The trace to subsets ofR n of Besov spaces in the general case

R ⁿ. , , , F R ⁿ, F , R ⁿ R ⁿ . ^p,q (Rⁿ), >0, 1, q, — ( ) Rⁿ. , ^p,q (Rⁿ) ^F Rⁿ. , q B ^p,q (F), = – (n–)/, >0, — « », ad — F, . , . : , F=R ^d,F— « » F— R ⁿ, « », F. .

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This work has been supported in part by the Swedish Natural Science Research Council.     

On equivalence of rearrangements of the Haar system in dyadic Hardy and BMO spaces

H={h ₁,I } — , . : , I ¦(I)¦=¦I¦, ¦I¦ — I. H H ={h _(I),I} . , , . L ^p .

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Dedicated to Professor B. Szökefalvi-Nagy on his 75th birthday

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This research was supported in part by MTA-NSF Grants INT-8400708 and 8620153.     

Differentiation of fractional order onp-groups,approximation properties

G- p- . [5] - (G) L _r(G) (1r<), . . , - . , , , . . , X. , . (. [1], [2] [4]).     

On a class of orthogonal series

, [0, 1], (n+1) n-. . [2]. — (. 5.4 5.6). . 6.4 2 [5]. , [4]. , , [6] [7]. [1].     

Theory of Eisenstein series for the group SL(3,R) and its application to a binary problem

On the basis of arithmetic considerations, a Fourier expansion for the leading Eisenstein series is obtained for the principal homogeneous space of the group SL(3,), which is automorphic with respect to the discrete group SL(3,). The main result is Theorem 1 in which an explicit form of the Fourier expansion is presented which generalizes the well-known formula of Selberg and Chowla. From this, in particular, there follows a proof of the analytic continuation and the functional equations for this Eisentein series which is independent of the work of Langlands. The arithmetic coefficients in the Fourier expansion which generalize the number-theoretic functions _s(n)=_d¦n,d>od⁵ make it possible to relate the Eisenstein series considered to the problem of finding the asymptotics as of the sum _n3(n)₃(n+), where ₃(n) is the number of solutions of the equation d₁d₂d₃=n in natural numbers. Part II of the present work will be devoted to this binary problem. At the end of the paper properties of special functions used in Theorem 1 are discussed.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 76, pp. 5–52, 1978.     

Regularity of ∫Ω|∇u|2 + λ∫Ω|u−f{2 and some gap phenomenon

We study the regularity of the minimizer u for the functional F (u,f)=|u|² + |u–f{² over all maps uH ¹(, S ²). We prove that for some suitable functions f every minimizer u is smooth in if ₀ and for the same functions f, u has singularities when is large enough.

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Résumé On étudie la régularité des minimiseurs u du problème de minimisation min_ueH ¹(,S²)(|u|² + |u–f{². On montre que pour certaines fonctions f, u est régulière lorsque ₀ et pour les mêmes f, si est assez grand, alors u possède des singularités.

     

On contiguity of probability measures corresponding to semimartingales

, (ⁿ), - (P ⁿ ), P ⁿ (A ⁿ )>0P ⁿ (A ⁿ )0,n. [15] - , . , P ⁿ P ⁿ T ⁿ T ⁿ .     

Produktsätze für Verfahren zur Limitierung von Doppelfolgen

{p _mn } - ₀₀>0, (1, 1) (1.1) (1.2). {s _mn } J _p - ( bJ _p -lims _mn =), (1.3) 0<x,y<1 p _s (, )/p(x, y) x, y 1-. {r _mn } - , (1.5) 0<, <1. N _rp - , (1.6). , bJ _p -lims _mn = bJ _q -lim(N _rps) _mn =. J _p - . , .     

Uniform boundedness of a family of set functions

Let be a ring of sets, X a normed space, : X ( ) a bounded family of triangular functions. The following generalized Nikodym theorem is established: the family {} is uniformly bounded on if and only if it is bounded on every sequence of pairwise disjoint sets of which the union is a part of some set in . An analogous criterion is established also for semiadditive functions. In addition, it is shown that uniform boundedness of a family of triangular functions is preserved in passing from a ring to the -ring it generates.Translated from Matematicheskie Zametki, Vol. 23, No. 6, pp. 855–861, June, 1978.     

On shifts of sequences

X ₂ ={x _k } _k=2 X={x _k } _k=1 . , , ( , ) . , , X— , , X ₂ H, , , , , X ₂ H.     

