Continuous Real-valued Functions Research Articles

Descriptive topology for analysts

We collect together old and recent research on Descriptive Topology, mainly related to the linear space $$C\left( X\right) $$ of real-valued continuous functions on a Tychonoff space X, equipped with the pointwise or with the compact-open topology.

On [formula omitted]-factorizable topological groups

A topological group G is called M-factorizable if for every continuous real-valued function f:G→R on G, there exist a continuous homomorphism φ of G onto a first-countable topological group H and a continuous real-valued function g on H such that f=g∘φ.It is shown that an R-factorizable topological group is exactly an M-factorizable ω-narrow topological group. It is also proved that a subgroup H of an M-factorizable group G is M-factorizable if and only if H is z-embedded in G.We show that for a P-group G, G is M-factorizable if and only if every continuous real-valued function on G is uniformly continuous and G is ω-balanced. We also show that for a locally compact group G, G is M-factorizable if and only if G is either metrizable or σ-compact.

Spectral representations of topological groups and near-openly generated groups

Near-openly generated groups are introduced. They form a topological and multiplicative subclass of -factorizable groups. Dense and open subgroups, quotients and the Raikov completion of a near-openly generated group are near-openly generated. Almost connected pro-Lie groups, Lindelöf almost metrizable groups and the spaces of all continuous real-valued functions on a Tychonoff space with pointwise convergence topology are near-openly generated. We provide characterizations of near-openly generated groups using methods of inverse spectra and topological game theory. Bibliography: 24 titles.

Simple Properties and Existence Theorem for the Henstock-Kurzweil-Stieltjes Integral of Functions Taking Values on C[a,b] Space-valued Functions

Henstock--Kurzweil integral, a nonabsolute integral, is a natural extension of the Riemann integral that was studied independently by Ralph Henstock and Jaroslav Kurzweil. This paper will introduce the Henstock--Kurzweil--Stieltjes integral of $\mathcal{C}[a,b]$-valued functions defined on a closed interval $[f,g]\subseteq\mathcal{C}[a,b]$, where $\mathcal{C}[a,b]$ is the space of all continuous real-valued functions defined on $[a,b]\subseteq\mathbb{R}$. Some simple properties of this integral will be formulated including the Cauchy criterion and an existence theorem will be provided.

Simple Properties and Existence Theorem for the Henstock-Kurzweil-Stieltjes Integral of Functions Taking Values on C[a,b] Space-valued Functions

Henstock--Kurzweil integral, a nonabsolute integral, is a natural extension of the Riemann integral that was studied independently by Ralph Henstock and Jaroslav Kurzweil. This paper will introduce the Henstock--Kurzweil--Stieltjes integral of $\mathcal{C}[a,b]$-valued functions defined on a closed interval $[f,g]\subseteq\mathcal{C}[a,b]$, where $\mathcal{C}[a,b]$ is the space of all continuous real-valued functions defined on $[a,b]\subseteq\mathbb{R}$. Some simple properties of this integral will be formulated including the Cauchy criterion and an existence theorem will be provided.

ON THE DECOMPOSITION OF CONTINUOUS FUNCTIONS AND DIMENSIONS

In this paper, we consider decomposition of continuous functions in [Formula: see text] in terms of Hausdorff dimension and lower box dimension. Precisely, we show that, given real numbers [Formula: see text], any real-valued continuous function in [Formula: see text] can be decomposed into a sum of two real-valued continuous functions each having a graph of Hausdorff dimension [Formula: see text] and lower box dimension [Formula: see text]. This generalizes a theorem of Wingren, also Wu and the present author. We also consider the arbitrary decomposition of continuous functions in terms of Hausdorff dimension and lower box dimension.

