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.     

Onp-Helson sets in R n

p- . E R ⁿ -, f () _p(R ⁿ)., ER ⁿ 2nq ₀, E— - q ₀(q ₀-1). : q₀>2 n1 E R ⁿ 2nq ₀, p- p<₀. , f-[-, ]ⁿ, f A _p(R ⁿ) , _p([-, ]ⁿ) (1 << ).     

The asymptotic behaviour of certain additive functionals of Brownian motion

We prove pathwise asymptotic stability for certain additive functionals of one- and two-dimensional Brownian motion.     

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.     

Verallgemeinerung eines Satzes von F. Riesz

Let X be a compact Hausdorff space, C(X) the class of all continuous functions f: XR, considered as an ordered vector space over R with respect to its canonical order; furthermore let F be an order complete ordered vector space over R. A linear operator A: C(X)F is called bounded, if it transforms each bounded set of C(X) into a bounded set of F.The purpose of this paper is to represent such a bounded linear operator A: C(X)F as an integral Af= d with respect to some content , defined on the algebraa(X) which is generated by the open sets UX.This representation is unique, if is required to be regular (p. 11).     

Leaves in Representation Diagrams of Bipartite Distance-Regular Graphs

Let denote a bipartite distance-regular graph with diameter D 3 and valency k 3. Let ₀ > ₁ ··· > _D denote the eigenvalues of and let q ^h _ij (0 h, i, j D) denote the Krein parameters of . Pick an integer h (1 h D – 1). The representation diagram = _h is an undirected graph with vertices 0,1,...,D. For 0 i, j D, vertices i, j are adjacent in whenever i j and q ^h _ij 0. It turns out that in , the vertex 0 is adjacent to h and no other vertices. Similarly, the vertex D is adjacent to D – h and no other vertices. We call 0, D the trivial vertices of . Let l denote a vertex of . It turns out that l is adjacent to at least one vertex of . We say l is a leaf whenever l is adjacent to exactly one vertex of . We show has a nontrivial leaf if and only if is the disjoint union of two paths.     

Grothendieck Institutions

We extend indexed categories, fibred categories, and Grothendieck constructions to institutions. We show that the 2-category of institutions admits Grothendieck constructions (in a general 2-categorical sense) and that any split fibred institution is equivalent to a Grothendieck institution of an indexed institution. We use Grothendieck institutions as the underlying mathematical structure for the semantics of multi-paradigm (heterogenous) algebraic specification. We recuperate the so-called extra theory morphisms as ordinary theory morphisms in a Grothendieck institution. We investigate the basic mathematical properties of Grothendieck institutions, such as theory colimits, liberality (free constructions), exactness (model amalgamation), and inclusion systems by globalisation from the local level of the indexed institution to the level of the Grothendieck institution.     

Mengenwertige Masse und Fortsetzungen

Let (, _i) be a probability space for i=1,2 with and : ^m a correspondence, i.e. () is a non-void subset of ^m for all . We give necessary and sufficient conditions under which it holds, that ₂ extends ₁. iff _A d₂ is equal to _A d₁ for all A, where _A d_i is the set of all integrals _A f d_i of functions f: ^m with f()() _i.-a.e.     

About generalized ideals in a distributive lattice

Let L be a distributive lattice characterized by a ternary operation (, ,), where (a,b,c)=(ab)(bc)(ac)=(ab)(ac)(bc), a,b,cL. The note considers convex sublattices of L, called generalized ideals of L generated by the operation (, ,). Some remarks have been stated about the graph of a distributive lattice.     

On a condition on the radical of a Banach algebra ensuring strong decomposability

The main result is the following theorem. Let be a commutative Banach algebra with radical R, where the factor algebra is isomorphic to the algebra of all continuous functions on a totally disconnected compact space. If rⁿ¹ /n 0 as n uniformly for r R, rl, then the algebra is strongly decomposable, i.e., there exists a closed subalgebra B isomorphic to such that =BR.This is a strengthening of the theorem of A. Ya. Khelemskii, who assumed . There are 4 references.Translated from Matematicheskie Zametki, Vol. 2, No. 6, pp. 589–592, December, 1967.     

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.     

