Birkhoff interpolation on some perturbed roots of unity: Revisited

We consider the regularity of Birkhoff interpolation on some non-uniformly distributed roots of unity. We determine the range of values of in the complex plane which makes the problem of lacunary interpolation on the zeros of (z ⁿ +1)(z–) regular.     

On the Decay of Infinite Products of Trigonometric Polynomials

We consider infinite products of the form f(=_k=1 m _k(2^-k), where {m _k} is an arbitrary sequence of trigonometric polynomials of degree at most n with uniformly bounded norms such that m _k(0)= 1 for all k. We show that f() can decrease at infinity not faster than O(^-n) and present conditions under which this maximal decay is attained. This result can be applied to the theory of nonstationary wavelets and nonstationary subdivision schemes. In particular, it restricts the smoothness of nonstationary wavelets by the length of their support. This also generalizes well-known similar results obtained for stable sequences of polynomials (when all m _k coincide). By means of several examples, we show that by weakening the boundedness conditions one can achieve exponential decay.     

On the Distribution of the Number of Calls in the Queuing System Dη|Dξkappa|1

For the queuing system D |D ^kappa|1, we determine the distribution of the number of calls in transient and stationary modes.     

Functional calculus under the Tadmor-Ritt condition, and free interpolation by polynomials of a given degree

For Banach space operators T satisfying the Tadmor-Ritt condition ||(zI−T)⁻¹||?C|z−1|⁻¹, |z|>1, we prove that the best-possible constant C_T(n) bounding the polynomial calculus for T, ||p(T)||?C_T(n)||p||_∞, deg(p)?n, behaves (in the worst case) as as n→∞. This result is based on a new free (Carleson type) interpolation theorem for polynomials of a given degree.     

The Bergman kernel: Explicit formulas,deflation, Lu Qi-Keng problem and Jacobi polynomials

We investigate the Bergman kernel function for the intersection of two complex ellipsoids {(z,w ₁,w ₂) ∈ C ⁿ⁺²: |z ₁|²+...+|z _n |²+|w ₁| ^q < 1, |z ₁|²+...+|z _n |²+|w ₂| ^r < 1}. We also compute the kernel function for {(z ₁,w ₁,w ₂) ∈ C³: |z ₁|^2/n + |w ₁| ^q < 1, |z ₁|^2/n + |w ₂| ^r < 1} and show deflation type identity between these two domains. Moreover in the case that q = r = 2 we express the Bergman kernel in terms of the Jacobi polynomials. The explicit formulas of the Bergman kernel function for these domains enables us to investigate whether the Bergman kernel has zeros or not. This kind of problem is called a Lu Qi-Keng problem.     

On the stability of rational approximations to the cosine with only imaginary poles

Letr(z) be a rational approximation to cosz with only imaginary poles ±i ₁ ^–1/2 , ±i ₂ ^–1/2 , ..., ±i _m ^–1/2 such that |cozz –r(z)| C|z|^2m+2 as |z| 0. If the degree of the numerator ofr(z) is less than or equal to 2m and _i m/4,i=1, ...,m, then we show that |r(z)|1 for all realz.     

On the Asymptotic Behavior of Faber Polynomials for Domains With Piecewise Analytic Boundary

Let φ(z) be an analytic function on a punctured neighborhood of ∞, where it has a simple pole. The nth Faber polynomial F _n (z) (n=0,1,2,…) associated with φ is the polynomial part of the Laurent expansion at ∞ of [φ(z)] ⁿ . Assuming that ψ (the inverse of φ) conformally maps |w|>1 onto a domain Ω bounded by a piecewise analytic curve without cusps pointing out of Ω, and under an additional assumption concerning the “Lehman expansion” of ψ about those points of |w|=1 mapped onto corners of ∂ Ω, we obtain asymptotic formulas for F _n that yield fine results on the limiting distribution of the zeros of Faber polynomials.      

Convergence of Sums of Appell Polynomials with Infinite Variance

We investigate the convergence of distributions of partial sums of Appell polynomials of a long-memory moving average process X _t with i.i.d. innovations _s in the case where the variance , and the distribution of #x03BE; ₀ ^m belongs to the domain of attraction of an -stable law with 1<< 2. We prove that the limit distribution of partial sums of Appell polynomials is either an -stable Lévy process, or an mth order Hermite process, or the sum of two mutually independent processes depending on the values of , m, and d, where 0X _t.     

Unconditional constants and polynomial inequalities

If P is a polynomial on R^m of degree at most n, given by , and P_n(R^m) is the space of such polynomials, then we define the polynomial |P| by . Now if is a convex set, we define the norm on P_n(R^m), and then we investigate the inequality providing sharp estimates on for some specific spaces of polynomials. These ’s happen to be the unconditional constants of the canonical bases of the considered spaces.     

Asymptotic Behavior of Sobolev-Type Orthogonal Polynomials on a Rectifiable Jordan Curve or Arc

Our object of study is the asymptotic behavior of the sequence of polynomials orthogonal with respect to the discrete Sobolev inner product <formula> \langle f, g \rangle = ∈t_{E} f(ξ) \overline{g(ξ)} ρ(ξ) |d ξ|+ f(Z) A g(Z)^H, </formula> where E is a rectifiable Jordan curve or arc in the complex plane f(Z) = (f(z_1), \ldots, f^{(l_1)}(z_1) , \ldots , f(z_m) , \ldots ,f^{(l_m)}(z_m)), A is an M \times M Hermitian matrix, M l ₁ + ⋅s + l _m + m , |d ξ| denotes the arc length measure, ρ is a nonnegative function on E , and z _i ∈Ω, i=1,2,\ldots,m , where Ω is the exterior region to E . July 23, 1999. Dates revised: September 11, 2000 and February 16, 2001. Date accepted: February 26, 2001.     

