Continuous Complex-valued Functions Research Articles

Projective freeness and Hermiteness of complex function algebras

The paper studies projective freeness and Hermiteness of algebras of complex-valued continuous functions on topological spaces, Stein algebras, and commutative unital Banach algebras. New sufficient cohomology conditions on the maximal ideal spaces of the algebras are given that guarantee the fulfilment of these properties. The results are illustrated by nontrivial examples. Based on the Borsuk theory of shapes, a new class C of commutative unital complex Banach algebras is introduced (an analog of the class of local rings in commutative algebra) such that the projective tensor product with algebras in C preserves projective freeness and Hermiteness. Some examples of algebras of class C and of other projective free and Hermite function algebras are assembled. These include, e.g., Douglas algebras, finitely generated algebras of symmetric functions, Bohr-Wiener algebras, algebras of holomorphic semi-almost periodic functions, and algebras of bounded holomorphic functions on Riemann surfaces.

Construction of a Banach algebra in summable infinite matrices and its realization of the Gelfand transformation

In this paper, we extend the construction of a Banach Algebra from summable sequences to summable matrices. We also find the representation of the Banach algebra as complex-valued continuous functions defined on a Hausdorff space.

On local isometries between algebras of C(Y)-valued differentiable maps

Let K be either the real unit interval [0, 1] or the complex unit circle {mathbb {T}} and let C(Y) be the space of all complex-valued continuous functions on a compact Hausdorff space Y. We prove that the isometry group of the algebra C^1(K,C(Y)) of all C(Y)-valued continuously differentiable maps on K, equipped with the Sigma -norm, is topologically reflexive and 2-topologically reflexive whenever the isometry group of C(Y) is topologically reflexive.

On Harmonic Univalent Functions Involving (p,q)-Poisson Distribution Series

Harmonic functions are a classic title in the class of geometric functions. Many researchers have studied these function classes from past to present, and since it has a wide range of applications, it is still a popular class. In this study, we will examine harmonic univalent functions, a subclass of harmonic functions. In this study, a subclass of harmonic univalent functions will be examined. Let H denote the class of continuous complex-valued harmonic functions which are harmonic in the open unit disk U={z ϵ C∶|z| |g'(z)| (see [3]). Throughout this paper, we will use introductory notations and delineations of the (p, q)- calculus. The aim of the present paper is to find connections between (p,q)-starlike harmonic univalent functions involving (p,q)-Poisson distribution series.

On boundary-value problems for semi-linear equations in the plane

The study of the Dirichlet problem with arbitrary measurable data for harmonic functions in the unit disk 𝔻 is due to the dissertation of Luzin. Later on, the known monograph of Vekua was devoted to boundary-value problems only with Hölder continuous data for generalized analytic functions, i.e., continuous complex-valued functions f(z) of the complex variable z = x + iy with generalized first partial derivatives by Sobolev satisfying equations of the form $$ {\partial}_{\overline{z}}f+ af+b\overline{f}=c, $$ where the complexvalued functions a; b, and c are assumed to belong to the class Lp with some p > 2 in smooth enough domains D in ℂ. Our last paper [12] contained theorems on the existence of nonclassical solutions of the Hilbert boundaryvalue problem with arbitrary measurable data (with respect to logarithmic capacity) for generalized analytic functions f : D → ℂ such that $$ {\partial}_{\overline{z}}f=g $$ with the real-valued sources. On this basis, the corresponding existence theorems were established for the Poincaré problem on directional derivatives and, in particular, for the Neumann problem to the Poisson equations △U = G ∈ Lp; p > 2, with arbitrary measurable boundary data over logarithmic capacity. The present paper is a natural continuation of the article [12] and includes, in particular, theorems on the existence of solutions for the Hilbert boundary-value problem with arbitrary measurable data for the corresponding nonlinear equations of the Vekua type $$ {\partial}_{\overline{z}}f(z)=h(z)q\left(f(z)\right). $$ On this basis, existence theorems were also established for the Poincar´e boundary-value problem and, in particular, for the Neumann problem for the nonlinear Poisson equations of the form △U(z) = H(z)Q(U(z)) with arbitrary measurable boundary data over logarithmic capacity. The Dirichlet problem was investigated by us for the given equations, too. Our approach is based on the interpretation of boundary values in the sense of angular (along nontangential paths) limits that are a conventional tool of the geometric function theory. As consequences, we give applications to some concrete semi-linear equations of mathematical physics arising from modelling various physical processes. Those results can also be applied to semi-linear equations of mathematical physics in anisotropic and inhomogeneous media.

