Upper Bounds of the Lebesgue Constants for Fourier–Jacobi Series Summation Methods Defined by a Multiplier Function

Let be the Jacobi polynomials and let C[a,b] be the space of continuous functions on [a,b] with the uniform norm. In this paper, we study sequences of Lebesgue constants, i.e., of the norms of linear operators generated by a multiplier matrix defined by the following relations:

On Invariance of Some Classes of Holomorphic Functions Under Integrodifferential Operators

The following classes of functions analytic in the unit disk are considered:

Wilson's functional equations on groups

We study properties of solutionsf, g, h C(G) of the functional equation

Sharp Estimates for Deviations of Linear Approximation Methods for Periodic Functions by Linear Combinations of Moduli of Continuity of Different Order

Estimates for deviations are established for a large class of linear methods of approximation of periodic functions by linear combinations of moduli of continuity of different orders. These estimates are sharp in the sense of constants in the uniform and integral metrics. In particular, the following assertion concerning approximation by splines is proved: Suppose that is odd, . Then

Approximating periodic functions by interpolation sums of Jackson type

Let

Interlineation of functions of 2 variables on M (M≥2) lines with the highest algebraic precision

A general algorithm is proposed for constructing interlineation operators , x=(x₁, x₂) with the properties

Commutativity of rings through a Streb's result

In this paper we investigate commutativity of rings with unity satisfying any one of the properties:

Sharp Constant in a Jackson-Type Inequality for Approximation by Positive Linear Operators

In what follows, C is the space of -periodic continuous real-valued functions with uniform norm, is the first continuity modulus of a function with step h, H _n is the set of trigonometric polynomials of order at most n, is the set of linear positive operators (i.e., of operators such that for every ), is the space of square-integrable functions on ,

Methods of oscillation theory of half-linear second order differential equations

In this paper we investigate oscillatory properties of the second order half-linear equation

The Limit of Lebesgue Constants for Summation Methods of the Fourier―Legendre Series Defined by a Multiplier Function

Let C[-1,1] be the space of continuous functions f:[-1,1] with the uniform norm, let _Pk be the Legendre polynomials such that _Pk (1)=1, and let _J0 be the Bessel function of zero index. We consider sequences of linear operators (summation methods) _Un:C [-1,1] C[-1,1] defined by a multiplier function as follows:

Localization Operators and Exponential Weights for Modulation Spaces

We study localization operators within the framework of ultradistributions. More precisely, given a symbol a and two windows φ₁, φ₂, we investigate the multilinear mapping from to the localization operator Results are formulated in terms of modulation spaces with weights which may have exponential growth. We give sufficient and necessary conditions for a to be bounded or to belong to a Schatten class. As an application, we study symbols defined by ultra-distributions with compact support, that give trace class localization operators.     

The Law of the Iterated Logarithm for Functionals of Harris Recurrent Markov Chains: Self Normalization

Let {X _n } _n0 be a Harris recurrent Markov chain with state space E, transition probability P(x, A) and invariant measure , and let f be a real measurable function on E. We prove that with probability one,

On a theorem of existence for scaling problems

We study the question of the existence of the global minimum of the functional

L ∞-solutions for some nonlinear degenerate elliptic and parabolic equations

In this paper we prove the existence of bounded solutions for equations whose prototype is:

On realizations related to Weyl operators

The paper deals with the minimal and the maximal realizations (L ^w )^~ and (L ^w ):L ₂L ₂ of linear operators of Weyl type

Best polynomial approximation in Sobolev-Laguerre and Sobolev-Legendre spaces

We investigate limiting behavior as γ tends to ∞ of the best polynomial approximations in the Sobolev-Laguerre space W^N,2([0, ∞); e^−x) and the Sobolev-Legendre space W^N,2([−1, 1]) with respect to the Sobolev-Laguerre inner product

The Order of Best Approximation in Some Classes of Functions

Starting from the equivalence between the Ditzian–Totik modulus and , where , in this article large classes of functions are introduced for which the modulus can be easily calculated. As a consequence, very good estimates for the bestapproximation are obtained. The attempts to estimate or calculate themodulus can be a very intricateproblem.     

Uniform Approximation of Nonperiodic Functions Defined on the Entire Axis

Using the following notation: C is the space of continuous bounded functions f equipped with the norm , V is the set of functions f such that , the set E consists of fCV and possesses the following property:

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.     

Kolmogorov-type inequalities for mixed derivatives of functions of many variables

Let = (₁,...,_d) be a vector with positive components and let D be the corresponding mixed derivative (of order _j with respect to the jth variable). In the case where d > 1 and 0 < k < r are arbitrary, we prove that