The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Convergence of probability measures

Introduction to Probability and its Applications. By R. L. Scheaffer. ISBN 0 534 91970 7. PWS–Kent, Boston, 1990. 364 pp. £16.95.

Introduction to Probability and its Applications

In this paper we give sufficient conditions over the differentiability of a function to assure existence of the best local approximant in Lp-spaces, 0 < p ≤ ∞. These conditions are weaker than those given in previous papers. For p = 2 we show that, in a certain way, they are also necessary. In addition, we characterize the best local approximant.

Differentiability and best local approximation

We get results in Orlicz spaces L φ about best local approximation on non-balanced neighborhoods when φ satisfies a certain asymptotic condition. This fact generalizes known previous results in L p spaces.

Best Local Approximation in Orlicz Spaces

C^p Condition and the Best Local Approximation

In this paper we prove the existence of best multipoint local ||·||−approximation
to a function f from an N−dimensional space SN for a suitable integer N. This
problem is considered in an arbitrary Orlicz space for both the Luxemburg and the
Orlicz norms when some bits of data are more important than others. For this
purpose, we introduce the concept of || · ||−balanced integer.

Weighted best local || · ||−approximation in Orlicz spaces

We study the convergence of a net of subspaces generated by dilations of polynomials in a finite dimensional subspace. As a consequence, we extend the results given by Z´o and Cuenya [Advanced Courses of Mathematical Analysis II (Granada, 2004), 193–213, World Scientific, 2007] on a general approach to the problems of best vector-valued approximation on small regions from a finite dimensional subspace of polynomials.

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On convergence of subspaces generated by dilations of polynomials. An application to best local approximation

In this article, we introduce the τ condition, which is weaker than the L2 differentiability. If a function satisfies the τ condition on two points of ℝ, we prove the existence and characterization of the best local polynomial approximation on these points.

Best L2 Local Approximation on Two Small Intervals

We introduce two classes of functions, one containing the class of L2 differentiable functions, and another containing the class of L2 lateral differentiable functions. For functions in these new classes we prove existence of best local approximation at several points. Moreover, we get results about the asymptotic behavior of the derivatives of the net of best approximations, which are unknown even for L2 differentiable functions.

Claudia V. Ridolfi

Papers

Weighted best local || · ||−approximation in Orlicz spaces

Best Local Approximation in Orlicz Spaces

On convergence of subspaces generated by dilations of polynomials. An application to best local approximation

Best L2 Local Approximation on Two Small Intervals

An extension of best L 2 local approximation