Continuous Real-valued Functions Research Articles

Maximum Boundaries for Cones of Continuous Functions on a Compact Space and Integral Representations for Linear Functionals

We present a simplified and easily accessible approach to the integral representation for continuous linear functionals on a cone of continuous real-valued functions on a compact set. The measures defining these integrals are supported by the maximum boundary of the respective cones.

Global convergence of discrete-time inhomogeneous Markov processes from dynamical systems perspective

Given the continuous real-valued objective function f and the discrete time inhomogeneous Markov process Xt defined by the recursive equation of the form Xt+1=Tt(Xt,Yt), where Yt is an independent sequence, we target the problem of finding conditions under which the Xt converges towards the set of global minimums of f. Our methodology is based on the Lyapunov function technique and extends the previous results to cover the case in which the sequence f(Xt) is not assumed to be a supermartingale. We provide a general convergence theorem. An application example is presented: the general result is applied to the Simulated Annealing algorithm.

The bi-compact-open topology on C(X)

For the set C(X) of real-valued continuous functions on a Tychonoff space X, the compact-open topology on C(X) is a “set-open topology”. This paper studies the separation and countability properties of the space C(X) having the topology given by the join of the compact-open topology and an “open-set topology” called the open-point topology, that was introduced in Jindal et al. (Topol. Appl. 187: 62–74, 2015).

On Filtered Polynomial Approximation on the Sphere

This paper considers filtered polynomial approximations on the unit sphere $\mathbb{S}^d\subset \mathbb{R}^{d+1}$, obtained by truncating smoothly the Fourier series of an integrable function $f$ with the help of a "filter" $h$, which is a real-valued continuous function on $[0,\infty)$ such that $h(t)=1$ for $t\in[0,1]$ and $h(t)=0$ for $t\ge2$. The resulting "filtered polynomial approximation" (a spherical polynomial of degree $2L-1$) is then made fully discrete by approximating the inner product integrals by an $N$-point cubature rule of suitably high polynomial degree of precision, giving an approximation called "filtered hyperinterpolation". In this paper we require that the filter $h$ and all its derivatives up to $\lfloor\tfrac{d-1}{2}\rfloor$ are absolutely continuous, while its right and left derivatives of order $\lfloor \tfrac{d+1}{2}\rfloor$ exist everywhere and are of bounded variation. Under this assumption we show that for a function $f$ in the Sobolev space $W^s_p(\mathbb{S}^d),\ 1\le p\le \infty$, both approximations are of the optimal order $ L^{-s}$, in the first case for $s>0$ and in the second fully discrete case for $s>d/p$.

The $\sigma$-property in $C(X)$

The $\sigma$-property of a Riesz space (real vector lattice) $B$ is: For each sequence $\{b_{n}\}$ of positive elements of $B$, there is a sequence $\{\lambda_{n}\}$ of positive reals, and $b\in B$, with $\lambda_{n}b_{n}\leq b$ for each $n$. This condition is involved in studies in Riesz spaces of abstract Egoroff-type theorems, and of the countable lifting property. Here, we examine when ``$\sigma$'' obtains for a Riesz space of continuous real-valued functions $C(X)$. A basic result is: For discrete $X$, $C(X)$ has $\sigma$ iff the cardinal $|X|< \mathfrak{b}$, Rothberger's bounding number. Consequences and generalizations use the Lindelöf number $L(X)$: For a $P$-space $X$, if $L(X)\leq \mathfrak{b}$, then $C(X)$ has $\sigma$. For paracompact $X$, if $C(X)$ has $\sigma$, then $L(X)\leq \mathfrak{b}$, and conversely if $X$ is also locally compact. For metrizable $X$, if $C(X)$ has $\sigma$, then $X$ \textit{is} locally compact.

A metrizable X with C p (X) not homeomorphic to C p (X) × C p (X)

We give an example of an infinite metrizable space $X$ such that the space $C_p(X)$, of continuous real-valued function on $X$ endowed with the pointwise topology, is not homeomorphic to its own square $C_p(X)\times C_p(X)$. The space $X$ is a zero-dimensional subspace of the real line. Our result answers a long-standing open question in the theory of function spaces posed by A.V. Arhangel'skii.

