f-Konvergenz undf-Stetigkeit von Operatorenfolgen auf metrischen Räumen

Summary Equicontinuity of the operators of a given sequence of nonlinear operators is one of the necessary and sufficient conditions for continuous convergence of this sequence. This lemma, which is due to Rinow, gives a generalization of the Banach-Steinhaus-theorem. Hence it led to some generalizations of the Lax-Richt-myer-theory of difference approximations for initial value problems. The equicontinuity in this case correspondends to numerical stability. But often continuous convergence is a too strong demand in the theory of nonlinear numerical problems (for instance in the case of difference schemes for quasilinear partial differential equations), whereas a restriction to only pointwise convergence possibly leads to numerical instability. Therefore in this paper a set of definitions of convergence is considered lying between pointwise and continuous convergence. Sorts of continuity are described which are as characteristic for these kinds of convergence as equicontinuity for continuous convergence. As an numerical application we study the connection between the solution-depending stability and the sensitiveness to perturbations of difference schemes for quasilinear initial value problems.     

Constructive notions of equicontinuity

In the informal setting of Bishop-style constructive reverse mathematics we discuss the connection between the antithesis of Specker’s theorem, Ishihara’s principle BD-N, and various types of equicontinuity. In particular, we prove that the implication from pointwise equicontinuity to uniform sequential equicontinuity is equivalent to the antithesis of Specker’s theorem; and that, for a family of functions on a separable metric space, the implication from uniform sequential equicontinuity to uniform equicontinuity is equivalent to BD-N.      

On statistical exhaustiveness

We study statistical versions of several types of convergence of sequences of functions between two metric spaces. Special attention is devoted to statistical versions of recently introduced notions of exhaustiveness (Gregoriades and Papanastassiou (2008) [4]) and strong uniform convergence on a bornology (Beer and Levi (2009) [3]). We obtain a few results about the continuity of the statistical pointwise limit of a sequence of functions.     

Modifications of sequence selection principles

We study modifications of the sequence selection principles for real functions obtained by allowing other than continuous functions and/or by replacing the pointwise convergence by quasi-normal or discrete convergence. We show that for important families of real functions we obtain already known sequence selection principles. Replacing the pointwise convergence we obtain plenty of sequence selection principles and we try to find equivalences or implications among them. Finally we attempt to show that these modifications are useful tools for proofs of known results or their strengthening.     

Convergence of set-valued mappings: Equi-outer semicontinuity

The concept of equi-outer semicontinuity allows us to relate the pointwise and the graphical convergence of set-valued-mappings. One of the main results is a compactness criterion that extends the classical Arzelà-Ascolì theorem for continuous functions to this new setting; it also leads to the exploration of the notion of continuous convergence. Equi-lower semicontinuity of functions is related to the outer semicontinuity of epigraphical mappings. Finally, some examples involving set-valued mappings are re-examined in terms of the concepts introduced here.Research supported in part by a grant of the National Science Foundation.     

Generalized Dini theorems for nets of functions on arbitrary sets

We characterize the uniform convergence of pointwise monotonic nets of bounded real functions defined on arbitrary sets, without any particular structure. The resulting condition trivially holds for the classical Dini theorem. Our vector-valued Dini-type theorem characterizes the uniform convergence of pointwise monotonic nets of functions with relatively compact range in Hausdorff topological ordered vector spaces. As a consequence, for such nets of continuous functions on a compact space, we get the equivalence between the pointwise and the uniform convergence. When the codomain is locally convex, we also get the equivalence between the uniform convergence and the weak-pointwise convergence; this also merges the Dini-Weston theorem on the convergence of monotonic nets from Hausdorff locally convex ordered spaces. Most of our results are free of any structural requirements on the common domain and put compactness in the right place: the range of the functions.     

