scholarly journals The Levi-Civita equation in function classes

2019 ◽  
Vol 94 (4) ◽  
pp. 689-701 ◽  
Author(s):  
Miklós Laczkovich
Keyword(s):  
Topological Semigroup ◽  
Classical Result ◽  
Continuous Functions ◽  
Exponential Polynomial ◽  
Additive Functions ◽  
Translation Invariant ◽  
Function Classes ◽  
Finite Dimensional ◽  
Special Cases ◽  
Complex Valued

Abstract Let G be an Abelian topological semigroup with unit. By a classical result (called Theorem A), if V is a finite dimensional translation invariant linear space of complex valued continuous functions defined on G, then every element of V is an exponential polynomial. More precisely, every element of V is of the form $$\sum _{i=1}^np_i \cdot m_i$$ ∑ i = 1 n p i · m i , where $$m_1 ,\ldots ,m_n$$ m 1 , … , m n are exponentials belonging to V, and $$p_1 ,\ldots ,p_n$$ p 1 , … , p n are polynomials of continuous additive functions. We generalize this statement by replacing the set of continuous functions by any algebra $${{\mathcal {A}}}$$ A of complex valued functions such that whenever an exponential m belongs to $${{\mathcal {A}}}$$ A , then $$m^{-1}\in {{\mathcal {A}}}$$ m - 1 ∈ A . As special cases we find that Theorem A remains valid even if the topology on G is not compatible with the operation on G, or if the set of continuous functions is replaced by the set of measurable functions with respect to an arbitrary $$\sigma $$ σ -algebra. We give two proofs of the result. The first is based on Theorem A. The second proof is independent, and seems to be more elementary than the existing proofs of Theorem A.

Higher Dimensional Spaces of Functions on the Spectrum of a Uniform Algebra

2007 ◽  
Vol 50 (1) ◽  
pp. 3-10
Author(s):  
Richard F. Basener
Keyword(s):  
Continuous Functions ◽  
Uniform Algebra ◽  
Finite Dimensional ◽  
Higher Dimensional ◽  
Spaces Of Continuous Functions ◽  
The Family ◽  
Continuous Complex ◽  
Finite Dimensional Case ◽  
Complex Valued ◽  
Shed Light

AbstractIn this paper we introduce a nested family of spaces of continuous functions defined on the spectrum of a uniform algebra. The smallest space in the family is the uniform algebra itself. In the “finite dimensional” case, from some point on the spaces will be the space of all continuous complex-valued functions on the spectrum. These spaces are defined in terms of solutions to the nonlinear Cauchy–Riemann equations as introduced by the author in 1976, so they are not generally linear spaces of functions. However, these spaces do shed light on the higher dimensional properties of a uniform algebra. In particular, these spaces are directly related to the generalized Shilov boundary of the uniform algebra (as defined by the author and, independently, by Sibony in the early 1970s).

Algebras of measures on C-distinguished topological semigroups

1978 ◽  
Vol 84 (2) ◽  
pp. 323-336 ◽  
Author(s):  
H. A. M. Dzinotyiweyi
Keyword(s):  
Total Variation ◽  
Topological Semigroup ◽  
Continuous Mapping ◽  
Continuous Functions ◽  
Compact Set ◽  
Locally Compact ◽  
Radon Measures ◽  
Topological Semigroups ◽  
Bounded Complex ◽  
Complex Valued

Let S be a (jointly continuous) topological semigroup, C(S) the set of all bounded complex-valued continuous functions on S and M (S) the set of all bounded complex-valued Radon measures on S. Let (S) (or (S)) be the set of all µ ∈ M (S) such that x → │µ│ (x-1C) (or x → │µ│(Cx-1), respectively) is a continuous mapping of S into ℝ, for every compact set C ⊆ S, and . (Here │µ│ denotes the measure arising from the total variation of µ and the sets x-1C and Cx-1 are as defined in Section 2.) When S is locally compact the set Ma(S) was studied by A. C. and J. W. Baker in (1) and (2), by Sleijpen in (14), (15) and (16) and by us in (3). In this paper we show that some of the results of (1), (2), (14) and (15) remain valid for certain non-locally compact S and raise some new problems for such S.