Statistical e-Convergence of Double Sequences and its Application to Korovkin-Type Approximation Theorem for Functions of Two Variables

In this paper, we have introduced the concept of statistical e-convergence and proved some fundamental properties of statistical e-convergence. In addition, we have introduced strongly \(e\texttt {-}|C_{10}|\)-summability and examined the relationship between statistical e-convergence and strongly \(e\texttt {-}|C_{10}|\)-summability of double sequences. Finally, we investigate a Korovkin-type approximation theorem for double sequences of positive linear operators on the space of all continuous real-valued functions defined on any compact subset of the real two-dimensional space. Then we display an application which shows that our new result is stronger than its classical version.

Existence of Continuous Functions That Are One-to-One Almost Everywhere

It is shown that given a metric space $X$ and a $\sigma$-finite positive regular Borel measure $\mu$ on $X$, there exists a bounded continuous real-valued function on $X$ that is one-to-one on the complement of a set of $\mu$ measure zero.

On the uniform approximation of Cauchy continuous functions

In the context of real-valued functions defined on metric spaces, it is known that the locally Lipschitz functions are uniformly dense in the continuous functions and that the Lipschitz in the small functions – the locally Lipschitz functions where both the local Lipschitz constant and the size of the neighborhood can be chosen independent of the point – are uniformly dense in the uniformly continuous functions. Between these two basic classes of continuous functions lies the class of Cauchy continuous functions, i.e., the functions that map Cauchy sequences in the domain to Cauchy sequences in the target space. Here, we exhibit an intermediate class of Cauchy continuous locally Lipschitz functions that is uniformly dense in the real-valued Cauchy continuous functions. In fact, our result is valid when our target space is an arbitrary Banach space.

Scale transformations for present position-dependent conditional expectations over continuous paths

Let C[0,t] denote a generalized Wiener space, the space of real-valued continuous functions on the interval [0,t], and define a random vector Zn:C[0,t]→Rn by Zn(x)=(∫0t1h(s)dx(s),…,∫0tnh(s)dx(s)), where 0<t1<⋯<tn=t is a partition of [0,t] and h∈L2[0,t] with h≠0 almost everywhere. Using a simple formula for a generalized conditional Wiener integral on C[0,t] with the conditioning function Zn, we evaluate the generalized analytic conditional Wiener and Feynman integrals of the cylinder function G(x)=f((e,x))ϕ((e,x)) for x∈C[0,t], where f∈Lp(R)(1≤p≤∞), e is a unit element in L2[0,t], and ϕ is the Fourier transform of a measure of bounded variation over R. We then express the generalized analytic conditional Feynman integral of G as two kinds of limits of nonconditional generalized Wiener integrals with a polygonal function and cylinder functions using a change-of-scale transformation. The choice of a complete orthonormal subset of L2[0,t] used in the transformation is independent of e.

On the prime ideals of C[0,1

It is well known that the set C[0, 1] of real-valued continuous functions on the closed interval [0, 1] has a natural ring structure. Its maximal ideals are known to be points. Here we show that although there exist prime ideals which are not maximal, the zero set is again a singleton. In particular, each prime ideal is contained in a unique maximal ideal.

Computation of the output of a function with fuzzy inputs based on a low-rank tensor approximation

We apply a derivative-free optimization method based on novel low-rank tensor methods to the problem of propagating fuzzy uncertainty through a continuous real-valued function. Adhering to Zadeh's extension principle, such a problem can be reformulated as a sequence of optimization problems over nested search spaces. The optimization method we use is based on a low-rank tensor approximation of the function sampled on a grid and a search for the minimal and maximal entries in this low-rank tensor. In contrast to classical fuzzy uncertainty propagation methods, such as the vertex method and the transformation method, the method we propose does not exhibit an inherent exponential scaling for increasing dimension of the search space. Obviously, no derivative-free optimization algorithm can exist which shows sub-exponential scaling with the dimension for all possible continuous functions. The algorithm that we present here, however, can exploit a specific type of structure and regularity (beyond continuity) that is often present in real-world optimization problems. We illustrate this with some high-dimensional numerical examples where the presented method clearly outperforms some established derivative-free optimization codes.