Topological reflexivity of isometries on algebras of C(Y)-valued Lipschitz maps

Let X be a compact metric space and let C(Y) be the space of all complex-valued continuous functions on a Hausdorff compact space Y. We prove that the isometry group of the algebra Lip(X,C(Y)) of all C(Y)-valued Lipschitz maps on X, equipped with the sum norm, is topologically reflexive and 2-topologically reflexive whenever the isometry group of C(Y) is topologically reflexive. The same results are established for the sets of isometric reflections and generalized bi-circular projections of Lip(X).

On the Univalence of Poly-analytic Functions

A continuous complex-valued function F in a domain $$D\subseteq \mathbf {C}$$ is poly-analytic of order $$\alpha $$ if it satisfies $$\partial ^{\alpha }F/\partial \overline{z}^{\alpha }=0$$ . One can show that F has the form $$F(z)=\sum _{k=0}^{\alpha -1}\overline{z}^{k}A_{k}(z)$$ , where each $$A_k$$ is an analytic function. In this paper, we prove the existence of a Landau constant for poly-analytic functions and the special bi-analytic case. We also establish Bohr’s inequality for poly-analytic and bi-analytic functions. In addition, we give an estimate for the arc-length over the class of poly-analytic mappings and consider the problem of minimizing moments of order p.

Intermediate rings of complex-valued continuous functions

<p>For a completely regular Hausdorff topological space X, let C(X, C) be the ring of complex-valued continuous functions on X, let C ∗ (X, C) be its subring of bounded functions, and let Σ(X, C) denote the collection of all the rings that lie between C ∗ (X, C) and C(X, C). We show that there is a natural correlation between the absolutely convex ideals/ prime ideals/maximal ideals/z-ideals/z ◦ -ideals in the rings P(X, C) in Σ(X, C) and in their real-valued counterparts P(X, C) ∩ C(X). These correlations culminate to the fact that the structure space of any such P(X, C) is βX. For any ideal I in C(X, C), we observe that C ∗ (X, C)+I is a member of Σ(X, C), which is further isomorphic to a ring of the type C(Y, C). Incidentally these are the only C-type intermediate rings in Σ(X, C) if and only if X is pseudocompact. We show that for any maximal ideal M in C(X, C), C(X, C)/M is an algebraically closed field, which is furthermore the algebraic closure of C(X)/M ∩C(X). We give a necessary and sufficient condition for the ideal CP (X, C) of C(X, C), which consists of all those functions whose support lie on an ideal P of closed sets in X, to be a prime ideal, and we examine a few special cases thereafter. At the end of the article, we find estimates for a few standard parameters concerning the zero-divisor graphs of a P(X, C) in Σ(X, C).</p>

Approximate Local Isometries on Spaces of Absolutely Continuous Functions

Let $$\mathrm {AC}(X)$$ be the Banach algebra of all absolutely continuous complex-valued functions f on a compact subset $$X\subset \mathbb {R}$$ with at least two points under the norm $$\left\| f\right\| _{\Sigma }=\left\| f\right\| _\infty +\mathrm {V}(f)$$ , where $$\mathrm {V}(f)$$ denotes the total variation of f. We prove that every approximate local isometry from $$\mathrm {AC}(X)$$ to $$\mathrm {AC}(Y)$$ admits a Banach–Stone type representation as an isometric weighted composition operator. Using this description, we prove that the set of linear isometries from $$\mathrm {AC}(X)$$ onto $$\mathrm {AC}(Y)$$ is algebraically reflexive and 2-algebraically reflexive. Moreover, it is shown that although the topological reflexivity and 2-topological reflexivity do not necessarily hold for the isometry group of $$\mathrm {AC}(X)$$ , but they hold for the sets of isometric reflections and generalized bi-circular projections of $$\mathrm {AC}(X)$$ .

