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.     

Über die Existenz gemeinsamer Elemente bester Approximation bezüglich zweier Normen II

The present paper deals with the possibility of existence of best approximation elements, simultaneously with respect to two norms ·_i,i=1,2, for all the elements of a class of subspaces. In case this class in any of the following: (a) All n-dimensional subspaces, (b) All ·₁-or ·||₂-closed, n-codimensional subspaces, (c) All ·₁-or ·₂-closed subspaces with infinite dimension and codimension, we prove that the two norms differ at most by a constant factor.     

Joint norm of operators and an investigation of nonlinear problems

Nonlinear operator equations of the form x=Fx in a real-valued Hilbert space H are studied. If the operator F is completely continuous and admits the bound Fx< Bx+b, where B is a continuous linear operator then for B<1 the Schauder principle is applicable to the equation x=Fx and this equation possesses at least one solution x H. If the bound Fx<,B₁x+B₂x+b is valid where B₁ and B₂ are bounded linear operators then the simplest conditions for solvability of the equation x=Fx is of the form B₁+B₂<1. This condition could be relaxed. The proposed method is applied to the investigation of a two-point boundary problem (cf., e.g., [1–3]). New conditions for the existence of solutions are obtained.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 12, pp. 1605–1616, December, 1990.     

The Spectral Theory of Amenable Actions and Invariants of Discrete Groups

Let G denote a semisimple group, a discrete subgroup, B=G/P the Poisson boundary. Regarding invariants of discrete subgroups we prove, in particular, the following:(1) For any -quasi-invariant measure on B, and any probablity measure on , the norm of the operator () on L ²(B,) is equal to (), where is the unitary representation in L ²(X,), and is the regular representation of .(2) In particular this estimate holds when is Lebesgue measure on B, a Patterson–Sullivan measure, or a -stationary measure, and implies explicit lower bounds for the displacement and Margulis number of (w.r.t. a finite generating set), the dimension of the conformal density, the -entropy of the measure, and Lyapunov exponents of .(3) In particular, when G=PSL₂() and is free, the new lower bound of the displacement is somewhat smaller than the Culler–Shalen bound (which requires an additional assumption) and is greater than the standard ball-packing bound.We also prove that ()=_G() for any amenable action of G and L ¹(G), and conversely, give a spectral criterion for amenability of an action of G under certain natural dynamical conditions. In addition, we establish a uniform lower bound for the -entropy of any measure quasi-invariant under the action of a group with property T, and use this fact to construct an interesting class of actions of such groups, related to 'virtual' maximal parabolic subgroups. Most of the results hold in fact in greater generality, and apply for instance when G is any semi-simple algebraic group, or when is any word-hyperbolic group, acting on their Poisson boundary, for example.     

Problem of instability in the first approximation

Let E be a Banach space, A be a continuous linear operator such that (A) ; Re>0 Ø, and F(t, x) be a continuous function on [0, )×E satisfying the condition F(t, x) q x (q= const). An example of a system dx/dt=Ax + F(t, x) is given which has an exponentially stable zero solution for certain F(t, x) with arbitrarily small q.Translated from Matematicheskie Zametki, Vol. 23, No. 5, pp. 721–723, May, 1978.     

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.

     

Замечания об ихтерполировании

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Stationary Gaussian processes with a finite correlation function

If the correlation function vanishes outside the segment [–R, R], then an upper estimate (uniform with respect to all such processes) is possible for the probability of the fact that on an other segment [–r, r] the process remains between – and . Such an estimate is obtained, decreasing for 0 asexp(–f(r/R ln ²+ ) and, moreover,r/R may be either 0 or +. The proof is based on an estimate of the form P_mQ_n c_mn P_m Q_n for norms of polynomials on a circle in the complex plane.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 184, pp. 279–288, 1990.     

L 2-approximation by the translates of a function and related attenuation factors

Summary Let be thek-dimensional subspace spanned by the translates (·–2j/k),j=0, 1, ...,k–1, of a continuous, piecewise smooth, complexvalued, 2-periodic function . For a given functionfL ²(–, ), its least squares approximantS _kf from can be expressed in terms of an orthonormal basis. Iff is continuous,S _kf can be computed via its discrete analogue by fast Fourier transform. The discrete least squares approximant is used to approximate Fourier coefficients, and this complements the works of Gautschi on attenuation factors. Examples of include the space of trigonometric polynomials where is the de la Valleé Poussin kernel, algebraic polynomial splines where is the periodic B-spline, and trigonometric polynomial splines where is the trigonometric B-spline.     