Continuous and Discrete Tight Frames of Orthogonal Polynomials for a Radially Symmetric Weight

This paper considers tight frame decompositions of the Hilbert space ℘ _n of orthogonal polynomials of degree n for a radially symmetric weight on ℝ ^d , e.g., the multivariate Gegenbauer and Hermite polynomials. We explicitly construct a single zonal polynomial p∈℘ _n with the property that each f∈℘ _n can be reconstructed as a sum of its projections onto the orbit of p under SO(d) (symmetries of the weight), and hence of its projections onto the zonal polynomials p _ξ obtained from p by moving its pole to ξ∈S:={ξ∈ℝ ^d :|ξ|=1}. Furthermore, discrete versions of these integral decompositions also hold where SO(d) is replaced by a suitable finite subgroup, and S by a suitable finite subset. One consequence of our decomposition is a simple closed form for the reproducing kernel for ℘ _n .      

An extremal property of Hermite polynomials

Let H_n be the nth Hermite polynomial, i.e., the nth orthogonal on polynomial with respect to the weight w(x)=exp(−x²). We prove the following: If f is an arbitrary polynomial of degree at most n, such that |f||H_n| at the zeros of H_n+1, then for k=1,…,n we have f^(k)H_n^(k), where · is the norm. This result can be viewed as an inequality of the Duffin and Schaeffer type. As corollaries, we obtain a Markov-type inequality in the norm, and estimates for the expansion coefficients in the basis of Hermite polynomials.     

On weak convergence of functionals of stochastic processes with continuously differentiable paths

We obtain a criterion for weak convergence of a sequence of stochastic processes _n(t), t [0, 1],n N, _n(t) R ^m in the spaceC _m ^k [0, 1] of continuously differentiable functions. We consider several examples of weakly convergent sequences of stochastic processes inC _m ^k [0, 1] and several integer functionals defined on these random variables.Translated fromTeoriya Sluchainykh Protsessov, Vol. 15, pp. 85–90, 1987.     

Asymptotic Behavior of Sobolev-Type Orthogonal Polynomials on the Unit Circle

We study the asymptotic behavior of the sequence of polynomials orthogonal with respect to the discrete Sobolev inner product on the unit circle

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where f(Z)=(f(z₁), …, f^(l₁)(z₁), …, f(z_m), …, f^(l_m)(z_m)), A is a M×M positive definite matrix or a positive semidefinite diagonal block matrix, M=l₁+…+l_m+m, dμ belongs to a certain class of measures, and |z_i|>1, i=1, 2, …, m.     

Non-Uniformly Elliptic Equations with Natural Growth in the Gradient

We prove the existence of bounded solutions for a class of nonlinear elliptic problems of type–div(a(x,u,Du))=H(x,u,Du)+f, uW ^1,p ₀()L (),where a(x,,)b(||)|| ^p , b is a continuous monotone decreasing function and |H(x,,)| k()|| ^p , k is a continuous monotone increasing function.     

Limit distribution function of inhomogeneities in regions with random boundary. I

A study is made of the interaction of systems of charged particles with a membrane consisting of inhomogeneities randomly distributed in accordance with the same law in the neighborhoods of corresponding sites of a planar crystal lattice. A system of equations for the self-consistent potentialU ₁(x, ⁰,..., ^N ,...) and density of surface changes (x, ⁰,..., ^N ,...) is derived and solved.Institute of Nuclear Physics, Uzbek Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 92, No. 1, pp. 98–112, July, 1992.     

Asymptotic behaviour of Verblunsky coefficients

Let , ζh≠ζk, h≠k and |ζj|=1, j=1,…,m, and consider the polynomials orthogonal with respect to , , where μ is a finite positive Borel measure on the unit circle with infinite points in its support, such that the reciprocal of its Szeg? function has an analytic extension beyond |z|<1. In this paper we deduce the asymptotic behaviour of their Verblunsky coefficients. By means of this result, an asymptotic representation for these polynomials inside the unit circle is also obtained.     

Spline polynomials with a prescribed sequence of extrema

In the present note a theorem about strong suitability of the space of algebraic polynomials of degree n in C[a,b] (Theorem A in [1]) is generalized to the space of spline polynomials _{[a, b} ^{]n, k} (n2, 0) in C[a, b]. Namely, it is shown that the following theorem is valid: for arbitrary numbers ₀, ₁, ..., _n+k, satisfying the conditions (_i–_i–1) (_{i+1{ i}< 0(i=1, ..., n +k–1), there is a unique polynomials _n,k (t) _{[a, b} ^]/n,k and pointsa=₀,<₁<...< _n+k– 1< _n+k = b (₁1 <_n, ..., _kk<_n+k–1), such that s_n,k(_i) = _i(i=0, ..., n + k), s_n,k(_i)=0 (i=1, ..., n + k–1).Translated from Matematicheskii Zametki, Vol. 11, No. 3, pp. 251–258, March, 1972.     

Estimate of a random field based on the trajectory of a point moving in it

We consider the stochastic differential equationd _t =( _t )dt+ _t ( _t )dw _t in Euclidean space, where (x) is a Gaussian random field andw _t is a standard Wiener process. Let f _t ={ _s ,st}. Equations are obtained for the conditional meansm _t (x)=f _t } andB _t (x, y)=M{(x)(y)|f _t }.Translated fromTeariya Sluchaínykh Protsessov, Vol. 14, pp. 7–9, 1986.