Convergence theorems for continuous functions on an arbitrary interval

In this paper, we introduce a new iterative scheme for finding a fixed points of continuous functions on an arbitrary interval. The convergence theorems are also established. Further, the numerical examples comparing with Mann, Ishikawa and Noor iterations are demonstrated. Main results generalize and unify the corresponding ones announced in the literature.     

Generalized matching theorems for closed coverings of convex sets

In this paper we use fixed point and coincidence theorems due to Park [8] to give matching theorems concerning closed coverings of nonempty convex sets in a real topological vector space. Our new results extend previously given ones due to Ky Fan [2], [3], Shih [10], Shih and Tan [11], and Park [7].     

Linearization of operator functions on arbitrary open sets

In this paper it is shown that after a suitable extension an operator function A, which is holomorphic on an open set in , is equivalent on to a linear pencil S-V. If is bounded, then V turns out to be right invertible, and in that case a further extension is equivalent on to a linear pencil T-I.     

Uniform and tangential harmonic approximation of continuous functions on arbitrary sets

Necessary and sufficient conditions which must be imposed on a set E are derived, such that functions continuous on E G can be approximated by functions harmonic in a region G (Rⁿ.Translated from Matematicheskie Zametki, Vol. 9, No. 2, pp. 131–142, February, 1971.In conclusion I wish to thank my scientific director S. N. Mergelyan and also A. A. Gonchar for their valuable advice.     

Separation theorems and minimax theorems for fuzzy sets

In this paper, we prove some theorems on fuzzy sets. We first show that, in order to demonstrate that the equality of shadows ofA andB implies the equality ofA andB, it is necessary to assume thatA andB are closed and thatS _H (A)=S _H (B) for any closed hyperplane hyperplaneH. We also obtain a separation theorem for two convex fuzzy sets in a Hilbert space. Finally, we investigate results relating to minimax theorems for fuzzy sets. We obtain a necessary and sufficient condition for compactness.The authors wish to express their sincere thanks to Professor Hisaharu Umegaki for his invaluable suggestions and advice.     

Empirical and Poisson processes on classes of sets or functions too large for central limit theorems

Summary Let P be the uniform probability law on the unit cube I ^d in d dimensions, and P _n the corresponding empirical measure. For various classes of sets AI ^d , upper and lower bounds are found for the probable size of sup {¦P _n –P) (A)¦ A }. If is the collection of lower layers in I ², or of convex sets in I ³, an asymptotic lower bound is ((log n)/n) ^1/2(log log n)^––1/2 for any >0. Thus the law of the iterated logarithm fails for these classes.If >0, is the greatest integer <, and 0 d f(x₁,...,x _d-1)} where f has all its partial derivatives of orders bounded by K and those of order satisfy a uniform Hölder condition ¦D ^p (f(x)–f(y))¦K¦x –y¦ ^–. For 0<–/(d–1+) for a constant = (d,)>0. When = d-1 the same lower bound is obtained as for the lower layers in I ² or convex sets in I ³. For 0 – 1 there is also an upper bound equal to a power of log n times the lower bound, so the powers of n are sharp.This research was partially supported by National Science Foundation Grant MCS-79-04474     

Some theorems on Chebyshev sets

We consider the class of -suns which is used in the study of Chebyshev sets. We give sufficient conditions for a set to be a -sun. We prove that, in a uniformly smooth Banach space, a weakly closed Chebyshev set is convex.Translated from Matematicheskie Zametki, Vol. 11, No. 2, pp. 135–144, February, 1972.The author expresses his thanks to V. I. Berdyshev and N. I. Chernykh for a lively discussion of this note.     

Pointwise ergodic theorems for arithmetic sets

With an appendix on return-time sequences     

Strong limit theorems for arbitrary stochastic sequences

In this paper, we establish two strong limit theorems for arbitrary stochastic sequences. As corollaries, we generalize some known results.     

Support theorems for unbounded convex sets

We present some generalizations of Helgason support theorem for functions with unbounded convex support.     

Fixed point theorems for convex sets

A conjecture of the author is verified by setting up a fundamental fixed point theorem so that some earlier results are unified and strengthened. Supported by National Foundation of Natural Science.     

Limit theorems for multidimensional renewal sets

Given a sequence \({\mathcal{U} =\{U_n: n \in \omega\}}\) of non-empty open subsets of a space X, a set \({\{x_n : n \in \omega\}}\) is a selection of \({\mathcal{U}}\) if \({x_n \in U_n}\) for every \({n \in \omega}\). We show that a space X is uncountable if and only if every sequence of non-empty open subsets of C _p (X) has a closed discrete selection. The same statement is not true for \({C_p(X,[0,1])}\) so we study when the above selection property (which we call discrete selectivity) holds in \({C_p(X,[0,1])}\). We prove, among other things, that \({C_p(X, [0,1])}\) is discretely selective if X is an uncountable Lindelöf \({\Sigma}\)-space. We also give a characterization, in terms of the topology of X, of discrete selectivity of \({C_p(X,[0,1])}\) if X is an \({\omega}\)-monolithic space of countable tightness.     

Uniform polynomial approximation of entire functions on arbitrary compact sets in the complex plane

For an arbitrary compact setK⊂ℂ, we relate the order and the type of an entire functionf to the sequenceE _n (f,K) of best polynomial approximations to this function onK. Translated fromMatematicheskie Zametki, Vol. 58, No. 3, pp. 355–364, September, 1995.