an elemantary proof of the Mar enko–Pastur and the semicircle law without relying on combinatorical arguments.     

Asymptotics of the averages of orthogonal polynomials on two intervals

We find uniform asymptotics for the averages of orthogonal polynomials on two intervals.     

Exact L 2-Small Ball Asymptotics of Gaussian Processes and the Spectrum of Boundary-Value Problems

We sharpen a classical result on the spectral asymptotics of boundary-value problems for self-adjoint ordinary differential operator. Using this result, we obtain the exact L ₂-small ball asymptotics for a new class of zero-mean Gaussian processes. This class includes, in particular, the integrated generalized Slepian process, integrated centered Wiener process, and integrated centered Brownian bridge. Partially supported by RFFR grant No.07-01-00159 and by grant NSh-227.2008.1.     

Bernstein—Szegő's Theorem for Sobolev Orthogonal Polynomials

In this paper we obtain the strong asymptotics for the sequence of orthogonal polynomials with respect to the inner product where ρ ₀ and ρ ₁ are weights which satisfy Szegő's condition, supported on a smooth Jordan closed curve or arc. December 14, 1997. Date revised: September 21, 1998. Date accepted: November 16, 1998.     

Backward Stability of Clenshaw's Algorithm

We study numerical properties of Clenshaw's algorithm for summing the series w = _{n = 0} ^N b _n p _n where p _n satisfy the linear three-term recurrence relation. We prove that under natural assumptions Clenshaw's algorithm is backward stable with respect to the data b _n, n = 0,N.     

On a Modification of the Jacobi Linear Functional: Asymptotic Properties and Zeros of the Corresponding Orthogonal Polynomials

The paper deals with orthogonal polynomials in the case where the orthogonality condition is related to semiclassical functionals. The polynomials that we discuss are a generalization of Jacobi polynomials and Jacobi-type polynomials. More precisely, we study some algebraic properties as well as the asymptotic behaviour of polynomials orthogonal with respect to the linear functional U U=J _,+A ₁(x–1)+B ₁(x+1)–A ₂(x–1)–B ₂(x+1), where J _, is the Jacobi linear functional, i.e. J _,,p›=_–1 ¹ p(x)(1–x)(1+x)dx,,>–1, pP, and P is the linear space of polynomials with complex coefficients. The asymptotic properties are analyzed in (–1,1) (inner asymptotics) and C[–1,1] (outer asymptotics) with respect to the behaviour of Jacobi polynomials. In a second step, we use the above results in order to obtain the location of zeros of such orthogonal polynomials. Notice that the linear functional U is a generalization of one studied by T. H. Koornwinder when A ₂=B ₂=0. From the point of view of rational approximation, the corresponding Markov function is a perturbation of the Jacobi–Markov function by a rational function with two double poles at ±1. The denominators of the [n–1/n] Padé approximants are our orthogonal polynomials.     

A simple characterization of weighted Sobolev spaces with bounded multiplication operator

In this paper we give a simple characterization of weighted Sobolev spaces (with piecewise monotone weights) such that the multiplication operator is bounded: it is bounded if and only if the support of μ₀ is large enough. We also prove some basic properties of the appropriate weighted Sobolev spaces. To have bounded multiplication operator has important consequences in Approximation Theory: it implies the uniform bound of the zeros of the corresponding Sobolev orthogonal polynomials, and this fact allows to obtain the asymptotic behavior of Sobolev orthogonal polynomials.     

Asymptotics of orthogonal polynomials and the numerical solution of ill-posed problems

The paper reviews the impact of modern orthogonal polynomial theory on the analysis of numerical algorithms for ill-posed problems. Of major importance are uniform bounds for orthogonal polynomials on the support of the weight function, the growth of the extremal zeros, and asymptotics of the Christoffel functions.     

