Pointwise convergence, the answer

Fourier series approximations of continuous but nonperiodic functions on an interval suffer the Gibbs phenomenon, which means there is a permanent oscillatory overshoot in the neighborhoods of the endpoints. Fourier extensions circumvent this issue by approximating the function using a Fourier series that is periodic on a larger interval. Previous results on the convergence of Fourier extensions have focused on the error in the L^2 norm, but in this paper we analyze pointwise and uniform convergence of Fourier extensions (formulated as the best approximation in the L^2 norm). We show that the pointwise convergence of Fourier extensions is more similar to Legendre series than classical Fourier series. In particular, unlike classical Fourier series, Fourier extensions yield pointwise convergence at the endpoints of the interval. Similar to Legendre series, pointwise convergence at the endpoints is slower by an algebraic order of a half compared to that in the interior. The proof is conducted by an analysis of the associated Lebesgue function, and Jackson- and Bernstein-type theorems for Fourier extensions. Numerical experiments are provided. We conclude the paper with open questions regarding the regularized and oversampled least squares interpolation versions of Fourier extensions.

A large class of function spaces, under the topology of pointwise convergence, are shown to be of first category. The question as to when a space of continuous functions is of first category (i.e., can be written as a countable union of nowhere dense subsets) seems to be relatively unanswered. When the topology on the function space is the supremum metric topology, then if the range space is completely metrizable, so is the function space. Thus by the Baire category theorem, there is a large class of function spaces having the supremum metric topology which are Baire spaces (i.e., no open subspace is of first category). However, an example is given in L 4] of a metrizable Baire space Y such that the space of continuous functions from I, the closed unit interval, into Y, under the supremum metric topology, is of first category. If the domain space is compact, the supremum metric topology agrees with the compact-open topology on the function space, so there is also a large class of function spaces having the compact-open topology which are Baire spaces. However, the situation changes dramatically when the topology of pointwise convergence is imposed on the function spaces. Under this topology, the function spaces are of first category for most nonpathological domain and range spaces. For example, it will follow from the Theorem in this paper that the space of real-valued continuous functions on /, with the topology of pointwise convergence, is of first category. The notation C (X, Y) will stand for the space of all continuous functions from X into Y under the topology of pointwise convergence. This topology is generated by the base $ = I H L*;. V.]|x. e X and V. is open in YK Presented to the Society, January 23, 1975; received by the editors April 13, 1974. AMS (MOS) subject classifications (1970). Primary 54C35; Secondary 54D99.

On H-sober spaces and H-sobrifications of T0 spaces

For functions f1, . . . , fn on a set D, we characterize their linear independence with an invertible matrix from their values at n distinct points in D. With the matrix, the pointwise convergence of a sequence {gk} of functions in the span{f1, · · · , fn} is shown to be equivalent to those of the sequences of the coordinates of gks in the span. When fis are bounded, a pointwise convergent sequence {gk} must uniformly converge to a function in the span. It turns out that the limit of a convergent sequence {gk} inherits the continuity, differentiability, and integrability of fis. Furthermore the (pointwise or uniform) convergence of a sequence of solutions of an n-th order constant coefficients linear differential equation is completely determined by that of the sequence of relevant initial conditions.

