Formulas for Lagranigian and orthogonal degeneracy loci; \widetilde Q-polynomial approach

The main goal of the paper is to give explicit formulas for the fundamental classes of Schubert subschemes in Lagrangian and orthogonal Grassmannians of maximal isotropic subbundles as well as some globalizations of them. The used geometric tools overlap appropriate desingularizations of such Schubert subschemes and Gysin maps for such Grassmannian bundles. The main algebraic tools are provided by the families of and -polynomials introduced and investigated in the present paper. The key technical result of the paper is the computation of the class of the (relative) diagonal in isotropic Grassmannian bundles based on the orthogonality property of and polynomials. Some relationships with quaternionic Schubert varieties and Schubert polynomials for classical groups are also discussed.     

Completeness in analytic spaces of subsequences of Laguerre and Jacobi polynomials

In the space of all single-valued functions analytic in the circle ¦z¦相似文献   

Criterion for the Denseness of Algebraic Polynomials in the Spaces $$L_p \left( {{\mathbb{R}},d {\mu }} \right)$$ , 1 ≤ p < ∞

The criterion for the denseness of polynomials in the space established by Hamburger in 1921 is extended to the spaces , 1 p < .     

On the order and type of basic and composite sets of polynomials in complete Reinhardt domains

In this paper we will define the order and type of basic and composite sets of polynomials of several complex variables in complete Reinhardt domains. Also, the property of basic and composite sets of polynomials of several complex variables in complete Reinhardt domains is discussed.     

Interpolation by Rational Functions with Nodes on the Unit Circle

From the Erds–Turán theorem, it is known that if f is a continuous function on and L _n (f, z) denotes the unique Laurent polynomial interpolating f at the (2 n + 1)th roots of unity, then Several years later, Walsh and Sharma produced similar result but taking into consideration a function analytic in and continuous on and making use of algebraic interpolating polynomials in the roots of unity.In this paper, the above results will be generalized in two directions. On the one hand, more general rational functions than polynomials or Laurent polynomials will be used as interpolants and, on the other hand, the interpolation points will be zeros of certain para-orthogonal functions with respect to a given measure on .     

Polynomial Interpolation of the Discrete Logarithm

Recently, Coppersmith and Shparlinski proved several results on the interpolation of the discrete logarithm in the finite prime field by polynomials modulo p and modulo p-1, respectively. In this paper most of these results are extended to arbitrary .     

Some Classical Polynomials Seen from Another Side

The sequence of orthogonal polynomials is said to be classical if is also orthogonal. The aim of this paper is to find the sequences which have the property that is also orthogonal. We prove that sequences, with this property have to be, classical and belong either to the set of Laguerre or Jacobi polynomials, where in the Laguerre case c has to be zero and in the Jacobi case c = ±1.     

Best “Continuous” n-Term Approximations in the Spaces $$S_\phi ^p $$

We find exact values of upper bounds for the best approximations of q-ellipsoids by polynomials of degree n in the spaces in the case where the approximating polynomials are constructed on the basis of n-dimensional subsystems chosen successively from a given orthonormal system .     

Classes of (ψ, β)-differentiable functions of a complex variable and approximation by linear averages of their Faber series

We introduce the notion of (, )-derivative of a function of one complex variable, and define on the basis of this the classes of (, )-differentiable analytic functions in a bounded domain G. The classes consist of the Cauchy-type integrals whose densities f() are such that the induced functions on the unit circle are periodic functions of classes . We consider approximation of functions by algebraic polynomials constructed from their series expansions in Faber polynomials.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 11, pp. 1556–1570, November, 1992.     

Approximation Characteristics of Spaces $$S_\phi ^p$$

We introduce linear vector spaces and study their approximation properties. By using the general results obtained, we establish statements concerning the approximation of classes of periodic functions of many variables by trigonometric polynomials.     

On Polynomial Bases for the Space C[-1,1]

For an arbitrary , we construct an orthogonal basis in the space consisting of algebraic polynomials such that their degrees do not exceed . It is not possible to decrease the order of growth of the degrees. Bibliography: 9 titles.     

