On the uniform (C, 1)-summability of Fourier series and integrability of series with respect to multiplicative orthonormal systems

H (G), f(g)H (G) , (, 1)- OHMC G. , OHMC, A. H. . , . , OHMC, lim supp _n=, , ,n .. . , 117 234 . . -      

Perfect splines of least uniform deviation

(v _k) ₁ ⁿ , 1v _kr, v₁+...+v_n r, p(;r) — =(x ₁,v ₁),...,(x _n,v _n), v _1+...+v _n-r ^(r(;·)=1. , (^*;t) [, ] (;t) _{x 1}<... n v ₁ ...,v _n. (^*;t) .     

Die Dupinschen Zykliden im pseudoeuklidischen Raum

In the pseudoeuclidean space¹ E ³ a surface of degree 3 or 4 is calledDupin-cyclide if there exists a (pseudoeuclidean) torus such that a (pseudoeuclidean) spherical inversion maps upon . If the axis of is respectively space-like, isotropic or time-like is calledDupin-r-,-l- or-z-cyclide. ADupin-cyclide is the envelope of two families of spheres, the caustic surfaces of degenerate in caustic curves and the lines of curvature of constitute two (orthogonal) families of circles.     

Critical points of essential norms of singular integral operators in weighted spaces

We show that for any simple piecewise Ljapunov contour there exists a power weight such that the essential norm |S | in the spaceL ₂(, ) does not depend on the angles of the contour and it is given by formula (2). All such weights are described. For the union =₁₂ of two simple piecewise Lyapunov curves we prove that the essential norm |S | inL ₂() is minimal if both ₁ and ₂ are smooth in some neighborhoods of the common points. It is the case when the norm |S | in the spaceL ₂() as well as inL ₂(, ) does not depend on the values of the angles and it can be calculated by formula (5).     

Converse theorems for multidimensional Kantorovich operators

L _p [0, l]. . . - .

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Supported by National Science Foundation, Zhejiang Provincial Science Foundation of China, and Alexander von Humboldt Foundation of Germany.     

Asymptotic behavior of constrained stochastic approximations via the theory of large deviations

Summary Let G be a bounded convex set, and _G the projection onto G, and a bounded random process. Projected algorithms of the types , where 0<a _n0, a _n =) occur frequently in applications (among other places) in control and communications theory. The asymptotic convergence properties of {X _n } as 0, n, have been well analyzed in the literature. Here, we use large deviations methods to get a more thorough understanding of the global behavior. Let be a stable point of the algorithm in the sense that X _n in distribution as 0, n. For the unconstrained case, rate of convergence results involve showing asymptotic normality of , and use linearizations about . In the constrained case is often on G, and such methods are inapplicable. But the large deviations method yields an alternative which is often more useful in the applications. The action functionals are derived and their properties (lower semicontinuity, etc.) are obtained. The statistics (mean value, etc.) of the escape times from a neighborhood of are obtained, and the global behavior on the infinite interval is described.Research has been supported in part by the US Army Research Office under Contract #DAAG 29-84-K-0082, and in part by the Office of Naval Research under Contract #N00014-83-K0542Research has been supported in part by the National Science Foundation Grant #ECS 82-11476, and the Air Force Office of Scientific Research under Contract #AF-AFOSR 81-0116     

A homology transfer for a class of simplicial maps

We construct a homology transfer _*H_*(B) H_*(E) for a certain class of proper simplicial maps pE B. Roughly, the important hypothesis is that there is a compact space F, so that all fibres F_x>=p^–1(x), xB, are quotient spaces of F, in a certain locally controlled manner. The composition p_** H_*(B)H_*(E)H_*(B) is multiplication by the Euler characteristic of F.     

Morera type theorems on the unit disc

, . . : , , , , .     

On power series with positive coefficients

M. . , . , ^p () (). , , .     

Minimality of root vectors of operator functions analytic in an angle

We study the minimality of elementsx _h,j,k of canonical systems of root vectors. These systems correspond to the characteristic numbers _k of operator functionsL() analytic in an angle; we assume that operators act in a Hilbert space . In particular, we consider the case whereL()=I+T()c, >0,I is an identity operator,C is a completely continuous operator, (I- C)^–1c for ¦arg¦, 0<<, the operator functionT() is analytic, and T()c for ¦arg¦<. It is proved that, in this case, there exists >0 such that the system of vectorsC ^v x _h,j,k is minimal in for arbitrary positive <1+, provided that ¦_k¦>.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 5, pp. 545–566, May, 1994.This research was partially supported by the Ukrainian State Committee of Science and Technology.     

Spectral synthesis on a system of unbounded domains starlike in a common direction

, . . - ₁, ..., ₄, — ; =(₁,)×...×H(₄), — H(₁, ..., H(₄), r H(₁) — , ₁ ; D: HH- . , D. , ₁..., ₄ , (.. z₁ z+te^ia ₁ t>0), W H .     

Concretely Locally Presentable and Locally Generated Categories

We give concrete versions of the characterizations of locally -presentable (resp. -generated) categories as categories of models of (resp. weakly) -ary limit-theories and of (resp. weakly) -ary essentially algebraic theories. By concrete we mean that starting with a category of -structures, the theories obtained are extensions of the original ones, and the equivalences of categories are concrete isomorphisms. -presentable and -generated objects are also investigated from this viewpoint.     

A generalization of the Heine limit for functions which converge on a base

. , , . . . .     

Growth of length of sequential derivation transformed into natural one

We consider the (&, )-fragment of the intuitionistic propositional calculus. It is proved that under the standard transformation of a Gentzen derivation into a natural derivation(), the length of (())2^2·length( ). There is constructed a sequence of Gentzen derivations of length _i, for which the length of (( _i))2^{1/3·length(i}), which shows that the upper bound obtained is not too weak.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 88, pp. 192–196, 1979.     

On the newly generalized absolute Riesz summability of orthogonal series

[3] , [1], ¦C,,(t)¦ _k - . , , ¦R, , (t)¦ _k - ¦R, , (t)¦ _k - . (t)=1,k=1 [7], [9].

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This research was partially supported by the Hungarian National Foundation for Scientific Research under Grant#T 016 393.     

On the limit superior of analytic sets

. , A ₀,A ₁,— - lim supA _j - H, . , - - . , , ; , , . - . - .     

On the Morse index of minimal surfaces in ℝp with polygonal boundaries

The minimal surfaces spanning a polygon in ^p (p2) correspond to the critical points of an analytic function in finitely many variables, namely Shiffman's function. We shall prove that the Morse index of the minimal surface coincides with the Morse index of at the corresponding critical point. Alternatively expressed, the Schwarz operator of the minimal surface and the Hessian of have the same number of negative eigenvalues. Finally we control the degeneration of the critical points.     

Convergence theorems for set-valued amarts and uniform amarts

( ) , — , - . .     

A high-indices theorem for strong Cesàro summability of double series

[4] , [C,(, )]_;- (, 0 1) s, [C,(+,+)]- s , 0 0< , (i) , ; > > 1 , (ii) ,>0; >=1 ,>1-1/, [C,(,)- [C,(+,+)]- .

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This work was done while the author was a visiting researcher at the Steklov Mathematical Institute, Moscow, U.S.S.R.