Asymptotic analysis of the Askey-scheme I: from Krawtchouk to Charlier

We analyze the Charlier polynomials C _n (χ) and their zeros asymptotically as n → ∞. We obtain asymptotic approximations, using the limit relation between the Krawtchouk and Charlier polynomials, involving some special functions. We give numerical examples showing the accuracy of our formulas.      

Erd s–Turán Type Theorems on Quasiconformal Curves and Arcs

The theorems of Erd s and Turán mentioned in the title are concerned with the distribution of zeros of a monic polynomial with known uniform norm along the unit interval or the unit disk. Recently, Blatt and Grothmann (Const. Approx.7(1991), 19–47), Grothmann (“Interpolation Points and Zeros of Polynomials in Approximation Theory,” Habilitationsschrift, Katholische Universität Eichstätt, 1992), and Andrievskii and Blatt (J. Approx. Theory88(1977), 109–134) established corresponding results for polynomials, considered on a system of sufficiently smooth Jordan curves and arcs or piecewise smooth curves and arcs. We extend some of these results to polynomials with known uniform norm along an arbitrary quasiconformal curve or arc. As applications, estimates for the distribution of the zeros of best uniform approximants, values of orthogonal polynomials, and zeros of Bieberbach polynomials and their derivatives are obtained. We also give a negative answer to one conjecture of Eiermann and Stahl (“Zeros of orthogonal polynomials on regularN-gons,” in Lecture Notes in Math.1574(1994), 187–189).     

Asymptotics of Orthogonal Polynomials: Some Old, Some New, Some Identities

We briefly review some asymptotics of orthonormal polynomials. Then we derive the Bernstein–Szeg, the Riemann–Hilbert (or Fokas–Its–Kitaev), and Rakhmanov projection identities for orthogonal polynomials and attempt a comparison of their applications in asymptotics.     

Asymptotics and weighted estimates of meixner polynomials orthogonal on the gird {0, δ, 2δ, ...}

Suppose that 0<δ≤1,N=1/δ, and α, ga≥0, is an integer. For the classical Meixner polynomials orthonormal on the gird {0, δ, 2δ, ...} with weight ρ(x)=(1-e ^−δ)^αг(Nx+α+ 1)/г(Nx+1), the following asymptotic formula is obtained: . The remainderv _n,N ^α (z) forn≤λN satisfies the estimate

Ratio and relative asymptotics of polynomials orthogonal with respect to varying Denisov-type measures

Let μ be a finite positive Borel measure with compact support consisting of an interval plus a set of isolated points in , such that μ^′>0 almost everywhere on [c,d]. Let , be a sequence of polynomials, , with real coefficients whose zeros lie outside the smallest interval containing the support of μ. We prove ratio and relative asymptotics of sequences of orthogonal polynomials with respect to varying measures of the form dμ/w_2n. In particular, we obtain an analogue for varying measures of Denisov's extension of Rakhmanov's theorem on ratio asymptotics. These results on varying measures are applied to obtain ratio asymptotics for orthogonal polynomials with respect to fixed measures on the unit circle and for multi-orthogonal polynomials in which the measures involved are of the type described above.     

Extensions of homogeneous polynomials on c 0(l 2 i )

We show that a 2-homogeneous polynomial on the complex Banach space c ₀ l ₂ ⁱ ) is norm attaining if and only if it is finite (i.e, depends only on finite coordinates). As the consequence, we show that there exists a unique norm-preserving extension for norm-attaining 2-homogeneous polynomials on c ₀(l ₂ ⁱ ). The second author was supported by FAPESP, Brazil, Research Grant 01/04220-8.     

Asymptotics of the parabolic Green function for an elliptic operator on a manifold with conical points

We construct the asymptotics ast→0 of the trace of the operator exp(−tP) for an elliptic operatorP on a manifold with conical points. Translated fromMatematicheskie Zametki, Vol. 63, No. 1, pp. 28–36, January, 1998. This research was supported by the Belarus Foundation for Basic Research under grant No. 96-01 00790.     