On a Characterization of Spaces of Differentiable Functions

In this paper, we generalize Bernstein's theorem characterizing the space by means of local approximations. The closed interval is partitioned into disjoint half-intervals on which best approximation polynomials of degree divided by the lengths of these half-intervals taken to the power are considered. The existence of the limits of these ratios as the lengths of the half-intervals tend to zero is a criterion for the existence of the th derivative of a function. We prove the theorem in a stronger form and extend it to the spaces .     

Matching Polynomials And Duality

Let G be a simple graph on n vertices. An r-matching in G is a set of r independent edges. The number of r-matchings in G will be denoted by p(G, r). We set p(G, 0) = 1 and define the matching polynomial of G by and the signless matching polynomial of G by .It is classical that the matching polynomials of a graph G determine the matching polynomials of its complement . We make this statement more explicit by proving new duality theorems by the generating function method for set functions. In particular, we show that the matching functions and are, up to a sign, real Fourier transforms of each other.Moreover, we generalize Foatas combinatorial proof of the Mehler formula for Hermite polynomials to matching polynomials. This provides a new short proof of the classical fact that all zeros of µ(G, x) are real. The same statement is also proved for a common generalization of the matching polynomial and the rook polynomial.     

On Polynomials over a Finite Field of Even Characteristic with Maximum Absolute Value of the Trigonometric Sum

We study trigonometric sums in finite fields . The Weil estimate of such sums is well known: , where f is a polynomial with coefficients from F(Q). We construct two classes of polynomials f, , for which attains the largest possible value and, in particular, .     

Approximation by Simplest Fractions

In this paper, a number of problems concerning the uniform approximation of complex-valued continuous functions on compact subsets of the complex plane by simplest fractions of the form are considered. In particular, it is shown that the best approximation of a function by the fractions is of the same order of vanishing as the best approximations by polynomials of degree .     

A set of orthogonal polynomials induced by a given orthogonal polynomial

Summary Given an integern 1, and the orthogonal polynomials _n (·; d) of degreen relative to some positive measured, the polynomial system induced by _n is the system of orthogonal polynomials corresponding to the modified measure . Our interest here is in the problem of determining the coefficients in the three-term recurrence relation for the polynomials from the recursion coefficients of the orthogonal polynomials belonging to the measured. A stable computational algorithm is proposed, which uses a sequence ofQR steps with shifts. For all four Chebyshev measuresd, the desired coefficients can be obtained analytically in closed form. For Chebyshev measures of the first two kinds this was shown by Al-Salam, Allaway and Askey, who used sieved orthogonal polynomials, and by Van Assche and Magnus via polynomial transformations. Here, analogous results are obtained by elementary methods for Chebyshev measures of the third and fourth kinds. (The same methods are also applicable to the other two Chebyshev measures.) Interlacing properties involving the zeros of _n and those of are studied for Gegenbauer measures, as well as the orthogonality—or lack thereof—of the polynomial sequence .Work supported in part by the National Science Foundation under grant DMS-9023403.Dedicated to the memory of Alexander M. Ostrowski on the occasion of the 100th anniversary of his birth.     

On sequences of Dirichlet polynomials uniformly bounded on half planes

Given and a sequence of Dirichlet polynomials estimates for the coefficientsa _n are proved if {_n} is uniformly bounded on a region containing a half plane. Thereby a result is obtained which is an analogue of a known result for polynomials, that is for theA-transforms of the geometric sequence; moreover a Jentzsch type theorem for {_n(z)} is derived.     

Uniform Convergence of the Generalized Bieberbach Polynomials in Regions with Zero Angles

Let C be the extended complex plane; G C a finite Jordan with 0 G; w= (z) the conformal mapping of G onto the disk normalized by . Let us set , and let be the generalized Bieberbach polynomial of degree n for the pair (G,0), which minimizes the integral in the class of all polynomials of degree not exceeding n with . In this paper we study the uniform convergence of the generalized Bieberbach polynomials with interior and exterior zero angles and determine its dependence on the properties of boundary arcs and the degree of their tangency.     

Criterion of Polynomial Denseness and General Form of a Linear Continuous Functional on the Space C w 0

For an arbitrary function , we determine the general form of a linear continuous functional on the space C _w ⁰ . The criterion for denseness of polynomials in the space established by Hamburger in 1921 is extended to the spaces C _w ⁰ .