On Paracompactness of the Space CLipk(Ω)

By the construction C^Lipk(Ω) we mean the space of all continuous complex-valued functions of class Lip^k, defined on an open subset Ω of Euclidean space ℝⁿ the present work, proved the paracompactness of the space C^Lipk(Ω), also many important results were discussed and proved.

Correcting for non-periodic behaviour in perturbative experiments: application to heat pulse propagation and modulated gas-puff experiments

This paper introduces a recent innovation in dealing with non-periodic behavior often referred to as transients in perturbative experiments. These transients can be the result from the unforced response due to the initial condition and other slow trends in the measurement data and are a source of error when performing and interpreting Fourier spectra. Fourier analysis is particularly relevant in system identification used to build feedback controllers and the analysis of various pulsed experiments such as heat pulse propagation studies. The basic idea behind the methodology is that transients are continuous complex-valued smooth functions in the Fourier domain which can be estimated from the Fourier data. Then, these smooth functions can be subtracted from the data such that only periodic components are retained. The merit of the approach is shown in two experimental examples, i.e. heat pulse propagation (core transport analysis) and radiation front movement due to gas puffing in the divertor. The examples show that the quality of the data is significantly improved such that it allows for new interpretation of the results even for non-ideal measurements.

KLOOSTERMAN PATHS OF PRIME POWERS MODULI, II

G. Ricotta and E. Royer (2018) have recently proved that the polygonal paths joining the partial sums of the normalized classical Kloosterman sums $S(a,b;p^n)/p^(n/2) converge in law in the Banach space of complex-valued continuous function on [0,1] to an explicit random Fourier series as (a,b) varies over (Z/p^nZ)^\times\times(Z/p^nZ)^\times, p tends to infinity among the odd prime numbers and n>=2 is a fixed integer. This is the analogue of the result obtained by E. Kowalski and W. Sawin (2016) in the prime moduli case. The purpose of this work is to prove a convergence law in this Banach space as only a varies over (Z/p^nZ)^\times, p tends to infinity among the odd prime numbers and n>=31 is a fixed integer.

Derivations and cohomologies of Lipschitz algebras

For a compact metric space (M, d), $${\mathrm {Lip}}M$$ denotes the Banach algebra of all complex-valued Lipschitz functions on (M, d). Motivated by a classical result of de Leeuw, we give a canonical construction of a compact Hausdorff space $${\hat{M}}$$ and a continuous surjection $$\pi :{\hat{M}} \rightarrow M$$ which may viewed as a metric analogue of the unit sphere bundle over a Riemannian manifold. It is shown that, for each $$n \ge 1$$ the continuous Hochschild cohomology $${\mathrm {H}}^{n}({\mathrm {Lip}}M, C({\hat{M}}))$$ has the infinite rank as a $${\mathrm {Lip}}M$$-module, if the metric space (M, d) admits a local geodesic structure, for example, if M is a compact Riemannian manifold or a non-positively curved metric space. Here $$C({\hat{M}})$$ denotes the algebra of all complex-valued continuous functions on $${\hat{M}}$$. On the other hand, if the coefficient $$C({\hat{M}})$$ is replaced with C(M), then it is shown that $${\mathrm {H}}^{1}({\mathrm {Lip}}M,C(M)) = 0$$ for each compact Lipschitz manifold M.

A Urysohn-type theorem and the Bishop-Phelps-Bollobás theorem for holomorphic functions

A Urysohn-type theorem is introduced for a subalgebra of the algebra Cb(Ω) of all bounded complex-valued continuous functions on a Hausdorff topological space Ω. With use of this theorem, it is shown that a type of the Bishop-Phelps-Bollobás theorem holds for certain classes of holomorphic functions on the unit ball of a complex Banach space X if X is either a locally uniformly convex space or a locally c-uniformly convex, order-continuous sequence space.