scholarly journals Intermediate rings of complex-valued continuous functions

2021 ◽  
Vol 22 (1) ◽  
pp. 47
Author(s):  
Amrita Acharyya ◽  
Sudip Kumar Acharyya ◽  
Sagarmoy Bag ◽  
Joshua Sack
Keyword(s):  
Algebraic Closure ◽  
Continuous Functions ◽  
Prime Ideals ◽  
Closed Field ◽  
Completely Regular ◽  
Closed Sets ◽  
Maximal Ideals ◽  
Special Cases ◽  
Necessary And Sufficient ◽  
Complex Valued

<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>

scholarly journals On Montel’s theorem in several variables

2015 ◽  
Vol 31 (1) ◽  
pp. 1-10
Author(s):  
J. M. ALMIRA ◽  
◽  
KH. F. ABU-HELAIEL ◽  
Keyword(s):  
Type Theorem ◽  
Invariant Subspaces ◽  
Real Variable ◽  
Translation Invariant ◽  
Finite Dimensional ◽  
Several Variables ◽  
Complex Functions ◽  
Montel’S Theorem ◽  
Complex Valued ◽  
Fréchet’S Functional Equation

Recently, the first author of this paper, used the structure of finite dimensional translation invariant subspaces of C(R, C) to give a new proof of classical Montel’s theorem, about continuous solutions of Frechet’s functional equation ∆m h f = 0, for real functions (and complex functions) of one real variable. In this paper we use similar ideas to prove a Montel’s type theorem for the case of complex valued functions defined over the discrete group Z d. Furthermore, we also state and demonstrate an improved version of Montel’s Theorem for complex functions of several real variables and complex functions of several complex variables.

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

2019 ◽  
Vol 6 (1) ◽  
pp. 228-264
Author(s):  
Hiroki Yagisita
Keyword(s):  
Cross Sections ◽  
Hausdorff Space ◽  
Banach Algebras ◽  
Continuous Functions ◽  
Continuous Family ◽  
Complex Manifolds ◽  
Finite Dimensional ◽  
Commutative Banach Algebras ◽  
Complex Valued ◽  
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.

scholarly journals A convolution semigroup of modular functions

1966 ◽  
Vol 6 (2) ◽  
pp. 251-255
Author(s):  
Y.-F. Lin
Keyword(s):  
Topological Semigroup ◽  
Present Note ◽  
Continuous Functions ◽  
Convolution Semigroup ◽  
Modular Functions ◽  
Star Topology ◽  
Borel Measures ◽  
Weak Star ◽  
Compact Topological Semigroup ◽  
Complex Valued

Let S be a compact topological semigroup, and let be the collection of all normalized non-negative Borel measures on S. It is well-known that , under convolution and the topology induced by the weak-star topology on the dual of the Benach space C(S) of all complex valued continuous functions on S, forms a compact topological semigroup which is known as the convolution semigroup of measures (see for instance, Glicksberg [3], Collins [1], Schwarz [5] and the author [4]). [1], Schwarz [5] and the author [4]). Professor A. D. Wallace asked if the process of forming the convolution semigroup of measures might be generalized to a more general class of set functions, the so-called “modular functions.” The purpose of the present note is to settle this question in the affirmative under a slight restriction. Before we are able to state the Wallace problem precisely, some preliminaries are necessary.

Some results in spectral synthesis

1965 ◽  
Vol 61 (2) ◽  
pp. 395-424 ◽  
Author(s):  
Robert J. Elliott
Keyword(s):  
Spectral Synthesis ◽  
Related Result ◽  
Convolution Equation ◽  
Continuous Functions ◽  
Exponential Polynomial ◽  
Space Of Continuous Functions ◽  
Translation Invariant ◽  
Famous Paper ◽  
Translation Invariant Subspace ◽  
Homogeneous Convolution Equation

For the group of real numbers R, an exponential monomial is defined as a function of the form xr(−ixz), for some non-negative integer r and some complex number z. Similarly, an exponential polynomial is a function P(x) exp (−ixz), for a polynomial P. In a now famous paper ((15)), Schwartz proved that every closed translation invariant subspace (variety) of the space of continuous functions on R is determined by the exponential monomials it contains. His techniques do not generalize to groups other than R as they use the theory of functions of one complex variable. A shorter proof of this result, using the Carleman transform of a function, was given by Kahane in his thesis ((9)). Ehrenpreis ((5)) proved results similar to those of Schwartz for certain varieties in the space of analytic functions of n complex variables, and Malgrange ((13)) proved the related result that any solution in ℰ(Rn) (for the notation see (16)) of the homogeneous convolution equation μ*f = 0, for some μ∈ℰ′, belongs to the closure of the exponential polynomial solutions of the equation.