Convergence of sequences of semi-continuous functions

In this paper we investigate how three well-known modes of convergence for (real-valued) functions are related to one another. In particular, we consider order convergence, pointwise convergence and continuous convergence of sequences of nearly finite normal lower semi-continuous functions. There is a natural comparison to be made between the results we obtain for convergence of sequences of semi-continuous functions, and classic results on the convergence of sequences of measurable functions.     

Properties of functions generalized convex with respect to a WT-system

Continuity and convergence properties of functions, generalized convex with respect to a continuous weak Tchebysheff system, are investigated. It is shown that, under certain non-degeneracy assumptions on the weak Tchebysheff system, every function in its generalized convex cone is continuous, and pointwise convergent sequences of generalized convex functions are uniformly convergent on compact subsets of the domain. Further, it is proved that, with respect to a continuous Tchebysheff system, L_p-convergence to a continuous function, pointwise convergence and uniform convergence of a sequence of generalized convex functions are equivalent on compact subsets of the domain.     

Arzelà's Theorem and strong uniform convergence on bornologies

In 1883 Arzelà (1983/1984) [2] gave a necessary and sufficient condition via quasi-uniform convergence for the pointwise limit of a sequence of real-valued continuous functions on a compact interval to be continuous. Arzelà's work paved the way for several outstanding papers. A milestone was the P.S. Alexandroff convergence introduced in 1948 to tackle the question for a sequence of continuous functions from a topological space (not necessarily compact) to a metric space. In 2009, in the realm of metric spaces, Beer and Levi (2009) [10] found another necessary and sufficient condition through the novel notion of strong uniform convergence on finite sets. We offer a direct proof of the equivalence of Arzelà, Alexandroff and Beer-Levi conditions. The proof reveals the internal gear of these important convergences and sheds more light on the problem. We also study the main properties of the topology of strong uniform convergence of functions on bornologies, initiated in Beer and Levi (2009) [10].     

Uniform convergence of a sequence of functions at a point

It is well known that the uniform limit of a sequence of continuous real-valued functions defined on an interval I is itself continuous. However, if the convergence is pointwise, the limit function need not be continuous (take ? _n (x) = x ⁿ on [0, 1], for example). Boas has shown that the pointwise limit function of a sequence of continuous real-valued functions defined on the compact interval [a,b] is, nonetheless, continuous on a dense subset of [a,b]. In this paper, the notion of uniform convergence at a point is offered as an alternative to the Boas approach in establishing this and, consequently, other results. The arguments stay within the realm of a first proof course in classical mathematical analysis.     

Spaces of continuous functions over a Ψ-space

The Lindelöf property of the space of continuous real-valued continuous functions is studied. A consistent example of an uncountable Ψ-like space is constructed for which the space of continuous real-valued functions with the pointwise convergence topology is Lindelöf.     

WEAK CONVERGENCE OF HENSTOCK INTEGRABLE SEQUENCES

Some relationships between pointwise and weak convergence of a sequence of Henstock integrable functions are studied, In particular it is provided an example of a sequence of Henstock integrable functions whose pointwise limit is different from the weak one. By introducing an asymptotic version of the Henstock equiintegrability notion it is given a necessary and sufficient condition in order that a pointwisely convergent sequence of Henstock integrable functions is weakly convergent to its pointwise limit.     

The Carathéodory topology for multiply connected domains II

We continue our exposition concerning the Carathéodory topology for multiply connected domains which we began in [Comerford M., The Carathéodory topology for multiply connected domains I, Cent. Eur. J. Math., 2013, 11(2), 322–340] by introducing the notion of boundedness for a family of pointed domains of the same connectivity. The limit of a convergent sequence of n-connected domains which is bounded in this sense is again n-connected and will satisfy the same bounds. We prove a result which establishes several equivalent conditions for boundedness. This allows us to extend the notions of convergence and equicontinuity to families of functions defined on varying domains.     