Angular and frequency behaviour of elastodynamic scattering from embedded scatterers

The elastodynamic scattering behaviour of a finite-sized scatterer in a homogeneous isotropic medium can be encapsulated in a scattering matrix (S-matrix) for each wave mode combination. In a 2-dimension (2D) space, each S-matrix is a continuous complex-valued function of 3 variables: incident wave angle, scattered wave angle and frequency. In this paper, the S-matrices for various 2D scatterer shapes (circular voids, straight cracks, rough cracks and a cluster of circular voids) are investigated to find general properties of their angular and frequency behaviour. For all these shapes, it is shown that the continuous data in the angular dimensions of their S-matrices can be represented to a prescribed level of accuracy by a finite number of complex-valued Fourier coefficients that are physically related to the angular orders of the incident and scattered wavefields. It is shown mathematically that the number of angular orders required to represent the angular dimensions of an S-matrix at a given frequency is a function of overall scatterer size to wavelength ratio, regardless of its geometric complexity. This can be interpreted as a form of the Nyquist sampling theorem and indicates that there is an upper bound on the sampling interval required in the angular domain to completely define an S-matrix. The variation of scattering behaviour with frequency is then examined. The frequency dependence of the S-matrix can be interpreted as the Fourier transform of the time-domain impulse response of the scatterer for each incident and scattering angle combination. Depending on the nature of the scatterer, these are typically decaying reverberation trains with no definite upper bound on their durations. Therefore, in contrast to the angular domain, there is no lower bound on the sampling interval in the frequency domain needed to completely define an S-matrix, although some pragmatic solutions are suggested. These observations may help for the direct problem (computing ultrasonic signals from known scatterers efficiently) and the inverse problem (characterising scatterers from measured ultrasonic signals).

Generalized Numerical Index of Function Algebras

Let X be a complex Banach space and Cb(Ω:X) be the Banach space of all bounded continuous functions from a Hausdorff space Ω to X, equipped with sup norm. A closed subspace A of Cb(Ω:X) is said to be an X-valued function algebra if it satisfies the following three conditions: (i) A≔{x⁎∘f:f∈A, x⁎∈X⁎} is a closed subalgebra of Cb(Ω), the Banach space of all bounded complex-valued continuous functions; (ii) ϕ⊗x∈A for all ϕ∈A and x∈X; and (iii) ϕf∈A for every ϕ∈A and for every f∈A. It is shown that k-homogeneous polynomial and analytic numerical index of certain X-valued function algebras are the same as those of X.

Tauberian conditions under which convergence follows from Cesàro summability of double integrals over R2

For a real- or complex-valued continuous function f over R2+:= [0,1) x [0,1), we denote its integral over [0,u] x [0,v] by s(u,v) and its (C,1, 1) mean, the average of s(u,v) over [0,u] x [0,v], by ?(u,v). The other means (C,1,0) and (C; 0; 1) are defined analogously. We introduce the concepts of backward differences and the Kronecker identities in different senses for double integrals over R2+. We give onesided and two-sided Tauberian conditions based on the difference between double integral of s(u, v) and its means in different senses for Ces?ro summability methods of double integrals over [0,u] x [0,v] under which convergence of s(u,v) follows from integrability of s(u,v) in different senses.

Passive Approximation and Optimization Using B-splines

A passive approximation problem is formulated where the target function is an arbitrary complex-valued continuous function defined on an approximation domain consisting of a finite union of closed ...

Finite-dimensional complex manifolds on commutative Banach algebras and continuous families of compact complex manifolds

AbstractLet Γ(M) be the set of all global continuous cross sections of a continuous family M of compact complex manifolds on a compact Hausdorff space X. In this paper, we introduce a C(X)-manifold structure on Γ(M). Especially, if X is contractible, then Γ(M) is a finite-dimensional C(X)-manifold. Here, C(X) denotes the Banach algebra of all complex-valued continuous functions on X.

Спектральный синтез на нульмерных локально компактных абелевых группах

Let G be a zero-dimensional locally compact Abelian group whose elements are compact, C(G) the space of continuous complex-valued functions on the group G. A closed linear subspace H⊆ C(G) is called invariant subspace, if it is invariant with respect to translations τ_y ∶ f(x) ↦ f(x + y), y ∈ G. We prove that any invariant subspace H admits spectral synthesis, which means that H coincides with the closure of the linear span of all characters of the group G contained in H.