On some class of nonlinear parabolic inequalities in Orlicz spaces

The paper deals with the existence of solutions of some unilateral problems in the framework of Orlicz spaces.     

Analytic norms in Orlicz spaces

It is shown that an Orlicz sequence space admits an equivalent analytic renorming if and only if it is either isomorphic to or isomorphically polyhedral. As a consequence, we show that there exists a separable Banach space admitting an equivalent -Fréchet norm, but no equivalent analytic norm.

     

On the rate of convergence of certain random series in terms of norms of Orlicz spaces

There are obtained conditions for the convergence and estimates of the rate of convergence in terms of norms of Orlicz spaces of random series of the form where f_k(¯t) is an orthonormal system of eigenfunctions of a certain integral equation, k are random variables in an Orlicz space, ¯t Rⁿ.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 6, pp. 841–848, June, 1991.     

On the uniform convergence of wavelet expansions of random processes from Orlicz spaces of random variables. II

We establish conditions under which wavelet expansions of random processes from Orlicz spaces of random variables converge uniformly with probability one on a bounded interval. Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 6, pp. 759–775, June, 2008.     

On the uniform convergence of wavelet expansions of random processes from Orlicz spaces of random variables. I

We establish conditions under which there exists a function c(t) > 0 such that {fx1850-01}, where X(t) is a random process from an Orlicz space of random variables. We obtain estimates for the probabilities {fx1850-02}. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 12, pp. 1647–1660, December, 2007.     

Convergence of trigonometric series in norms of Orlicz spaces

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 8, pp. 1088–1094, August, 1989.     

On n-widths of a Sobolev function class in Orlicz spaces

This paper considers the problem of n-widths of a Sobolev function class Ωr∞ determined by Pr(D) = Dσ∏lj =1(D2- t2jI) in Orlicz spaces. The relationship between the extreme value problem and width theory is revealed by using the methods of functional analysis. Particularly, as σ = 0 or σ = 1, the exact values of Kolmogorov's widths, Gelfand's widths, and linear widths are obtained respectively, and the related extremal subspaces and optimal linear operators are given.     

On n-widths of some periodic convolution classes in Orlicz spaces

In this paper we extend the some accurate results on n-width of Sobolevclass W_∞~γ in L_p spaces to Orlicz spaces,and establish the correspondingresults for dual cases.     

Korn type inequalities in Orlicz spaces

We exhibit balance conditions between a Young function A and a Young function B   for a Korn type inequality to hold between the L^B$L^B$ norm of the gradient of vector-valued functions and the L^A$L^A$ norm of its symmetric part. In particular, we extend a standard form of the Korn inequality in L^p$L^p$, with 1<p<∞$1<p<\infty$, and an Orlicz version involving a Young function A   satisfying both the _Δ2${\mathrm{\Delta }}_2$ and the _∇2${\mathrm{\nabla }}_2$ condition.     

On the exponential inequalities for negatively dependent random variables

A number of exponential inequalities for identically distributed negatively dependent and negatively associated random variables have been established by many authors. The proofs use the truncation technique together with the control of the bounded terms and unbounded terms. In this paper, we improve essentially the control of bounds for the unbounded terms and obtain exponential inequalities for negatively dependent random variables which include negatively associated random variables. Our results improve on the corresponding ones in the literature.     

On discrete and continuous norms in Paley-Wiener spaces and consequences for exponential frames

It is well known that for certain sequences {t_n}_n the usual L^p norm ·_p in the Paley-Wiener space PW ^p is equivalent to the discrete norm f_p,{tn}:=( _n=– |f(t_n)|^p)^1/p for 1 p = < and f_,{tn}:=sup _n|f(t_n| for p=). We estimate f_p from above by Cf_p, _n and give an explicit value for C depending only on p, , and characteristic parameters of the sequence {t_n}_n. This includes an explicit lower frame bound in a famous theorem of Duffin and Schaeffer.     

On approximation in Weighted Orlicz spaces

An inverse theorem of the trigonometric approximation theory in Weighted Orlicz spaces is proved and the constructive characterization of the generalized Lipschitz classes defined in these spaces is obtained.     

Probabilistic norms and convergence of random variables

We prove that the probabilistic norms of suitable Probabilistic Normed spaces induce convergence in probability, L^p convergence and almost sure convergence.