Tauberian conditions for the $$(C, \alpha )$$ ( C , α ) integrability of functions

For a real-valued continuous function f(x) on \([0,\infty )\), we define $$\begin{aligned} s(x)=\int _{0}^{x} f(u)du\quad \text {and}\quad \sigma _{\alpha } (x)= \int _{0}^{x}\left( 1-\frac{u}{x}\right) ^{\alpha }f(u)du \end{aligned}$$ for \(x>0\). We say that \(\int _{0}^{\infty } f(u)du\) is \((C, \alpha )\) integrable to L for some \(\alpha >-1\) if the limit \(\lim _{x \rightarrow \infty } \sigma _{\alpha } (x)=L\) exists. It is known that \(\lim _{x \rightarrow \infty } s(x) =L\) implies \(\lim _{x \rightarrow \infty }\sigma _{\alpha } (x) =L\) for all \(\alpha >-1\). The aim of this paper is twofold. First, we introduce some new Tauberian conditions for the \((C, \alpha )\) integrability method under which the converse implication is satisfied, and improve classical Tauberian theorems for the \((C,\alpha )\) integrability method. Next we give short proofs of some classical Tauberian theorems as special cases of some of our results.

On strongly separately continuous functions on sequence spaces

We study strongly separately continuous real-valued function defined on the Banach spaces ℓp. Determining sets for the class of strongly separately continuous functions on ℓp are characterized. We prove that for every 2≤α<ω1 there exists a strongly separately continuous function which belongs to the α'th Baire class and does not belong to the β'th Baire class on ℓp for β<α. We show that any open set in ℓp is the set of discontinuities of a strongly separately continuous real-valued function.

Criteria for the density of the graph of the entropy map restricted to ergodic states

We consider a non-uniquely ergodic dynamical system given by a$\mathbb{Z}^{l}$-action (or$(\mathbb{N}\cup \{0\})^{l}$-action)$\unicode[STIX]{x1D70F}$on a non-empty compact metrisable space$\unicode[STIX]{x1D6FA}$, for some$l\in \mathbb{N}$. Let (D) denote the following property: the graph of the restriction of the entropy map$h^{\unicode[STIX]{x1D70F}}$to the set of ergodic states is dense in the graph of$h^{\unicode[STIX]{x1D70F}}$. We assume that$h^{\unicode[STIX]{x1D70F}}$is finite and upper semi-continuous. We give several criteria in order that (D) holds, each of which is stated in terms of a basic notion: Gateaux differentiability of the pressure map$P^{\unicode[STIX]{x1D70F}}$on some sets dense in the space$C(\unicode[STIX]{x1D6FA})$of real-valued continuous functions on$\unicode[STIX]{x1D6FA}$, level-two large deviation principle, level-one large deviation principle, convexity properties of some maps on$\mathbb{R}^{n}$for all$n\in \mathbb{N}$. The one involving the Gateaux differentiability of$P^{\unicode[STIX]{x1D70F}}$is of particular relevance in the context of large deviations since it establishes a clear comparison with another well-known sufficient condition: we show that for each non-empty$\unicode[STIX]{x1D70E}$-compact subset$\unicode[STIX]{x1D6F4}$of$C(\unicode[STIX]{x1D6FA})$, (D) is equivalent to the existence of an infinite dimensional vector space$V$dense in$C(\unicode[STIX]{x1D6FA})$such that$f+g$has a unique equilibrium state for all$(f,g)\in \unicode[STIX]{x1D6F4}\times V\setminus \{0\}$; any Schauder basis$(f_{n})$of$C(\unicode[STIX]{x1D6FA})$whose linear span contains$\unicode[STIX]{x1D6F4}$admits an arbitrary small perturbation$(h_{n})$so that one can take$V=\text{span}(\{f_{n}+h_{n}:n\in \mathbb{N}\})$. Taking$\unicode[STIX]{x1D6F4}=\{0\}$, the existence of an infinite dimensional vector space dense in$C(\unicode[STIX]{x1D6FA})$constituted by functions admitting a unique equilibrium state is equivalent to (D) together with the uniqueness of the measure of maximum entropy.

Nonseparable closed vector subspaces of separable topological vector spaces

In 1983 P. Domanski investigated the question: For which separable topological vector spaces E, does the separable space Open image in new window have a nonseparable closed vector subspace, where \(\hbox {c}\) is the cardinality of the continuum? He provided a partial answer, proving that every separable topological vector space whose completion is not q-minimal (in particular, every separable infinite-dimensional Banach space) E has this property. Using a result of S.A. Saxon, we show that for a separable locally convex space (lcs) E, the product space Open image in new window has a nonseparable closed vector subspace if and only if E does not have the weak topology. On the other hand, we prove that every metrizable vector subspace of the product of any number of separable Hausdorff lcs is separable. We show however that for the classical Michael line \(\mathbb M\) the space of all continuous real-valued functions on \(\mathbb M\) endowed with the pointwise convergence topology, \(C_p(\mathbb M)\) contains a nonseparable closed vector subspace while \(C_p(\mathbb M)\) is separable.

