Empirical spectral processes for stationary state space models

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.

In this note an analytic representation is given for continuous linear operators from C ( X ) C(X) into a linear normed space Y where C ( X ) C(X) is the space of continuous functions on [0, 1] with values in a linear normed space X.

In this paper, the chaotic dynamics of composition operators on the space of real-valued continuous functions is investigated. It is proved that the hypercyclicity, topologically mixing property, Devaney chaos, frequent hypercyclicity and the specification property of the composition operator are equivalent to each other and are stronger than dense distributional chaos. Moreover, the composition operator [Formula: see text] exhibits dense Li–Yorke chaos if and only if it is densely distributionally chaotic, if and only if the symbol [Formula: see text] admits no fixed points. Finally, the long-time behaviors of the composition operator with affine symbol are classified in detail.

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.

Confidence intervals for ARMA–GARCH Value-at-Risk: The case of heavy tails and skewness

On the universal completion of pointfree function spaces

On continuous selections for metric projections in spaces of continuous functions

Extension of operators on spaces of vector-valued continuous functions

Let K be a compact Hausdorff space and let E be a Banach space. We denote by C(K, E) the Banach space of all E-valued continuous functions defined on K, endowed with the supremum norm. We study in this paper Banach-Saks operators defined on C(K, E) spaces. We characterize these operators for a large class of compacts K (the scattered ones), or for a large class of Banach spaces E (the superreflexive ones). We also show by some examples that our theorems can not be extended directly.

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.

On the density of polyharmonic splines

We show that continuous bilinear forms on spaces of continuous functions can be approximated by norm attaining bilinear forms.

A system of fractional differential equations (FDEs) with fractional derivatives of diverse orders is called an incommensurate system of FDEs. In this paper, the well-posedness of the initial value problem for incommensurate systems of FDEs is obtained on the space of continuous functions. Three different methods for this analysis are used and compared. The complexity of such analysis is reduced by new techniques. Strong existence results are obtained by weaker conditions. The uniqueness and the continuous dependency of the solution on initial values are investigated using the Gronwall inequality.

Differential equations with discrete state-dependent delay: Uniqueness and well-posedness in the space of continuous functions