Asymmetric approximations in the spaceL p(t)

We introduce the notion of (α,β)-norm in the spaceL _p(t) of functionsx(t) for which

The sharp Jackson inequality in the spaceL p on the sphere

The sharp Jackson inequality in the spaceL _p, 1≤p<2, on the unit Euclidean sphereS ⁿ⁻¹ ,n≥3, is proved. Forn=2, it was established by N. I. Chernykh. Translated fromMatematicheskie Zametki, Vol. 66, No. 1, pp. 50–62, July, 1999.     

On linear topologies determined by a family of subsets of a topological vector space

We provide a general framework for the study of the finest linear (locally convex) topology which coincides on a family of subsets with a given linear (locally convex) topology. It is proved that the formation of such topologies always commutes with linear direct sums. We characterize the corresponding situation for products and prove a result about locally convex direct sums sufficiently general to cover the examples which already occurred in the literature. Moreover the 0-nbhd. filters of such topologies are characterized, and several examples are considered.     

On I-fuzzy topologies derived from a decreasing family of I-topologies

The aim of this paper is to study I-fuzzy topologies (fuzzy topologies in ostak's sense) derived from decreasing families of I-topologies (fuzzy topologies in Chang's sense), defined on a set X.     

On the boundedness of Cauchy singular operator from the spaceL p toL q,p>q>-1

It is proved that for a Cauchy type singular operator, given by equality (1), to be bounded from the Lebesgue spaceL _p () tol _q (), as = _n=1 ^Ȟ _n , _n ={z:|z|=r _n }, it is necessary and sufficient that either condition (4) or (5) be fulfilled.     

On a decomposition of the Hilbert spaceL 2 and its applications to Stokes problem

The Helmholtz-Weyl decomposition of the spaceL ² is strictly related with the solutions of Stokes problem. We state the main properties of this decomposition and we apply them to Stokes problem: suitable boundary conditions reduce its solution to simpler problems; the techniques involved also allow to solve two Stokes-type problems.

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Sunto La scomposizione di Helmholtz-Weyl dello spazioL ² è particolarmente adatta allo studio delle soluzioni del problema di Stokes. Si stabiliscono le principali proprietà della scomposizione e le loro applicazioni al problema di Stokes: opportune condizioni di bordo consentono di semplificare il problema; le tecniche utilizzate consentono anche lo studio di due problemi di tipo Stokes.

     

Compact sets in the spaceL p (O,T; B)

Summary A characterization of compact sets in L^p (0, T; B) is given, where 1P and B is a Banach space. For the existence of solutions in nonlinear boundary value problems by the compactness method, the point is to obtain compactness in a space L^p (0,T; B) from estimates with values in some spaces X, Y or B where XBY with compact imbedding XB. Using the present characterization for this kind of situations, sufficient conditions for compactness are given with optimal parameters. As an example, it is proved that if {f_n} is bounded in L^q(0,T; B) and in L _loc ¹ (0, T; X) and if {f_n/t} is bounded in L _loc ¹ (0, T; Y) then {f_n} is relatively compact in L^p(0,T; B), p相似文献   

Approximation of compressions of periodic functions in the spaceL p,p<1

We investigate the behavior of the best approximation of compressions of functions by trigonometric polynomials in the spaceL _p, p<1.Translated from Ukrainskii Matematicheskii Zhurnal, Vol.47, No. 12, pp. 1708–1711, December, 1995.This work was financially supported by the Foundation for Fundamental Research of the Ukrainian State Committee on Science and Technology.     

On bestm-term approximations and the entropy of sets in the spaceL 1

Mathematical Notes -     

On the cardinalities of finite topologies

Following the ideas of Sharp [2,3], we will give a partial answer to the question: “Let k be an integer, k ? 2. What is the smallest integer m for which there is a topology on m points with k open sets.” We state several results in the theory of finite topologies by introducing the idea of generating topologies. Using this concept, it is possible to derive existence theorems and get numerical results in an easy manner.     

