On the quotient ring by diagonal invariants

For a finite Coxeter group, W, and its reflection representation , we find the character and Hilbert series for a quotient ring of [^*] by an ideal containing the W–invariant polynomials without constant term. This confirms conjectures of Haiman.     

Lp-convergence for expansions in terms of the eigenfunctions of a Sturm — Liouville problem

For the operator L_v=–(x^2ay). x [0, 1], y(0)=y(1)=0 with 0 < 1/2, or ¦y¦ < , y(1)=0 with 1/2 <1, we investigate the effect which the singularity of the Sturm-Liouville operator derived from this self-adjoint expression has on L_p-convergence of expansions in terms of the eigenfunctions of this operator. We will prove that the orthonormalized system of eigenfunctions forms a basis in L_p [0, 1] for 2/(2–) < p < 2/.Translated from Matematicheskii Zametki, Vol. 3, No. 6, pp. 683–692, June, 1968.The author is grateful to V. M. Tikhomirov for his many valuable remarks and his constant attention to this work.     

On term by term dyadic differentiability of Walsh series

. . . . : {ja _j },j=1,2,... — , f(x) , , f ^[1](x) — f .     

Symmetrization of functions in Sobolev spaces and the isoperimetric inequality

A positive measurable function f on R^d can be symmetrized to a function f^* depending only on the distance r, and with the same distribution function as f. If the distribution derivatives of f are Radon measures then we have the inequality f^*f, where f is the total mass of the gradient. This inequality is a generalisation of the classical isoperimetric inequality for sets. Furthermore, and this is important for applications, if f belongs to the Sobolev space H¹,P then f^* belongs to H¹,P and f^*_pf_p.     

Functions with a finite dirichlet integral in a domain with cusp points on the boundary

Let be a domain in ⁿ, n >2, the boundary of which has a cusp point, pointing inside or outside the domain. The purpose of the paper is to characterize the traces on of the elements of the space H¹() of functions with a finite Dirichlet integral. As a consequence one establishes the existence of a linear continuous extension operator H¹ () H¹(ⁿ) under the presence of an interior cusp point on . Theorems on domains with cusps are proved with the aid of results on cylindrical domains. In the space of functions with a finite Dirichlet integral in the exterior or the interior of the cylinder one introduces the norm, depending on a small parameter and generating a norm of the trace on as an element of the quotient space. The latter is placed in correspondence with an explicitly described norm of functions on the boundary, uniformly equivalent relative to . One constructs an operator of extension of functions from the exterior of the cylinder to Rⁿ, preserving H¹, whose norm is uniformly bounded relative to . For the optimal operator of extension from the inside of the cylinder one finds the asymptotic behavior of the norm as 0. From these results there follow similar theorems on functions with a finite Dirichlet integral inside and outside a thin closed tube (of width ).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 126, pp. 117–137, 1983.     

Number of quasimodes of “bouncing-ball” type

A new two-scale expansion is proposed for eigenfunctions of bouncing-ball type and the corresponding eigenvalues of the Laplace operator with the Dirichlet condition in a domain of the plane. The eigenfunctions are concentrated in a neighborhood of a stable diameter of the domain and are numbered with two indices (p, q) where p is the number of longitudinal nodes and q the number of nodes in a direction orthogonal to the diameter. The validity of the asymptotic expansions is guaranteed for 0 q const p^–1 and >0 as p +.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 117, pp. 172–182, 1981.     

Bergman-Vekua theory in differential form

We develop Bergman-Vekua integral operator theory on the basis of a new kind of series suggested by polynomial operators (classP _j operators); in a sense, this is an analog of the Weierstrass approach as opposed to the Cauchy-Riemann approach in classical complex analysis, which also has advantages over the original form in boundary value problems and other applications.Dedicated to the memory of my friend, Peter Henrici     

Sets of uniqueness and closed subgroups in Vilenkin groups

, ( ) . , : , , .

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This research was partially supported by National Science Foundation under grant INT-8400708.     

On the summability of Fourier series with the method of lacunary arithmetic means

. f- ,S _n (f)— . {n _k }, n _k+1/n _k >1+ck ^– ,— , 0<1/2, f ₀, .     

On the Riemann derivatives ofC sP-integrable functions

[4] , .     

Dilation-analytic wave operators for three particles

Let H(0) be a dilation-analytic three-particle Schrödinger operator with analytic continuation H() (>0). Let a be zero or the energy of a two-particle bound state. Let- (a) be the Laplace operator representing the kinetic energy of the relative motion of fragments scattered in channel a. By recent results, wave operators W (±, a, ) with conjugates W (±, a, ) exist such that W (±, a, ) W (±, a, ) is a projection P (a, ) commuting with H () while [H ()-a]W (±, a, ) equals-W(±, a, ) (a) e²ⁱ. This paper shows that the wave operators transform dilation-analytic functions of particle coordinates into dilation-analytic functions. Specifically, if the left shoulder of the spectrum of P (a,) H () does not sweep across eigenvalues of H() when , then W(-, a, ) and W (+, a, ) are dilation analytic in [, ]. If the right shoulder does not sweep across eigenvalues, W(+, a, ) and W(-, a, ) are dilation analytic in [,]. A semisimple eigenvalue of H () embedded in the spectrum of P (a, ) H () does not prevent the wave operators from being dilation analytic in an interval [, ] with as an interior point.This work was supported in part by the National Science Foundation under grant DMS-8301096.     

Sharpening of Stečkin's theorem to strong approximation with exponentp

— , B _n , B p1., , p=1 , - . , , , p.     

On Gibbs' phenomenon for Riesz spherical means of multiple Fourier integrals and Fourier series

- - , .     

The conditions of selfadjointness of the operator of singular integration

It is proved that the operator of singular integration along a bounded rectifiable curve is self-adjoint in the weighted spaceL ₂(, ) if and only if is a circle and (t)=const.     

Lipschitz and darbo conditions for the superposition operator in ideal spaces

Summary In this paper we give necessary and sufficient conditions for the superposition operator Fx(s)=f(s, x(s)) to satisfy a Lipschitz condition Fx₁ - Fx₂kx₁ - x₂ or a Darbo condition (FN)k(N) in ideal spaces of measurable functions, where is the Hausdorff measure of noncompactness. Moreover, we characterize a large class of spaces in which the above mentioned two conditions are equivalent.

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Sunto In questo lavoro diamo delle condizioni necessarie e sufficienti perchè l'operatore di sovrapposizione Fx(s)=f (s, x(s)) soddisfi alla condizione di Lipschitz Fx₁–Fx₂ kx₁–x₂ o quella di Darbo (FN)k(N) in spazi ideali di funzioni misurabili, ove è la misura di non compattezza di Hausdorff. Inoltre, caratterizziamo un'ampia classe di spazi in cui le suddette due condizioni sono equivalenti.

     

The Cantor-Lebesgue and Denjoy-Luzin properties for double systems of functions

{ _mn ():, =1, 2, ...}, (X, , ). , ( ) , . { _mn }. . — — ( ) .     

Transferring estimates for zonal convolution operators

_p - , . Ad- .     

Notes on Interpolation. X (Two Boundary Value Problems)

In this paper we use (0, 2) interpolational polynomials to give an approximate solution of the differential equation y(x) + A(x)y(x) = F(x), x I := [-1, 1] j in case when the boundary values are y(-1) = and y(1) = , , R.     

Жизнь и деятельность яноша Бояи

: . BOLYAI.     

Investigation of Haar and Franklin series in Hardy spaces

(L ¹,H) (, ) , ; H — . , , L ¹ . [13] , . , , , .