The basic fuzzy wavelet type operators \(A_k; B_k; C_k; D_k; k \in {\mathbb Z}\) were studied in [14], [16], see also Chapters 12, 16, for their pointwise and uniform convergence with rates to the fuzzy unit operator. Also they were studied in [23], see also Chapter 13, in terms of estimating their fuzzy differences and giving their pointwise convergence with rates to zero.KeywordsCompact SupportPointwise ConvergenceFuzzy EnvironmentFuzzy EstimateErentiable FunctionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Let $T\subset\mathbb{R}$, $M$ be a metric space with metric $d$, and $M^T$ be the set of all functions mapping $T$ into $M$. Given $f\in M^T$, we study the properties of the approximate variation $\{V_\varepsilon(f)\}_{\varepsilon>0}$, where $V_\varepsilon(f)$ is the greatest lower bound of Jordan variations $V(g)$ of functions $g\in M^T$ such that $d(f(t),g(t))\le\varepsilon$ for all $t\in T$. The notion of $\varepsilon$-variation $V_\varepsilon(f)$ was introduced by Fra\'nkov\'a [Math. Bohem. 116 (1991), 20-59] for intervals $T=[a,b]$ in $\mathbb{R}$ and $M=\mathbb{R}^N$ and extended to the general case by Chistyakov and Chistyakova [Studia Math. 238 (2017), 37-57]. We prove directly the following basic pointwise selection principle: If a sequence of functions $\{f_j\}_{j=1}^\infty$ from $M^T$ is such that the closure in $M$ of the set $\{f_j(t):j\in\mathbb{N}\}$ is compact for all $t\in T$ and $\limsup_{j\to\infty}V_\varepsilon(f_j)$ is finite for all $\varepsilon>0$, then it contains a subsequence, which converges pointwise on $T$ to a bounded regulated function $f\in M^T$. We establish several variants of this result for sequences of regulated and nonregulated functions, for functions with values in reflexive separable Banach spaces, for the almost everywhere convergence and weak pointwise convergence of extracted subsequences, and comment on the necessity of assumptions in the selection principles. The sharpness of all assertions is illustrated by examples.

Fil: Medina, Juan Miguel. Consejo Nacional de Investigaciones Cientificas y Tecnicas. Oficina de Coordinacion Administrativa Saavedra 15. Instituto Argentino de Matematica Alberto Calderon; Argentina

<abstract><p>In this paper, we mainly study function spaces related to some kinds of weakly sober spaces, such as bounded sober spaces, $ k $-bounded sober spaces and weakly sober spaces. For $ T_{0} $ spaces $ X $ and $ Y $, it is proved that $ Y $ is bounded sober iff the function space $ {\bf{Top}}(X, Y) $ of all continuous functions $ f : X\longrightarrow Y $ equipped with the pointwise convergence topology is bounded sober iff $ {\bf{Top}}(X, Y) $ equipped with the Isbell topology is bounded sober. But for a $ k $-bounded sober space $ X $, the function space $ {\bf{Top}}(X, Y) $ equipped with the pointwise convergence topology or the Isbell topology may not be $ k $-bounded sober. It is shown that if the function space $ {\bf{Top}}(X, Y) $ equipped with the pointwise convergence topology or the Isbell topology is weakly sober (resp., a cut space), then $ Y $ is weakly sober (resp., a cut space). Relationships among some kinds of (weakly) sober spaces are also investigated.</p></abstract>

Introduction. Martingale theory for a linearly ordered index set in a finite measure space has been systematically discussed by Doob [4]. As convergence theory and system theory are the two main topics in martingale theory, Helms [6] has extended Doob's results on mean convergence to martingales in a finite measure space, indexed by directed sets, and Krickeberg [8; 9] extended the results of pointwise convergence and stochastic convergence to martingales in a a-finite measure space, indexed by directed sets. Krickeberg uses measure algebra and lattice theoretic methods, and in [8] makes the assumption in pointwise convergence that the martingales have some covering property, which is called Vo in his paper. Dieudonne [3] has given a martingale which is not pointwise convergent and has non-negative, bounded functions in a finite measure space, indexed by a countable directed set. Therefore, some additional conditions like Krickeberg's Vo to the usual ones seem to be necessary for pointwise convergence. However, the condition Vo is too weak to be the only additional condition needed for pointwise convergence of martingales indexed by noncountable directed sets. Some conditions such as terminal separability are needed. For the system theory, Bochner [2] has given some extensions to martingales indexed by directed sets, but without proofs. Some of his results are, unfortunately, not true. Snell [14] has defined regularity (which is closely related to system theorems) of martingales in a finite measure space, indexed by positive integers, and has given some sufficient conditions. He has not discussed the necessary conditions, nor the relation between regularity and the system theorems. In this paper, ?1 gives preliminary definitions and notation. ?2 is devoted to the definition of conditional expectation in a a-finite measure space W, and ?3 to that of martingales indexed by directed sets in W as well as to individual system theorems and inequalities. In ?4, a general pointwise convergence theorem is proved, and a new kind of convergence theorem, based on a differentiation theorem of interval functions [13, p. 192] is given. We