Utilization of the Discrete Differential Evolution for Optimization in Multidimensional Point Clouds.

The Differential Evolution (DE) is a widely used bioinspired optimization algorithm developed by Storn and Price. It is popular for its simplicity and robustness. This algorithm was primarily designed for real-valued problems and continuous functions, but several modified versions optimizing both integer and discrete-valued problems have been developed. The discrete-coded DE has been mostly used for combinatorial problems in a set of enumerative variants. However, the DE has a great potential in the spatial data analysis and pattern recognition. This paper formulates the problem as a search of a combination of distinct vertices which meet the specified conditions. It proposes a novel approach called the Multidimensional Discrete Differential Evolution (MDDE) applying the principle of the discrete-coded DE in discrete point clouds (PCs). The paper examines the local searching abilities of the MDDE and its convergence to the global optimum in the PCs. The multidimensional discrete vertices cannot be simply ordered to get a convenient course of the discrete data, which is crucial for good convergence of a population. A novel mutation operator utilizing linear ordering of spatial data based on the space filling curves is introduced. The algorithm is tested on several spatial datasets and optimization problems. The experiments show that the MDDE is an efficient and fast method for discrete optimizations in the multidimensional point clouds.

Arhangel’skiĭ sheaf amalgamations in topological groups

We consider amalgamation properties of convergent sequences in topological groups and topological vector spaces. The main result of this paper is that, for arbitrary topological groups, Nyikos's property $\alpha_{1.5}$ is equivalent to Arhangel'ski\u{\i}'s formally stronger property $\alpha_1$. This result solves a problem of Shakhmatov (2002), and its proof uses a new perturbation argument. We also prove that there is a topological space $X$ such that the space $C_p(X)$ of continuous real-valued functions on $X$, with the topology of pointwise convergence, has Arhangel'ski\u{\i}'s property $\alpha_1$ but is not countably tight. This result follows from results of Arhangel'ski\u{\i}--Pytkeev, Moore and Todor\v{c}evi\'c, and provides a new solution, with remarkable properties, to a problem of Averbukh and Smolyanov (1968) concerning topological vector spaces. The Averbukh--Smolyanov problem was first solved by Plichko (2009), using Banach spaces with weaker locally convex topologies.

The Bohr–Pál theorem and the Sobolev space $W_2^{1/2}$

The well-known Bohr--P\'al theorem asserts that for every continuous real-valued function $f$ on the circle $\mathbb T$ there exists a change of variable, i.e., a homeomorphism $h$ of $\mathbb T$ onto itself, such that the Fourier series of the superposition $f\circ h$ converges uniformly. Subsequent improvements of this result imply that actually there exists a homeomorphism that brings $f$ into the Sobolev space $W_2^{1/2}(\mathbb T)$. This refined version of the Bohr--P\'al theorem does not extend to complex-valued functions. We show that if $\alpha<1/2$, then there exists a complex-valued $f$ that satisfies the Lipschitz condition of order $\alpha$ and at the same time has the property that $f\circ h\notin W_2^{1/2}(\mathbb T)$ for every homeomorphism $h$ of $\mathbb T$.

Nonlinear order isomorphisms on function spaces

Let X be a topological space. A subset of C(X), the space of continuous real-valued functions on X, is a partially ordered set in the pointwise order. Suppose that X and Y are topological spaces, and A(X) and A(Y ) are subsets of C(X) and C(Y ) respectively. We consider the general problem of characterizing the order isomorphisms (order preserving bijections) between A(X) and A(Y ). Under some general assumptions on A(X) and A(Y ), and when X and Y are compact Hausdorff, it is shown that existence of an order isomorphism between A(X) and A(Y ) gives rise to an associated homeomorphism between X and Y . This generalizes a classical result of Kaplansky concerning linear order isomorphisms between C(X) and C(Y ) for compact Hausdorff X and Y . The class of near vector lattices is introduced in order to extend the result further to noncompact spaces X and Y . The main applications lie in the case when X and Y are metric spaces. Looking at spaces of uniformly continuous functions, Lipschitz functions, little Lipschitz functions, spaces of differentiable functions, and the bounded, “local” and “bounded local” versions of these spaces, characterizations of when spaces of one type can be order isomorphic to spaces of another type are obtained.