Ergodic properties of semi-Markovian operators on the Z1-part

Summary In this note we consider a semi-Markovian operator, that is a positive linear mapping T: L ¹ L ¹ such that sup T ⁿ <. We study the behavior of T ⁿ on the Z ¹-part of the space (the disappearing part in Sucheston's terminology). We show in particular, that if the operator T has a non-trivial conservative part in Z ¹, then the ratio theorem must fail.Research supported by the U.S.Army Research Office (Durham) under contract DA-31-124-ARO(D)-288.     

Isometries of symmetric operator spaces associated with AFD factors of typeII and symmetric vector-valued spaces

If is a surjective isometry of the separable symmetric operator spaceE(M, ) associated with the approximately finite-dimensional semifinite factorM and if · _E(M,) is not proportional to · _{L 2}, then there exist a unitary operatorUM and a Jordan automorphismJ ofM such that(x)=UJ(x) for allxME(M, ). We characterize also surjective isometries of vector-valued symmetric spacesF((0, 1), E(M, )).Research supported by the Australian Research Council     

Theorem on a convergence condition in the spaces

For a given -function (u), a condition on a -function (u) is found such that it is necessary and sufficient for the following to hold: if f_n(x) f(x) and f _n (x)M (n=1, 2, ...) where M>0 is an absolute constant, then f _n (x)–f(x)0(n). An analogous condition for convergence in Orlicz spaces is obtained as a corollary.Translated from Matematicheskie Zametki, Vol. 21, No. 5, pp. 615–626, May, 1977.The author thanks V. A. Skvortsov for his constant attention and guidance on this paper.     

О лагранжевом интерполировании с узлами в корнях многочленов сонина-маркова

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Some Landau's type inequalities for infinitesimal generators

Summary Lett T(t) be a strongly continuous contraction semigroup on a complex Banach space and letA be its infinitesimal generator. We prove that, forx D(A ³), the following inequalities hold true: Ax³ 243/8 x²A ³ x, A ² x 24 xA ³ x². Ift T(t) is a contraction group (resp. cosine function) we get the analogous but better inequalities with constants 9/8 and 3 (resp. 81/40 and 72/25) instead of 243/8 and 24. We consider also uniformly bounded semigroups, groups and cosine functions.     

Some extensions of the Hewitt-Savage zero-one law

Summary Let denote the class of infinite product probability measures = ₁× ₂× defined on an infinite product of replications of a given measurable space (X, A), and let denote the subset of for which (A) =0 or 1 for each permutation invariant event A. Previous works by Hewitt and Savage, Horn and Schach, Blum and Pathak, and Sendler (referenced in the paper) discuss very restrictive sufficient conditions under which a given member , of belongs to . In the present paper, the class is shown to possess several closure properties. E.g., if and _{0 n} for some n 1, then ₀× ₁× ₂×.... While the current results do not permit a complete characterization of they demonstrate conclusively that is a much larger subset of than previous results indicated. The interesting special case X={0,1} is discussed in detail.Research supported by the National Science Foundation under grant No. MCS75-07556     

Convergence of simultaneous Padé approximants to two dilogarithms

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Best approximation and best unilateral approximation of the kernel of a biharmonic equation and optimal renewal of the values of operators

For the classB _p , 0 < 1, 1p , of 2-periodic functions of the form f(t)=u(,t), whereu (,t) is a biharmonic function in the unit disk, we obtain the exact values of the best approximation and best unilateral approximation of the kernel K(t) of the convolution f= K *g, gl, with respect to the metric of L₁. We also consider the problem of renewal of the values of the convolution operator by using the information about the values of the boundary functions.Translated from Ukrainskii Matematicheskii Zhurnal, Vol.47, No. 11, pp. 1549–1557, November, 1995.     

Morera type theorems on the unit disc

, . . : , , , , .     

Geometric properties of some special cones in a Banach space

Let E be a Banach space partially ordered by a cone K. Let B be a closed linear operator in E with domain (B). In this paper certain cones in the Banach space (B) with norm x =x+Bx are singled out for study; a number of their geometric properties are established under the assumption that the cone K in the space E has analogous properties.Translated from Matematicheskie Zametki, Vol. 5, No. 1, pp. 63–70, 1969.The author is indebted to M. A. Krasnosel'skii and V. Ya. Stetsenko for helpful discussions and for their interest in the results of the present paper.     

On the rank of a spectral function

Let P(x), 0 x 1, be an absolutely continuous spectral function in the separable Hilbert spacesS. If the vectors h_j, j=1, 2, ..., s; s are such that the set P(x)h_j is complete inS, then the rank of the function P(x) equals the general rank of the matrix-function d/dxP(x)h_i,h_js₁.Translated from Matematicheskie Zametki, Vol. 5, No. 4, pp. 457–460, April, 1969.