Realcompact and Lindelöf Extensions with Compact Remainder

An extension of a space X is a space Y which contains X as a dense subspace; $$Y{\setminus } X$$ is called the remainder of Y. In this paper, for a locally realcompact (resp., locally Lindelof) space X, we construct all realcompact (resp., Lindelof) extensions of X with compact remainder by means of some particular free ideals of C(X), the ring of all real-valued continuous functions on X. In fact, we are going to give a new approach different from that of similar works in the literature. For an ideal I of C(X), let $$\upsilon _IX$$ be the largest subspace of $$\beta X$$ on which every function in I can be extended continuously. We observe that every realcompact (resp., Lindelof) extension Y of X with compact remainder is precisely of the form $${\bar{i}}(\upsilon _IX)$$ , for a free ideal I contained in the ideal $$C_\mathbb {R}(X)$$ (resp., $$C_L(X)$$ ) consisting of all functions in C(X) with realcompact (resp., Lindelof) cozero-sets, where $${\bar{i}}:\beta X\rightarrow \beta Y$$ is the Stone extension of the identity mapping $$i:X\rightarrow X\subseteq \beta Y$$ . It turns out that locally realcompact (resp., locally Lindelof) spaces are the only spaces which have realcompact (resp., Lindelof) extensions with compact remainder. We conclude by constructing the largest (with respect to the standard partial order) one-point realcompact (resp., Lindelof) extension.

Fractal approximation of a function from a countable sample set and associated fractal operator

In the literature of fractal approximation theory, the notion of fractal interpolation function is used to construct a family of fractal functions, called the $$\alpha $$-fractal functions, corresponding to a fixed real-valued continuous function on a compact interval. The said $$\alpha $$-fractal function that simultaneously interpolates and approximates a prescribed continuous function is established using a finite set of sampled values of the original function. Closer in this spirit, this note aims to enunciate the fractal approximate reconstruction of a continuous function from a countably infinite sample set. Some elementary properties of the fractal operator that maps the given Lipschitz continuous function to the reconstructed fractal counterpart is expounded from the view point of perturbation theory of operators. This may act as a first step to perceive the notion of fractal operator beyond the familiar terrain of bounded linear operators. Some sidelights are also presented.

A remark on the ring of integer-valued continuous functions in pointfree topology

This paper presents the counterpart of a recent result by O. Ighedo concerning the ring RL of real-valued continuous functions on a completely regular frame L (Ighedo, 2018 [3]) for the ring ZL of integer-valued continuous functions on a 0-dimensional frame L.

On the maximal regular ideal of pointfree function rings, and more

Let L be a completely regular frame and, as customary, let RL denote the ring of continuous real-valued functions on L. In the first part of the paper we characterize the maximal regular ideal of RL. We show that it consists precisely of the functions α such that the open sublocale of L associated with coz α is clopen and is a P-frame. We also give a characterization of this ideal in terms of the notion of the localic remainder. In contrast, the maximal regular ideal of the ring ZL of continuous integer-valued functions on L is shown to be always the zero ideal. Purity and regularity of ideals are germane in the description and characterization of the maximal regular ideal of RL. It is for this reason that in the last part of the paper we consider ideals of ideals of RL because, as we prove, it is precisely the pure ideals of RL all of whose ideals are ideals in the whole ring.

Dunford–Pettis type properties and the Grothendieck property for function spaces

For a Tychonoff space X, let $$C_k(X)$$ and $$C_p(X)$$ be the spaces of real-valued continuous functions C(X) on X endowed with the compact-open topology and the pointwise topology, respectively. If X is compact, the classic result of A. Grothendieck states that $$C_k(X)$$ has the Dunford–Pettis property and the sequential Dunford–Pettis property. We extend Grothendieck’s result by showing that $$C_k(X)$$ has both the Dunford–Pettis property and the sequential Dunford–Pettis property if X satisfies one of the following conditions: (1) X is a hemicompact space, (2) X is a cosmic space (= a continuous image of a separable metrizable space), (3) X is the ordinal space $$[0,\kappa )$$ for some ordinal $$\kappa $$ , or (4) X is a locally compact paracompact space. We show that if X is a cosmic space, then $$C_k(X)$$ has the Grothendieck property if and only if every functionally bounded subset of X is finite. We prove that $$C_p(X)$$ has the Dunford–Pettis property and the sequential Dunford–Pettis property for every Tychonoff space X, and $$C_p(X) $$ has the Grothendieck property if and only if every functionally bounded subset of X is finite.