Uniform Boundedness and Closed Graph Theorems for Convex Operators

We are concerned with convex operators mapping a convex subset of a certain topological vector space into an ordered topological vector space, whose positive cone is assumed to be normal. Under the appropriate topological assumptions, we prove the equicontinuity of every pointwise bounded family of continuous convex operators as well as the continuity of every closed convex operator at every algebraically interior point of the domain. We also show that some weak kind of monotonicity implies the continuity of a convex operator.     

Recurrence and Almost Periodicity in Transformation Groups

In [5] Gottschalk comments that an interesting question is to find conditions under which pointwise recurrence in a transformation group (X,T) implies pointwise almost periodicity. In this paper we show that if in a transformation group (X,T), the phase space X is connected locally compact and Hausdorff and T is generative and each point xεX is of P-characteristic 0 for each replete semi-group P in T, then the existence of a single recurrent point implies that (X,T) is pointwise recurrent and each orbit closure is minimal. This implies of course from a general result of Gottschalk and Hedlund that the transformation group is pointwise almost periodic. The condition P-characteristic 0 is weaker than pointwise equicontinuity even when the phase space is compact--see example (4.6) at the end.     

The convergence space of minimal USCO mappings

A convergence structure generalizing the order convergence structure on the set of Hausdorff continuous interval functions is defined on the set of minimal usco maps. The properties of the obtained convergence space are investigated and essential links with the pointwise convergence and the order convergence are revealed. The convergence structure can be extended to a uniform convergence structure so that the convergence space is complete. The important issue of the denseness of the subset of all continuous functions is also addressed.      

Homogeneous approximation property for continuous wavelet transforms

The homogeneous approximation property (HAP) for frames is useful in practice and has been developed recently. In this paper, we study the HAP for the continuous wavelet transform. We show that every pair of admissible wavelets possesses the HAP in L² sense, while it is not true in general whenever pointwise convergence is considered. We give necessary and sufficient conditions for the pointwise HAP to hold, which depends on both wavelets and functions to be reconstructed.     

POINTWISE CONVERGENCE FOR EXPANSIONS IN SPHERICAL MONOGENICS

We offer a new approach to deal with the pointwise convergence of FourierLaplace series on the unit sphere of even-dimensional Euclidean spaces. By using spherical monogenics defined through the generalized Cauchy-Riemann operator, we obtain the spherical monogenic expansions of square integrable functions on the unit sphere. Based on the generalization of Fueter＇s theorem inducing monogenic functions from holomorphic functions in the complex plane and the classical Carleson＇s theorem, a pointwise convergence theorem on the new expansion is proved. The result is a generalization of Carleson＇s theorem to the higher dimensional Euclidean spaces. The approach is simpler than those by using special functions, which may have the advantage to induce the singular integral approach for pointwise convergence problems on the spheres.     

Convergence for Sequences of Functions and an Egorov Type Theorem

For every ordinal <₁ we define a new type of convergence for sequences of functions (-uniform pointwise) which is intermediate between uniform and pointwise convergence. Using this type of convergence we obtain an Egorov type theorem for sequences of measurable functions.     

Quasicontinuous functions,minimal usco maps and topology of pointwise convergence

In [HOLá, Ľ.—HOLY, D.: Pointwise convergence of quasicontinuous mappings and Baire spaces, Rocky Mountain J. Math.] a complete answer is given, for a Baire space X, to the question of when the pointwise limit of a sequence of real-valued quasicontinuous functions defined on X is quasicontinuous. In [HOLá, Ľ.—HOLY, D.: Minimal USCO maps, densely continuous forms and upper semicontinuous functions, Rocky Mountain J. Math. 39 (2009), 545–562], a characterization of minimal USCO maps by quasicontinuous and subcontinuous selections is proved. Continuing these results, we study closed and compact subsets of the space of quasicontinuous functions and minimal USCO maps equipped with the topology of pointwise convergence. We also study conditions under which the closure of the graph of a set-valued mapping which is the pointwise limit of a net of set-valued mappings, is a minimal USCO map.