Monotonic analysis: convergence of sequences of monotone functions

In this article we examine various kinds of convergence of sequences of increasing positively homogeneous (IPH) functions and nonnegative decreasing functions defined on the interior of a pointed closed solid convex cone K. We show that five different types of convergency (including pointwise and epi-convergence) coincide for IPH functions. If the space under consideration is finite dimensional then the sixth type can be added: uniform convergence on bounded subsets of itn K. Using IPH functions, we study epi-convergence of sequences of lower semi-continuous (lsc) nonnegative decreasing functions.     

On the strong almost convergence of double sequences

In this paper, necessary and sufficient conditions that the real doubly infinite matrixA sums every strongly almost convergent double sequence, leaving the limit invariant, have been determined.     

On the topological entropy of continuous and almost continuous functions

We prove some results concerning the entropy of continuous and almost continuous functions. We first introduce the notions of bundle entropy and (strong) entropy points and then we study properties of these notions in connection with the theory of multifunctions. Based on these facts we give theorems about approximation of functions defined and assuming their values on compact manifold by functions having strong entropy points.     

On fuzzy almost continuous convergence in fuzzy function spaces

In this paper, we study the fuzzy almost continuous convergence of fuzzy nets on the set FAC(X, Y) of all fuzzy almost continuous functions of a fuzzy topological space X into another Y. Also, we introduce the notions of fuzzy splitting and fuzzy jointly continuous topologies on the set FAC(X, Y) and study some of its basic properties.     

On convergence of Fourier series of Besicovitch almost periodic functions

The paper deals with convergence of the Fourier series of q-Besicovitch almost periodic functions of the form $$ f(t)\sim \mathop{\sum}\limits_{m=1}^{\infty }{a_m}{{\mathrm{e}}^{{-\mathrm{i}{\uplambda_m}t}}}, $$ where {λm} is a Dirichlet sequence, that is, a strictly increasing sequence of nonnegative numbers tending to infinity. In particular, we show that, for 1 < q < ∞, the Fourier series of f(t) converges in norm to the function f(t) itself with usual order, which is analogous to the convergence in norm of the Fourier series of a function on [0, 2π]. A version of the Carleson–Hunt theorem is also investigated.     

Unconditional convergence almost everywhere of Fourier series of continuous functions in the Haar system

A continuous function is constructed whose Haar-Fourier series, after a definite rearrangement of its terms, diverges almost everywhere. A function is also constructed which has the maximum degree of smoothness in the sense that if its smoothness is increased its Haar-Fourier series becomes unconditionally convergent almost everywhere.Translated from Matematicheskie Zametki, Vol. 4, No. 2, pp. 211–220, August, 1968.     

On the asymptotic convergence of sequences of analytic functions

We consider sequences {f _n } of analytic self mappings of a domain and the associated sequence {Θ _n } of inner compositions given by . The case of interest in this paper concerns sequences {f _n } that converge assymptotically to a function f, in the sense that for any sequence of integers {n _k } with n ₁ < n ₂ < ... one has that locally uniformly in Ω. Most of the discussion concerns the case where the asymptotic limit f is the identity function in Ω. Received: 16 December 2006     

Measurable functions and almost continuous functions

I show that if (X, ) is a Radon measure space and Y is a metric space, then a function from X to Y is -measurable iff it is almost continuous (=Lusin measurable). I discuss other cases in which measurable functions are almost continuous.Part of the work of this paper was done during a visit to Japan supported by the United Kingdom Science Research Council and Hokkaido University     

On the almost everywhere convergence of double Fourier series of integrable functions

, . . Q _k [0,2],k=1,2, — . F(x, y)L(T), T=[0, 2]², G(x, y)L(T) , G(x,y)=F(x,y) Q=Q ₁ ×Q ₂ - .     

On almost everywhere convergence

Suppose thatf is an element ofL ²(R ⁿ ) whose orbit under the action ofSO(n) spans a finite-dimensional subspace. Then the spherical partial sums of the inverse Fourier transform off converge almost everywhere.     

On almost sure convergence of Cesaro averages of subsequences of vector-valued functions

A remarkable theorem proved by Komlòs [4] states that if {f_n} is a bounded sequence in L¹(R), then there exists a subsequence {f_nk} and f L¹(R) such that f_nk (as well as any further subsequence) converges Cesaro to f almost everywhere. A similar theorem due to Révész [6] states that if {f_n} is a bounded sequence in L²(R), then there is a subsequence {f_nk} and f L²(R) such that Σ_k=1^∞ a_k(f_nk − f) converges a.e. whenever Σ_k=1^∞ | a_k |² < ∞. In this paper, we generalize these two theorems to functions with values in a Hilbert space (Theorems 3.1 and 3.3).     

On the convergence of maximal monotone operators

We study the convergence of maximal monotone operators with the help of representations by convex functions. In particular, we prove the convergence of a sequence of sums of maximal monotone operators under a general qualification condition of the Attouch-Brezis type.

     

Completely monotone sequences and universally prestarlike functions

We introduce universally convex, starlike and prestarlike functions in the slit domain ℂ [1, ∞), and show that there exists a very close link to completely monotone sequences and Pick functions. S. R. and L. S. received partial support from FONDECYT (grants 1070269 and 7080064) and DGIP-UTFSM (grant 240954). S. R. acknowledges partial support also from the German-Israeli Foundation (grant G-809-234.6/2003) and a JSPS Grant-in-Aid for Scientific Research (A), 17204010. T. S. was partially supported by JSPS Grant-in-Aid for Scientific Research (B), 17340039.     

Graphical convergence of continuous functions

Let X and Y be metrizable spaces. We show that convergence of a net of continuous functions 〈f _λ 〉 to a continuous function f in the graph topology for C(X,Y) is equivalent to the uniform convergence of the net of associated distance functionals for the graphs with respect to each compatible metric on X×Y. Remarkably, no weaker convergence results if uniform convergence is replaced by pointwise convergence in the last statement.