The statistical distribution of the zeros of random paraorthogonal polynomials on the unit circle

The orthogonal polynomials on the unit circle are defined by the recurrence relation

where for any k0. If we consider n complex numbers and , we can use the previous recurrence relation to define the monic polynomials Φ₀,Φ₁,…,Φ_n. The polynomial Φ_n(z)=Φ_n(z;α₀,…,α_n-2,α_n-1) obtained in this way is called the paraorthogonal polynomial associated to the coefficients α₀,α₁,…,α_n-1.We take α₀,α₁,…,α_n-2 i.i.d. random variables distributed uniformly in a disk of radius r<1 and α_n-1 another random variable independent of the previous ones and distributed uniformly on the unit circle. For any n we will consider the random paraorthogonal polynomial Φ_n(z)=Φ_n(z;α₀,…,α_n-2,α_n-1). The zeros of Φ_n are n random points on the unit circle.We prove that for any the distribution of the zeros of Φ_n in intervals of size near e^iθ is the same as the distribution of n independent random points uniformly distributed on the unit circle (i.e., Poisson). This means that, for large n, there is no local correlation between the zeros of the considered random paraorthogonal polynomials.     

Relative asymptotics and Fourier series of orthogonal polynomials with a discrete Sobolev inner product

Let μ be a finite positive Borel measure supported in [−1,1] and introduce the discrete Sobolev-type inner product

where the mass points a_k belong to [−1,1], M_k,i0, i=0,…,N_k−1, and M_k,Nk>0. In this paper, we study the asymptotics of the Sobolev orthogonal polynomials by comparison with the orthogonal polynomials with respect to the measure μ and we prove that they have the same asymptotic behaviour. We also study the pointwise convergence of the Fourier series associated to this inner product provided that μ is the Jacobi measure. We generalize the work done by F. Marcellán and W. Van Assche where they studied the asymptotics for only one mass point in [−1,1]. The same problem with a finite number of mass points off [−1,1] was solved by G. López, F. Marcellán and W. Van Assche in a more general setting: they consider the constants M_k,i to be complex numbers. As regards the Fourier series, we continue the results achieved by F. Marcellán, B. Osilenker and I.A. Rocha for the Jacobi measure and mass points in .     

Error Bounds for Asymptotic Expansion of the Conditional Variance of the Scale Mixtures of the Multivariate Normal Distribution

Let X= A ^1/2 G be a scale mixture of a multivariate normal distribution with X, G ⁿ , Gis a multivariate normal vector, and A is a positive random variable independent of the multivariate random vector G. This study presents asymptotic results of the conditional variance-covariance, Cov(X ₂|X ₁), X ₁ ^m , m < n, under some moment expressions. A new representation form is also presented for conditional expectation of the scale variable on the random vector X ₁ ^m , m < n. Both the asymptotic expression and the representation are manageable and in computable form. Finally, an example is presented to illustrate how the computations are carried out.     

Asymptotics of multinomial sums and identities between multi-integrals

We calculate the asymptotics of combinatorial sums ∑ _α f(α)( _α ⁿ ) ^β , whereα = (α ₁, …,α _h ) withα _i =α _j for certaini, j. Hereh is fixed and theα _i ’s are natural numbers. This implies the asymptotics of the correspondingS _n -character degrees ∑_λ f(λ)d _λ ^β . For certain sequences ofS _n characters which involve Young’s rule, the latter asymptotics were obtained earlier [1] by a different method. Equating the two asymptotics, we obtain equations between multi-integrals which involve Gaussian measures. Special cases here give certain extensions of the Mehta integral [5], [6]. Supported by the Weizmann Institute of Science, Rehovot, Israel; by the Institute for Advanced Study, Princeton, New Jersey, USA; NSF grant number DMS 9304580; and by the Centre National de Recherche Scientifique, Lille, France. This work was partially supported by an NSF grant number DMS 94-01197.