Strongly sequentially separable function spaces, via selection principles

A separable space is strongly sequentially separable if, for each countable dense set, every point in the space is a limit of a sequence from the dense set. We consider this and related properties, for the spaces of continuous and Borel real-valued functions on Tychonoff spaces, with the topology of pointwise convergence. Our results solve a problem stated by Gartside, Lo, and Marsh.

Natural extension of EF-spaces and EZ-spaces to the pointfree context

Abstract Two problems concerning EF-frames and EZ-frames are investigated. In [Some new classes of topological spaces and annihilator ideals, Topology Appl. 165 (2014), 84–97], Tahirefar defines a Tychonoff space X to be an EF (resp., EZ)-space if disjoint unions of clopen sets are completely separated (resp., every regular closed subset is the closure of a union of clopen subsets). By extending these notions to locales, we give several characterizations of EF and EZ-frames, mostly in terms of certain ring-theoretic properties of 𝓡 L, the ring of real-valued continuous functions on L. We end by defining a qsz-frame which is a pointfree context of qsz-space and, give a characterization of these frames in terms of rings of real-valued continuous functions on L.

Further thoughts on the ring $${\mathcal {R}}_c (L)$$ in frames

Let C(X) denote the ring of all real-valued continuous functions on a topological space X and $${\mathcal {R}} (L)$$ be as the pointfree topology version of C(X), i.e., the ring of real-valued continuous functions on a frame L. The ring $${\mathcal {R}}_c (L)$$ is introduced as a sub-f-ring of $${\mathcal {R}} ( L)$$ as a pointfree analogue to the subring $$C_c(X)$$ of C(X) consisting of elements with the countable image. In this paper, we will study the concept of pointfree countable image in a way which will enable us to study the ring $${\mathcal {R}}_c (L)$$. In order to do so we introduce the set $$R_{\alpha } := \{ r \in {\mathbb {R}} : {{\,\mathrm{coz}\,}}(\alpha - \mathbf{r}) \not = \top \} $$ for every $$\alpha \in {\mathcal {R}} (L)$$. We prove that $$R_{\alpha } $$ is a countable subset of $${\mathbb {R}}$$ for every $$\alpha \in {\mathcal {R}}_c (L)$$. Next, we show that if L is a compact frame, then $$R_{\alpha } $$ is a finite subset of $${\mathbb {R}}$$ for every $$\alpha \in {\mathcal {R}}_c (L)$$. Also, we study the result which says that for any topological space X there is a zero-dimensional space Y which is a continuous image of X and $$C_c(X) \cong C_c(Y )$$ in pointfree topology. Finally, we prove that, for some frame L, the ring $${\mathcal {R}}_c (L)$$ may not be isomorphic to $${\mathcal {R}} (M)$$, for any given frame M.

The contractivity of cone-preserving multilinear mappings

With the notion of the mode-j Birkhoff contraction ratio, we prove a multilinear version of the Birkhoff–Hopf and Perron–Fronenius theorems, which provide conditions on the existence and uniqueness of a solution to a large family of systems of nonlinear equations of the type , with xi being an element of a cone Ci in a Banach space . We then consider a family of nonlinear integral operators f i with positive kernel, acting on the product of spaces of continuous real-valued functions. In this setting we provide an explicit formula for the mode-j contraction ratio, which is particularly relevant in practice, as this type of operator plays a central role in numerous models and applications.

Optimal Cognitive Radio Spectrum Access With Joint Spectrum Sensing and Power Allocation

Spectrum access policy in cognitive radio (CR) systems is classically based on a binary-valued function, where the secondary users (SUs) can access the spectrum of a primary user (PU) with the probabilities ‘0’ or ‘1’, according the measured energy. In this letter, following the concept of probabilistic spectrum access , which assigns a probability for CR signal transmission over the spectrum of a PU, we show that the optimal spectrum access approach belongs to the family of real-valued continuous functions where the probability of accessing or not accessing the spectrum by SUs is a real number between zero and one. Moreover, we show that the spectrum access probability is always greater than or equal to the tolerable miss detection probability. For comparison, we consider the well-known interweave (i.e., no transmission when PU is present) and underlay (i.e., always transmission even when PU is present) approaches where the spectrum access probability is either equal to ‘0’ or ‘1’. Numerical results confirm that the proposed spectrum access outperforms the interweave and underlay schemes in terms of the achievable CR data rate.