In the present article we study the monoid of continuous endomorphisms, in the topology of pointwise convergence, of a topological universlalgebra. Theorem 3.1 affirms that any semi-topological monoid (semigroup with unity) is isomorphic to a semigroup of continuous endomorphisms

Transformation monoids carry a canonical topology --- the topology of point-wise convergence. A closed transformation monoid $\mathfrak{M}$ is said to have automatic homeomorphicity with respect to a class $\mathcal{K}$ of structures, if every monoid-isomorphism of $\mathfrak{M}$ to the endomorphism monoid of a member of $\mathcal{K}$ is automatically a homeomorphism. In this paper we show automatic homeomorphicity-properties for the monoid of non-decreasing functions on the rationals, the monoid of non-expansive functions on the Urysohn space and the endomorphism-monoid of the countable universal homogeneous poset.

COROLLARY 2.4.For any space X, C π (X) is a Baire space if and only if C π (X) is of the second Baire category.

Let {mathbb {K}} be a non-trivially valued non-Archimedean complete field. Let ell _{infty }({mathbb {N}}, {mathbb {K}}) [ell _c({mathbb {N}}, {mathbb {K}});c_0({mathbb {N}}, {mathbb {K}})] be the space of all sequences in {mathbb {K}} that are bounded [relatively compact; convergent to 0] with the topology of pointwise convergence (i.e. with the topology induced from {mathbb {K}}^{{mathbb {N}}}). Let X be an infinite ultraregular space and let C_p(X,{mathbb {K}}) be the space of all continuous functions from X to {mathbb {K}} endowed with the topology of pointwise convergence. It is easy to see that C_p(X,{mathbb {K}}) is metrizable if and only if X is countable. We show that for any X [with an infinite compact subset] the space C_p(X,{mathbb {K}}) has an infinite-dimensional [closed] metrizable subspace isomorphic to c_0({mathbb {N}}, {mathbb {K}}). Next we prove that C_p(X,{mathbb {K}}) has a quotient isomorphic to c_0({mathbb {N}}, {mathbb {K}}) if and only if it has a complemented subspace isomorphic to c_0({mathbb {N}}, {mathbb {K}}). It follows that for any extremally disconnected compact space X the space C_p(X,{mathbb {K}}) has no quotient isomorphic to the space c_0({mathbb {N}}, {mathbb {K}}); in particular, for any infinite discrete space D the space C_p(beta D, {mathbb {K}}) has no quotient isomorphic c_0({mathbb {N}}, {mathbb {K}}). Finally we investigate the question for which X the space C_p(X,{mathbb {K}}) has an infinite-dimensional metrizable quotient. We show that for any infinite discrete space D the space C_p(beta D, {mathbb {K}}) has an infinite-dimensional metrizable quotient isomorphic to some subspace ell _c^0({mathbb {N}}, {mathbb {K}}) of {mathbb {K}}^{{mathbb {N}}}. If {mathbb {K}} is locally compact then ell _c^0({mathbb {N}}, {mathbb {K}})=ell _{infty }({mathbb {N}}, {mathbb {K}}). If |n1_{{mathbb {K}}}|ne 1 for some nin {mathbb {N}}, then ell _c^0({mathbb {N}}, {mathbb {K}})=ell _c ({mathbb {N}}, {mathbb {K}}). In particular, C_p(beta D, {mathbb {Q}}_q) has a quotient isomorphic to ell _{infty }({mathbb {N}}, {mathbb {Q}}_q) and C_p(beta D, {mathbb {C}}_q) has a quotient isomorphic to ell _c({mathbb {N}}, {mathbb {C}}_q) for any prime number q.

A SELECTION-THEORETIC APPROACH TO CERTAIN EXTENSION THEOREMS

Quasicontinuous functions and the topology of pointwise convergence