Building Blocks in Function Spaces

This chapter is about metric spaces, an abstract generalization of the real line that allows discussion of open and closed sets, limits, convergence, continuity, and similar properties. The usual distance function for the real line becomes an example of a metric. The other notions are defined in terms of the metric. The advantage of the generalization is that proofs of certain properties of the real line immediately go over to all other examples. Section 1 gives the definition of metric space and open set, and it lists a number of important examples, including Euclidean spaces and certain spaces of functions. Sections 2 through 4 develop properties of open and closed sets, continuity, and convergence of sequences that are simple generalizations of known facts about $\mathbb{R}$. Section 5 shows how a subset of a metric space can be made into a metric space so that the restriction of a continuous function from the whole space to the subset remains continuous. It also shows that three natural metrics for the product of two metric spaces lead to the same open sets, continuous functions, and convergent sequences. Section 6 shows that any metric space is “Hausdorff,” “regular,” and “normal,” and it goes on to exhibit three different countability hypotheses about a metric space as equivalent. A metric space with these properties is called “separable.” Section 7 concerns compactness and completeness. A metric space is defined to be “compact” if every open cover has a finite subcover. This property is equivalent to the condition that every sequence has a convergent subsequence. The Heine–Borel Theorem says that the compact sets of $\mathbb{R}^n$ are exactly the closed bounded sets. A number of the results early in Chapter I that were proved by the Bolzano–Weierstrass Theorem in the context of the real line are seen to extend to any compact metric space. A metric space is “complete” if every Cauchy sequence is convergent. A metric space is compact if and only if it is complete and “totally bounded.” Section 8 concerns connectedness, which is an abstraction of the property of an interval of the line that accounts for the Intermediate Value Theorem. Section 9 proves a fundamental result known as the Baire Category Theorem. A sample consequence of the theorem is that the pointwise limit of a sequence of continuous complex-valued functions on a complete metric space must have points where it is continuous. Section 10 studies the spaces of real-valued and complex-valued continuous functions on a compact metric space. A generalization of Ascoli's Theorem from the setting of Chapter I provides a characterization of compact sets in either of these spaces of continuous functions. A generalization of the Weierstrass Approximation Theorem, known as the Stone–Weierstrass Theorem, gives sufficient conditions for a subalgebra of either of these spaces of continuous functions to be dense. One consequence is that these spaces of continuous functions are separable. Section 11 constructs the “completion” of a metric space out of Cauchy sequences in the given space. The result is a complete metric space and a distance-preserving map of the given metric space into the completion such that the image is dense.

We prove that the existence of a solution operator for a convolution operator from the space of ultradifferntiable functions to the corresponding space of ultradistributions is equivalent to the existence of a continuous solution operator in the space of functions. Our results are in the spirit of a classical characterization of the surjectivity of convolution operators due to Hormander. The behaviour of a fixed convolution operator in different classes of ultradifferentiable functions of Beurling type concerning the existence of a continuous linear right inverse is also considered.

We investigate the two-point in time boundary value problem for the partial differential equations of the second-order with one spatial variable and constant coefficients. The problem is considered in in the spaces of functions which Fourier coefficients are characterized by exponential behavior on the Cartesian product of the time interval and spatial domain $\mathbb{R}/2\pi\mathbb{Z}$. The correct solvability of the problem is established, the formulas for solutions are presented, the kernel is described and the smoothness of the solution is established in the spaces of functions that are periodic in one spatial variable. We have established the conditions which are close to the necessary conditions of solvability of the problem in scale of spaces of functions with exponentially increasing (or decreasing) Fourier coefficients.We also found the asymptotic estimates demonstrating the absence of the problem of small denominators, which arises of many spatial variables and makes the boundary value problem incorrect. We have established sufficient conditions of the finite-dimensionality of the kernel of the problem and found upper bounds for its dimension. The results are obtained under the condition of minimum smoothness on the right-hand sides of two-point conditions, which is close to the necessary condition.

In due course we endow sets with particular properties and on the basis of these assumed properties construct a theory for special kinds of sets such as Hilbert spaces. In the development of this theory it is not necessary to appeal to the precise character of a set: the basic axioms, and the theorems that follow from these axioms, apply equally to sets whose members are numbers or matrices or functions. Before embarking on the task of describing this general framework, however, we first introduce two important examples of sets, or spaces (as they are usually called when endowed with additional properties) of functions: these are the spaces of continuous functions, and the L p spaces of functions whose pth powers are integrable. With these at our disposal it is possible in subsequent chapters to illustrate aspects of the general theory, using as special examples sets such as ℝ or ℝn which were introduced in the last chapter, as well as spaces of functions.KeywordsContinuous FunctionMeasurable FunctionLebesgue MeasureSimple FunctionMeasure ZeroThese 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.