Some Notes on z-Ideal and d-Ideal in $$\mathcal {R}L$$

As usual, the ring of continuous real-valued functions on a completely regular frame L is denoted by $$\mathcal {R}L$$. It is well known that an ideal Q in $$\mathcal {R}L$$ is a z-ideal if and only if $$\sqrt{Q}$$ is a z-ideal in which case $$Q=\sqrt{Q}$$. We show the same fact in d-ideal context and then it turns out that the sum of a primary ideal and a z-ideal (d-ideal) in $$\mathcal {R}L$$ which are not in a chain is a prime z-ideal (d-ideal). For an ideal (a non-regular ideal) Q in $$\mathcal {R}L$$, we characterize the greatest z-ideal (d-ideal) contained in Q and the smallest z-ideal (d-ideal) containing Q in terms of basic z-ideals (d-ideals). We characterize frames L for which prime z-ideals and d-ideals coincide in $$\mathcal {R}L$$, or equivalently the sum of any two ideals consisting entirely of zero-divisors consists entirely of zero-divisors. Some characterizations of the quasi F-frame, extremally disconnected frames and P-frames in terms of d-ideals are provided. Finally, we show that a reduced ring R is regular if and only if every prime d-ideal in R is maximal.

Annihilator-stability and unique generation in C(X)

Using the equivalence of unique generation and cleanness of [Formula: see text], we give affirmative answers to questions raised in [I. Kaplansky, Elementary divisors and modules, Trans. Amer. Math. Soc. 66 (1949) 464–491] and [D.D. Anderson et al., When are associates unit multiples? Rocky Mountain J. Math. 34 (2004) 811–828] for rings of real-valued continuous functions. In fact, we show that if [Formula: see text] is [Formula: see text] (uniquely generated) then [Formula: see text] is too, and [Formula: see text] is strongly associate if and only if [Formula: see text] is, where [Formula: see text]. We give topological characterizations of [Formula: see text] and [Formula: see text] (annihilator-stable) elements of [Formula: see text] for continuum spaces [Formula: see text] and using this, we observe that the product of two UG elements need not be [Formula: see text]. It is shown that the set of elements in [Formula: see text] which have stable range 1 and the set of [Formula: see text] elements of [Formula: see text] coincide and several examples are given which show that the set of [Formula: see text] elements, the set of [Formula: see text] elements and the set of clean elements of [Formula: see text] can differ. Finally, we characterize spaces [Formula: see text] for which every clean element of [Formula: see text] or every element of [Formula: see text] which has stable range 1 is [Formula: see text].

A Banach algebra of series of functions over paths

Let $C[0,T]$ denote the space of continuous real-valued functions on $[0,T]$. On the space $C[0,T]$, we introduce a Banach algebra of series of functions which are generalized Fourier-Stieltjes transforms of measures of finite variation on the product of simplex and Euclidean space. We evaluate analytic Feynman integrals of the functions in the Banach algebra which play significant roles in the Feynman integration theory and quantum mechanics.

Split continuity

In the context of functions between metric spaces, we introduce a class of functions that simultaneously provides characteristic answers to the following two questions: (1) What property must a function f between metric spaces X and Y have so that when it is followed by an arbitrary continuous real-valued function, the composition is either upper semicontinuous or lower semicontinuous at each point of X? (2) What property must a function f between metric spaces X and Y have so that there exists a second function g from X to Y such that the multifunction x↦{f(x),g(x)} is globally upper semicontinuous? Particular attention is given to the classical Thomae function and arbitrary characteristic functions as members of this class.