scholarly journals Inversion of Tridiagonal Matrices Using the Dunford-Taylor’s Integral

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 870
Author(s):  
Diego Caratelli ◽  
Paolo Emilio Ricci
Keyword(s):  
Functional Analysis ◽  
Computer Algebra ◽  
Toeplitz Matrices ◽  
Test Cases ◽  
Recursive Procedure ◽  
Tridiagonal Matrices ◽  
Special Cases ◽  
The Matrix ◽  
Matrix Eigenvalues ◽  
Complex Valued

We show that using Dunford-Taylor’s integral, a classical tool of functional analysis, it is possible to derive an expression for the inverse of a general non-singular complex-valued tridiagonal matrix. The special cases of Jacobi’s symmetric and Toeplitz (in particular symmetric Toeplitz) matrices are included. The proposed method does not require the knowledge of the matrix eigenvalues and relies only on the relevant invariants which are determined, in a computationally effective way, by means of a dedicated recursive procedure. The considered technique has been validated through several test cases with the aid of the computer algebra program Mathematica©.

APPROXIMATIVE PROPERTIES OF ABEL–POISSON-TYPE OPERATORS ON THE GENERALIZED HÖLDER CLASSES

2021 ◽  
Vol 1 ◽  
pp. 76-83
Author(s):  
Yuri I. Kharkevich ◽  
◽  
Alexander G. Khanin ◽  
Keyword(s):  
Modulus Of Continuity ◽  
Applied Mathematics ◽  
Continuous Functions ◽  
Elliptic Type ◽  
Periodic Functions ◽  
French Mathematician ◽  
First Order ◽  
Function Classes ◽  
In The Beginning ◽  
Poisson Type

The paper deals with topical issues of the modern applied mathematics, in particular, an investigation of approximative properties of Abel–Poisson-type operators on the so-called generalized Hölder’s function classes. It is known, that by the generalized Hölder’s function classes we mean the classes of continuous -periodic functions determined by a first-order modulus of continuity. The notion of the modulus of continuity, in turn, was formulated in the papers of famous French mathematician Lebesgue in the beginning of the last century, and since then it belongs to the most important characteristics of smoothness for continuous functions, which can describe all natural processes in mathematical modeling. At the same time, the Abel-Poisson-type operators themselves are the solutions of elliptic-type partial differential equations. That is why the results obtained in this paper are significant for subsequent research in the field of applied mathematics. The theorem proved in this paper characterizes the upper bound of deviation of continuous -periodic functions determined by a first-order modulus of continuity from their Abel–Poisson-type operators. Hence, the classical Kolmogorov–Nikol’skii problem in A.I. Stepanets sense is solved on the approximation of functions from the classes by their Abel–Poisson-type operators. We know, that the Abel–Poisson-type operators, in partial cases, turn to the well-known in applied mathematics Poisson and Jacobi–Weierstrass operators. Therefore, from the obtained theorem follow the asymptotic equalities for the upper bounds of deviation of functions from the Hölder’s classes of order from their Poisson and Jacobi–Weierstrass operators, respectively. The obtained equalities generalize the known in this direction results from the field of applied mathematics.

scholarly journals Green's relations for regular elements of sandwich semigroups, II; semigroups of continuous functions

1978 ◽  
Vol 25 (1) ◽  
pp. 45-65 ◽  
Author(s):  
K. D. Magill ◽  
S. Subbiah
Keyword(s):  
Continuous Functions ◽  
Maximal Subgroups ◽  
Green’S Relations ◽  
Boolean Ring ◽  
Regular Elements ◽  
Ring Homomorphisms ◽  
Special Cases ◽  
Green's Relations ◽  
Sandwich Semigroup ◽  
Sandwich Semigroups

AbstractA sandwich semigroup of continuous functions consists of continuous functions with domains all in some space X and ranges all in some space Y with multiplication defined by fg = foαog where α is a fixed continuous function from a subspace of Y into X. These semigroups include, as special cases, a number of semigroups previously studied by various people. In this paper, we characterize the regular elements of such semigroups and we completely determine Green's relations for the regular elements. We also determine the maximal subgroups and, finally, we apply some of these results to semigroups of Boolean ring homomorphisms.

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