Functional quantum theory is defined by an isomorphism of the state space H of a conventional quantum theory into an appropriate functional state space D It is a constructive approach to quantum theory in those cases where the state spaces H of physical eigenstates cannot be calculated explicitly like in nonlinear spinor field quantum theory. For the foundation of functional quantum theory appropriate functional state spaces have to be constructed which have to be representation spaces of the corresponding invariance groups. In this paper, this problem is treated for the spinor field. Using anticommuting source operator, it is shown that the construction problem of these spaces is tightly connected with the construction of appropriate relativistic function spaces. This is discussed in detail and explicit representations of the function spaces are given. Imposing no artificial restrictions it follows that the resulting functional spaces are indefinite. Physically the indefiniteness results from the inclusion of tachyon states. It is reasonable to assume a tight connection of these tachyon states with the ghost states introduced by Heisenberg for the regularization of the nonrenormalizable spinor theory

This chapter is devoted to some aspects of distribution theory. In Section 1.1 we define various spaces of smooth functions as well as some differential operators; we insist on an invariant definition of these objects, since this will be important in the study of N-body Hamiltonians. Various definitions and facts from distribution theory are reviewed in Section 1.2. The reader should be familiar with the contents of these two sections because we fix notations and terminology which are in some cases not quite standard. Sections 1.3 and 1.4 are more technical and may be skipped temporarily. We prove results relating local regularity properties of a distribution with the behaviour at infinity of its Fourier transform. We also establish an identity due to A. P. Calderon giving a representation of a distribution in terms of its derivatives of a fixed order plus a regular term. Finally, we use this representation in order to prove several facts that will be needed in later chapters.

Let L2 be the space of 2π-periodic square-summable functions and E(f, X)2 be the best approximation of f by the space X in L2. For n ∈ ℕ and B ∈ L2, let $${{\Bbb S}_{B,n}}$$ be the space of functions s of the form $$s\left( x \right) = \sum\limits_{j = 0}^{2n - 1} {{\beta _j}B\left( {x - \frac{{j\pi }}{n}} \right)} $$ . This paper describes all spaces $${{\Bbb S}_{B,n}}$$ that satisfy the exact inequality $$E{\left( {f,{S_{B,n}}} \right)_2} \leqslant \frac{1}{{^{{n^r}}}}\parallel {f^{\left( r \right)}}{\parallel _2}$$ . (2n–1)-dimensional subspaces fulfilling the same estimate are specified. Well-known inequalities are for approximation by trigonometric polynomials and splines obtained as special cases.

For a real semisimple group of matrices, every unitary principal series representation splits into inequivalent irreducible representations [5]. This multiplicity-one result was conjectured [7] in 1971 because one expected in general and knew in some special cases that all reducibility was accounted for by canonical geometric constructions, such as the imbedding of a space of analytic functions on the disc in the space of functions on the circle by passage to boundary values. We shall give an example to show that the corresponding multiplicity-one statement is false for a semisimple group of matrices defined over a locally compact, totally disconnected, nondiscrete field of characteristic 0. This example is summarized in §2, and its properties are verified in §5. It is ultimately motivated by the work of Langlands [11] on classification of irreducible admissible representations. Langlands [12] was able to give a formulation of the results of [8] that suggests that straightforward generalization of the multiplicityone theorem to other fields is not likely to succeed. An exposition of [12] is given in [9], and the way in which this work motivates our example is explained in § 6. Verification of the properties of our example depends on a suitable development of intertwining operators for split groups. Most of such a development has been carried out by Sally [15] and Winarsky [18], Some small modifications and elaborations of their work are the subject of § 3 and 4. The work in this paper grew out of conversations at the American Mathematical Society Summer Institute in 1977 with Langlands, Lusztig, Schiffmann, Shelstad, and Wallach. The work by I. Muller [13] on intertwining operators was also of influence; Muller developed a variation on Winarsky's work and was able to push through analogs of the results of [8]. In addition, she came close to discovering the example of § 2. We thank all these people* for their help. The results in this paper were announced in a talk at Hiroshima University on August 30, 1977, and later in Notices of the American Mathematical Society 25 (1978), abstract 753-G6, in connection with a meeting of the American Mathematical Society.