Poincaré duality for algebraic de rham cohomology

We discuss in some detail the algebraic notion of De Rham cohomology with compact supports for singular schemes over a field of characteristic zero. We prove Poincaré duality with respect to De Rham homology as defined by Hartshorne [H.75], so providing a generalization of some results of that paper to the non proper case. In order to do this, we work in the setting of the categories introduced by Herrera and Lieberman [HL], and we interpret our cohomology groups as hyperext groups. We exhibit canonical morphisms of cospecialization from complex-analytic De Rham (resp. rigid) cohomology groups with compact supports to the algebraic ones. These morphisms, together with the specialization morphisms [H.75, IV.1.2] (resp. [BB, 1]) going in the opposite direction, are shown to be compatible with our algebraic Poincaré pairing and the analogous complex-analytic (resp. rigid) one (resp. [B.97, 3.2]).Mathematics Subject Classification (2000):Primary 14FXX     

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 .     

Limiting behavior of weighted central paths in linear programming

We study the limiting behavior of the weighted central paths{(x(), s())} > 0 in linear programming at both = 0 and = . We establish the existence of a partition (B ,N ) of the index set { 1, ,n } such thatx _i() ands _j () as fori B , andj N , andx _N (),s _B () converge to weighted analytic centers of certain polytopes. For allk 1, we show that thekth order derivativesx ^(k) () ands ^(k) () converge when 0 and . Consequently, the derivatives of each order are bounded in the interval (0, ). We calculate the limiting derivatives explicitly, and establish the surprising result that all higher order derivatives (k 2) converge to zero when .     

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-.     

Integrability of multiple Walsh series and Parseval's formula

f(x,y) _jk . , {c _jk} , f(x, )(, ) [0,1)&#x0445;[0,1) , - (0,0). , , f, - f. , , , [1] . . - [5] [6].

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This research is supported by National Science Council, Taipei, R.O.C. under Grant #NSC 84-2121-M-007-026.     

A coherence theorem for canonical morphisms in cartesian closed categories

In this paper there is proved a coherence theorem in proof-theoretic formulation: all derivations of a balanced sequent are equivalent. (A sequent is called balanced if each variable appears in it no more than twice.) Canonical morphisms in a Cartesian closed category are morphisms which can be obtained from those explicitly mentioned in the definition of a Cartesian closed category (i.e., the left and right projectionsl: A x B A and r:A x B B, :A x hom (A, B) B, etc.) with the help of composition of functors x, hom and the operation +. Let the objects A and B be constructed from the objects C₁,..., C_n with the help of the functors x and hom. Then, generally speaking, not all canonical morphisms from A to B will be equal. For example, if A is C₁ x C₁, and B is C₁, then the left and right projections are different morphisms. The coherence theorem asserts that if one does not make superfluous identifications of objects, then all canonical morphisms from A to B will be equal, i.e., all diagrams of canonical morphisms beginning in A and ending in B will commute. There is a familiar translation of certain concepts of the theory of categories into the language of proof theory, under which to objects correspond formulas, and the functors x and hom are interpreted as the connectives & and . Under this translation, to canonical morphisms from A to B correspond derivations in the (&, )-fragment of the intuitionistic prepositional calculus of the sequent A B. Morphisms are equal if and only if the derivations corresponding to them are equivalent, i.e., certain of their normal forms coincide, or, what is the same thing, their deductive terms are equivalent. The theorem proved in this paper is equivalent with the coherence theorem in the algebraic formulation. There are given two proofs of this theorem, obtained independently by the authors, in one of which there are considered natural derivations and the apparatus of deductive terms is used, and the other is based on reduction of the depth of formulas preserving equivalence of derivations, specialization of forms of inference in Gentzen L-systems, and analysis of links in sequences.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 88, pp. 3–29, 1979.     

Waterman classes and spherical partial sums of double Fourier series

f(x,y) ₀BV(T²), ={1/n} _n=1 .

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Dedicated to Professor Károly Tandori, the outstanding mathematician and academician on his seventieth birthday

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This work was done under the financial support of the Russian Foundation for Fundamental Scientific Research, Grant 93-01-00240.     

Coherent Homotopical Algebras: “Special Gamma-Categories”

We introduce a theory of coherence for symmetric monoidal categories inthe spirit of Segal and show that it is equivalent, in an appropriate sense,to MacLanes original notion. More precisely, we prove thatspecial categories, the analogue ofspecial spaces, and coherently symmetric monoidalcategories are one and the same. This is analogous to the situation intopology where special spaces are precisely homotopicalcommutative monoids. In light of the obervation that the category of smallcategories Cat bears a functorial Quillen model structure with respect tothe class of categorical equivalences: in fact, is a homotopy theory in thesense of Heller, we may reinterpret the theorem as stating that coherentlysymmetric monoidal categories are precisely the homotopical commutativemonoids within this new homotopy theory.     

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 - . , .