On the compact-open and admissible topologies

It is known (see, for example, [H. Render, Nonstandard topology on function spaces with applications to hyperspaces, Trans. Amer. Math. Soc. 336 (1) (1993) 101-119; M. Escardo, J. Lawson, A. Simpson, Comparing cartesian closed categories of (core) compactly generated spaces, Topology Appl. 143 (2004) 105-145; D.N. Georgiou, S.D. Iliadis, F. Mynard, Function space topologies, in: Open Problems in Topology 2, Elsevier, 2007, pp. 15-23]) that the intersection of all admissible topologies on the set C(Y,Z) of all continuous maps of an arbitrary space Y into an arbitrary space Z, is always the greatest splitting topology (which in general is not admissible). The following, interesting in our opinion, problem is arised: when a given splitting topology (for example, the compact-open topology, the Isbell topology, and the greatest splitting topology) is the intersection of k admissible topologies, where k is a finite number. Of course, in this case this splitting topology will be the greatest splitting.In the case, where a given splitting topology is admissible the above number k is equal to one. For example, if Y is a locally compact Hausdorff space, then k=1 for the compact-open topology (see [R.H. Fox, On topologies for function spaces, Bull. Amer. Math. Soc. 51 (1945) 429-432; R. Arens, A topology for spaces of transformations, Ann. of Math. 47 (1946) 480-495; R. Arens, J. Dugundji, Topologies for function spaces, Pacific J. Math. 1 (1951) 5-31]). Also, if Y is a corecompact space, then k=1 for the Isbell topology (see [P. Lambrinos, B.K. Papadopoulos, The (strong) Isbell topology and (weakly) continuous lattices, in: Continuous Lattices and Applications, in: Lect. Notes Pure Appl. Math., vol. 101, Marcel Dekker, New York, 1984, pp. 191-211; F. Schwarz, S. Weck, Scott topology, Isbell topology, and continuous convergence, in: Lect. Notes Pure Appl. Math., vol. 101, Marcel Dekker, New York, 1984, pp. 251-271]).In [R. Arens, J. Dugundji, Topologies for function spaces, Pacific J. Math. 1 (1951) 5-31] a non-locally compact completely regular space Y is constructed such that the compact-open topology on C(Y,S), where S is the Sierpinski space, coincides with the greatest splitting topology (which is not admissible). This fact is proved by the construction of two admissible topologies on C(Y,S) whose intersection is the compact-open topology, that is k=2.In the present paper improving the method of [R. Arens, J. Dugundji, Topologies for function spaces, Pacific J. Math. 1 (1951) 5-31] we construct some other non-locally compact spaces Y such that the compact-open topology on C(Y,S) is the intersection of two admissible topologies. Also, we give some concrete problems concerning the above arised general problem.     

On ordinal-metric intersection topologies

We consider the intersection topology formed by the ordinal topology on ω₁ and a separable metric topology. Such a space is never perfectly normal, but it can be normal or perfect; which one depends on the model of set theory and the choice of the metric topology.     

On the lattice of principal generalized topologies

Periodica Mathematica Hungarica - In this paper we investigate the structure of the complete lattice of principal generalized topologies, employing the notion of ultratopology. On any partially...     

On similarity between topologies

Let T ₁ and T ₂ be topologies defined on the same set X and let us say that (X, T ₁) and (X, T ₂) are similar if the families of sets which have nonempty interior with respect to T ₁ and T ₂ coincide. The aim of the paper is to study how similar topologies are related with each other.     

On the generalization of density topologies on the real line

The paper concerns the density points with respect to the sequences of intervals tending to zero in the family of Lebesgue measurable sets. It shows that for some sequences analogue of the Lebesgue density theorem holds. Simultaneously, this paper presents proof of theorem that for any sequence of intervals tending to zero a relevant operator ? _J generates a topology. It is almost but not exactly the same result as in the category aspect presented in [WIERTELAK, R.: A generalization of density topology with respect to category, Real Anal. Exchange 32 (2006/2007), 273–286]. Therefore this paper is a continuation of the previous research concerning similarities and differences between measure and category.