Riesz representation theorem and weak compactness

. , .     

An isolation theorem for decomposable forms of purely real algebraic fields of degree n⩾3

Let M be a complete module of a purely algebraic field of degree n3, let be the lattice of this module and let F(X) be its form. By we denote any lattice for which we have = , where is a nondiagonal matrix satisfying the condition ¦-I¦ , I being the identity matrix. The complete collection of such lattices will be denoted by {}. To each lattice we associate in a natural manner the decomposable form F(X). The complete collection of forms, corresponding to the set {}, will be denoted by {F} It is shown that for any given arbitrarily small interval (N–, N+), one can select an such that for each F(X) from {F} there exists an integral vector X₀ such that N– < F(X₀) < N+.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 112, pp. 167–171, 1981.     

Pointwise convergence of multiple Fourier series using different convergence notions

, . , L ^p (²) p>1 . , C(T³), - , . - .

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This work was completed while the first named author was a visiting professor at Indiana University, Bloomington, Indiana; and the second named author was a visiting professor at Ohio State University, Columbus, Ohio, U.S.A.     

Bilateral difference method for solving the boundary value problem for an ordinary differential equation

A method is proposed for calculating the bilateral approximations of the solution of the boundary value problem on [0, 1] for the equation y+p(x)y-q(x)y=f(x) and the derivative of the solution having the maximum deviation O(h² (h)+h³) on {kh} _k ^N =0, where(t) is the sum of the continuity moduli of the functions p, q,f, on the set of points {kh} _k ^N =0, h=1/N by means of O(N) operations. The data obtained for fairly smooth p, q,f allow interpolation to be used for calculating the bilateral approximations of the solution and its higher derivatives having the maximum deviation O(h³) on [0, 1].Translated from Matematicheskie Zametkii, Vol. 11, No. 4, pp. 421–430, April, 1972.     

Existence of Incomplete Transversal Designs with Block Size Five and Any Index λ

The basic necessary condition for the existence of a TD(5, ; v)-TD(5, ; u), namely v 4u, is shown to be sufficient for any 1, except when (v, u) = (6, 1) and = 1, and possibly when (v, u) = (10, 1) or (52, 6) and = 1. For the case = 1, 86 new incomplete transversal designs are constructed. Several construction techniques are developed, and some new incomplete TDs with block size six and seven are also presented.     

Rate of approximation by rectangular partial sums of double orthogonal series

( ) . .

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Dedicated to Professor K. Tandori on his seventieth birthday

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This research was supported in part by Grant # K41 100 of the Joint Fund of the Government of Ukraine and the International Science Foundation.     

On convergence and summability of Fourier series

. , , –1<<0. .

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The present work was written on the basis of two earlier works received byAnalysis Mathematica on January 16, 1979, and July 20, 1979.     

On the growth order of partial sums of non-orthogonal series

a _k f _k , f _k L ₂, w-, (2), w(n) — . a _k f _k N {a _k }l ₂, {a _k }l ₂ ( 1, 2, 1a, 2a). ( 2) [8]. , {a _k } w-.     

Boundary control of heat conduction

The paper considers control of the heat conduction process u_t — u = g from the initial state u(x, 0) to the final state u(x, t₁) in a fixed (finite) time t₁ via the coefficient (z) in the boundary condition Bu = (u/n) + (x)u. A uniqueness theorem is proved for the problem to find the process—control pair (u, ). The control problem is posed in terms of the coefficient in a boundary condition of the form Bu = (u/n) + (t)u.Translated from Nelineinye Dinamicheskie Sistemy: Kachestvennyi Analiz i Upravlenie — Sbornik Trudov, No. 3, pp. 93–97, 1993.     

A Comparison of Methods for Estimating the Extremal Index

The extremal index, (01), is the key parameter when extending discussions of the limiting behavior of the extreme values from independent and identically distributed sequences to stationary sequences. As measures the limiting dependence of exceedances over a threshold u, as u tends to the upper endpoint of the distribution, it may not always be informative about the extremal dependence at levels of practical interest. Therefore we also consider a threshold-based extremal index, (u). We compare the performance of a range of different estimators for and (u) covering processes with < 1 and = 1. We find that the established methods for estimating actually estimate (u), so perform well only when (u) . For Markov processes, we introduce an estimator which is as good as the established methods when (u) but provides an improvement when (u) < = 1. We illustrate our methods using simulated data and daily rainfall measurements.