The Sturm-Liouville operator on the space of functions with discontinuity conditions

Both the theory of functions of a complex variable, and the approximation theory, are finding more and more applications in modern engineering sciences. In the present report we consider the spaces of analytic functions inside the unit circle satisfying on the boundary a strong Hölder condition for a rather wide class of the modulus of continuity. An analog of Smirnov's classical theorem relating to Hardy spaces has been obtained. The result is generalized to spaces of functions that are inductive limits of the spaces stated above.

A commutative Banach algebra is called symmetric if, regarded as a function algebra on its maximal ideal space, it is closed under complex conjugation. Varopoulos, [5], proved the asymmetry of the tensor algebra , where T is the unit circle. It is the object of this paper to prove the asymmetry of the Schur multipliers of the space , where m is the Lebesgue measure. In the second part of the paper we study the Hankel multipliers of the space and give an application to it.1. The asymmetry of. Let C(T) denote the space of continuous functions on T and A(T) be the space of those functions in C(T) that have absolutely convergent Fourier series. Consider the mapping F: C(T) → C(T × T) defined by F(f)(x, y) = f (x + y).

We examine translation-invariant measures on Banach spaces lp, where p ∈ [1,∞]. We construct analogs of the Lebesgue measure on Borel σ-algebras generated by the topology of pointwise convergence (σ-additive, invariant under shifts by arbitrary vectors, regular measures). We show that these measures are not σ-finite. We also study spaces of functions integrable with respect to measures constructed and prove that these spaces are not separable. We consider various dense subspaces in spaces of functions that are integrable with respect to a translation-invariant measure. We specify spaces of continuous functions, which are dense in the functional spaces considered. We discuss Borel σ-algebras corresponding to various topologies in the spaces lp, where p ∈ [1,∞]. For p ∈ [1,∞), we prove the coincidence of Borel σ-algebras corresponding to certain natural topologies in the given spaces of sequences and the Borel σ-algebra corresponding to the topology of pointwise convergence. We also verify that the space l∞ does not possess similar properties.

This chapter gives an introduction to the theory of normed linear spaces. A skeptical reader may wonder why this topic in pure mathematics is useful in applied mathematics. The reason is quite simple: Many problems of applied mathematics can be formulated as a search for a certain function, such as the function that solves a given differential equation. Usually the function sought must belong to a definite family of acceptable functions that share some useful properties. For example, perhaps it must possess two continuous derivatives. The families that arise naturally in formulating problems are often linear spaces. This means that any linear combination of functions in the family will be another member of the family. It is common, in addition, that there is an appropriate means of measuring the “distance” between two functions in the family. This concept comes into play when the exact solution to a problem is inaccessible, while approximate solutions can be computed. We often measure how far apart the exact and approximate solutions are by using a norm. In this process we are led to a normed linear space, presumably one appropriate to the problem at hand. Some normed linear spaces occur over and over again in applied mathematics, and these, at least, should be familiar to the practitioner. Examples are the space of continuous functions on a given domain and the space of functions whose squares have a finite integral on a given domain.KeywordsBanach SpaceLinear SpaceUnit BallCauchy SequenceNormed Linear SpaceThese 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.

For disposing numerous practical application problems involving expert systems, decision-making, image processing, classifications and etc, the investigations on the constructions and basic properties of n-ary aggregation operators (nAAOs) have always been a hot research topic with important research value and significance at theoretical investigations on aggregation operators (AOs). Herein, first, we propose a method for constructing nAAOs on function spaces via a family of known ones defined on a bounded poset, where those function spaces are composed by all fuzzy sets with that bounded poset as the truth values set. This method is different from the existing construction methods of nAAOs on bounded posets and provides a unified way of constructing usual nAAOs (like t-norms, uninorms, overlap functions, etc.) on function spaces via a family of known ones. Second, we present notion of representable nAAOs on function spaces and afford their equivalent characterization. Third, we discuss some vital properties of representable nAAOs on function spaces. Fourth, it is worth noticing that the obtained results cover the cases of nAAOs on function spaces composed of all interval-valued fuzzy sets and type-2 fuzzy sets when underlying bounded poset is taken as the corresponding truth values set, respectively. As a consequence, the theoretical results obtained herein have certain promotion and basic theoretical value for the mining of new potential applications of nAAOs in real problems, especially in expert systems, decision-making, image processing and etc.

The Geometric Theory of Analytic Functions was initially developed for the space of functions that are analytic in the unit disk. The convexity and starlikeness of functions are the first geometric ideas considered in this theory. We can notice a symmetry between the subjects considered in the space of analytic functions and those in the space of harmonic functions. In the presented paper, we consider the starlikeness of functions in the space of meromorphic harmonic functions.