Spaces of Densely Continuous Forms, USCO and Minimal USCO Maps

Our paper studies the topology of uniform convergence on compact sets on the space of densely continuous forms (introduced by Hammer and McCoy (1997)), usco and minimal usco maps. We generalize and complete results from Hammer and McCoy (1997) concerning the space D(X,Y) of densely continuous forms from X to Y. Let X be a Hausdorff topological space, (Y,d) be a metric space and D _k (X,Y) the topology of uniform convergence on compact sets on D(X,Y). We prove the following main results: D _k (X,Y) is metrizable iff D _k (X,Y) is first countable iff X is hemicompact. This result gives also a positive answer to question 4.1 of McCoy (1998). If moreover X is a locally compact hemicompact space and (Y,d) is a locally compact complete metric space, then D _k (X,Y) is completely metrizable, thus improving a result from McCoy (1998). We study also the question, suggested by Hammer and McCoy (1998), when two compatible metrics on Y generate the same topologies of uniform convergence on compact sets on D(X,Y). The completeness of the topology of uniform convergence on compact sets on the space of set-valued maps with closed graphs, usco and minimal usco maps is also discussed.     

Uniform convergence on spaces of multifunctions

We study spaces of multifunctions with closed values, multifunctions with closed graphs, USCO multifunctions, minimal USCO multifunctions and the space of densely continuous forms as metric spaces, equipped with the topology of uniform convergence. We give conditions under which these metric spaces are complete.      

Densely continuous forms, pointwise topology and cardinal functions

We consider the space D(X, Y) of densely continuous forms introduced by Hammer and McCoy [5] and investigated also by Holá [6]. We show some additional properties of D(X, Y) and investigate the subspace D*(X) of locally bounded real-valued densely continuous forms equipped with the topology of pointwise convergence τ- _p . The largest part of the paper is devoted to the study of various cardinal functions for (D*(X), τ _p ), in particular: character, pseudocharacter, weight, density, cellularity, diagonal degree, π-weight, π-character, netweight etc. This work was supported by Science and Technology Assistance Agency under the contract No. APVT-51-006904. The authors would like to thank Ľubica Holá for suggestions and comments.     

Estimates for the Lebesgue functions and the Nevai formula for the sinc-approximations of continuous functions on an interval

We obtain some upper and lower estimates for the sequences of the Lebesgue functions and constants of the Whittaker operators

$L_n (f,x) = \sum\limits_{k = 0}^n {\frac{{\sin (nx - k\pi )}}{{nx - k\pi }}} f\left( {\frac{{k\pi }}{n}} \right)$

for continuous functions. We give an analog of Nevai’s formula for the Lagrange-Chebyshev and Lagrange-Laguerre interpolation polynomials for the operators under consideration. Its “local” version is established.