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.     

Functional properties of rearrangement invariant spaces defined in terms of oscillations

Function spaces whose definition involves the quantity f^**-f^*, which measures the oscillation of f^*, have recently attracted plenty of interest and proved to have many applications in various, quite diverse fields. Primary role is played by the spaces S_p(w), with 0<p<∞ and w a weight function on (0,∞), defined as the set of Lebesgue-measurable functions on R such that f^*(∞)=0 and

     

Boundedness of the Riesz Potential in Local Morrey-Type Spaces

The problem of boundedness of the Riesz potential in local Morrey-type spaces is reduced to the problem of boundedness of the Hardy operator in weighted L _p -spaces on the cone of non-negative non-increasing functions. This allows obtaining sharp sufficient conditions for boundedness for all admissible values of the parameters, which, for a certain range of the parameters wider than known before, coincide with the necessary ones.     

Noncoincidence of geodesic lengths and hearing elliptic quantum billiards

Assume that the planar region has aC ¹ boundary and is strictly convex in the sense that the tangent angle determines a point on the boundary. The lengths of invariant circles for the billiard ball map (or caustics) accumulate on ||. It follows from direct calculations and from relations between the lengths of invariant circles and the lengths of trajectories of the billiard ball map that under mild assumptions on the lengths of some geodesics the region satisfies the strong noncoincidence condition. This condition plays a role in recovering the lengths of closed geodesics from the spectrum of the Laplacian. Asymptotics for the lengths of invariant circles and an application to ellipses are discussed. In addition; some examples regarding strong non coincidence are given.     

On bilinear Hardy inequality and corresponding geometric mean inequality

Ricerche di Matematica - The main aim of this paper to provide several scales of equivalent conditions for the bilinear Hardy inequalities in the case $$1&lt; q, p_1, p_2&lt;infty $$ with...     

Cobalt(II) complexes with hydantoin

Complexes of Co(II) with hydantoin (L, C₃H₄N₂O₂) have been synthesized. The complexes had the following compositions: [CoL₂(OH₂)₂](NO₃)₂ · 2H₂O, [CoL₂(OH₂)₂]Cl₂ · 3H₂O, and [CoL₂(OH₂)₂]SO₄ · 2H₂O. The individual character of the synthesized compounds are proved by the study of the IR absorption spectra (400–4000 cm^?1) of all the compounds and the initial ligand, as well as the X-ray diffraction patterns, thermograms, and thermogravigrams of the synthesized compounds. The coordination modes of the ligand and acido groups are revealed. The properties of the synthesized compounds are characterized.     

Construction of Function Spaces Close to $$L^\infty $$ with Associate Space Close to $$L^1$$L1

The paper introduces a variable exponent space X which has in common with \(L^{\infty }([0,1])\) the property that the space C([0, 1]) of continuous functions on [0, 1] is a closed linear subspace in it. The associate space of X contains both the Kolmogorov and the Marcinkiewicz examples of functions in \(L^{1}\) with a.e. divergent Fourier series.     

Convergence Rates of Padé and Padé-Type Approximants

A comparison is made between Padé and Padé-type approximants. LetQ_nbe thenth orthonormal polynomial with respect to a positive measureμwith compact support inC. We show that for functions of the form[formula]wherewis an analytic function on the support ofμ, Padé-type approximants with denominatorQ_ngive a successful and, in general, better approximation procedure than Padé approximation.     

Dual spaces of local Morrey-type spaces

In this paper we show that associated spaces and dual spaces of the local Morrey-type spaces are so called complementary local Morrey-type spaces. Our method is based on an application of multidimensional reverse Hardy inequalities.