Daniel Waterman

Approximately continuous transformations

1. An interesting class of real functions of a single real variable, the approximately continuous functions, was introduced by Denjoy [l] in his work on derivatives. The two principal facts discovered by Denjoy are that these functions are of Baire class 1 and have the Darboux property. Ridder [2] showed that the arguments of Denjoy apply to real functions of n variables. In this paper we discuss approximately continuous transformations from euclidean spaces into arbitrary metric spaces. We show that the image under such a transformation is always separable, that the transformation is of Baire class 1, and that it has a Darboux property. In §2 the Darboux property for real functions of a real variable is discussed. In §3 the notions of approximate continuity and metric density are defined in our context. In §4 it is shown that the image of En under an approximately continuous transformation is separable and such transformations are of Baire class 1. In §5 we introduce the notion of homogeneity of sets relative to metric density. En may be topologized by taking the homogeneous sets as open sets.1 We show that the open connected subsets of En are connected in this topology. The approximately continuous transformations are the continuous transformations in the new topology. In §6, we define ¿-regular sets, closed sets with connected interior and certain boundary restrictions. It is shown that approximately continuous transformations take irregular sets into connected sets, i.e., they have a Darboux property. Further remarks on connectivity are made in §7.

THE STRUCTURE OF REGULATED FUNCTIONS

It is shown that there is a nontrivial class of regulated functions each of which is a representable as the sum of a continuous function and a uniformly convergent series of jump functions whose jumps are those of the given function. The set of regulated functions is the union of the classes of functions of bounded -variation for convex

Basic sequences in the space of measurable functions

. The regulated functions on a closed interval are those functions whose right and left limits exist at each point. Every regulated function is bounded, has a countable set of discontinuities, and is the limit of a uniformly convergent sequence of step functions. The regulated functions are of importance in the theory of stochastic processes and in the theory of everywhere convergence of Fourier series. Functions of bounded variation are regulated, as are the functions of bounded 4>-variation (1) and the functions of bounded A-variation (3). In §1 we shall show that each regulated function is of bounded -variation for some 4>. A function of bounded variation has a canonical representation as the sum of a continuous function of bounded variation and a sum of jump functions, the jumps being those of the given function of bounded variation. In §2 we shall investigate the possibility that a regulated function have a representation as the sum of a continuous function and a uniformly convergent series of jump functions whose jumps are those of the given function. We shall see that, although we can characterize a nontrivial class of functions for which this representation is possible, we can construct functions which have no such representation. 1. Let (0) = 0,

FUNCTIONS WHOSE FOURIER SERIES CONVERGE FOR EVERY CHANGE OF VARIABLE

(x) > 0 for x > 0. A function / defined on an interval / is said to be of bounded

Estimating functions by partial sums of their Fourier series

- variation ( - BV) if the «^-variation of/,

Bounded variation in the mean

1. In a topological vector space X, a basic sequence [xn] is one whose finite linear combinations are dense in X. In a recent work, [l], A. A. Talalyan has observed that the space of measurable functions has a distinctly different character, with respect to the behavior of basic sequences, from, for example, the Lp spaces, p^l. A striking result of Talalyan is the fact that if { „} is basic, i.e., for every measurable , there are finite linear combinations of the , then if any finite number of functions is deleted from { „}, the remaining sequence is basic. This readily implies the existence of universal expansions, and the existence of a subsequence \<pnk\ which is basic even though the complement of the sequence {«*} is infinite. The proof given by Talalyan necessitates the use of considerable machinery from the theory of orthonormal systems in L2, and is quite involved. Our purpose is to show that the result follows almost immediately from the fact that the space M of measurable functions, with the topology of convergence in measure, has a trivial dual.

Convergence of double Fourier series with coefficients of generalized bounded variation

1. A theorem of Pal and Bohr, [l], [2], asserts that for every continuous/ of period 27r there is a homeomorphism g of [ — ir, ir] with itself such that the Fourier series of/og converges uniformly. Salem [3] has given a powerful test for the uniform convergence of a Fourier series. On the other hand, there is no criterion which gives necessary and sufficient conditions for the Fourier series of a continuous function to converge everywhere. In this note we show that the method of Salem may be used to determine a necessary and sufficient condition that a continuous function / be such that the Fourier series of / o g should converge everywhere for every homeomorphism g of [ — ir, ir] with itself. It is clear that this condition must be weaker than bounded variation since the continuous functions of bounded ^-variation with 4> = ^, p>l, have uniformly convergent Fourier series, and this class of functions is preserved by composition with homeomorphisms [3], [4].

Conjugate functions and the bohr—pal theorem

For classes of functions with convergent Fourier series. the problem of estimating the rate of convergence of the Fourier series has always been of interest. A classical theorem like that of Dirichlet and Jordan for functions of bounded variation (BV) assures the convergence of the Fourier series but gives no estimate of the rate of convergence. Such an estimate was recently provided by Bojanic [ 11. For certain classes of functions of generalized bounded variation the conclusions of the Dirichlet-Jordan theorem also hold. Waterman 15, 7 1 has shown that the class of functions of harmonic bounded variation (HBV) is, in a certain sense, the largest such class. An estimate of the rate of convergence of the Fourier series has been made for certain classes which lie between BV and HBV [2]. Here we consider this problem in greater generality for /IBV classes in that range to obtain a result which includes the previous estimates and allows us to make an estimate for a particular class which is closer to HBV than the classes previously con sidered. If f is a real valued function on the interval (a. 61 and .I = {A,,/ is a nondecreasing sequence of positive numbers such that x l/i,, diverges. we say that f is of /i-bounded variation (,4BV) if the sums

Segmentally Alternating Series

It is shown that the concept of bounded variation in the mean is not a meaningful generalization of ordinary bounded variation. In fact, it is a characterization of functions which differ from functions of bounded variation on a zero set. Let f be a real-valued function in L1 on the circle group T. We define the corresponding interval function by f (I) = f (b) -f (a), where I denotes the interval [a, b]. Let 0 = to < tj < ... < t4 = 27r be a partition of [0, 27r], and Ikx = [x + tkl-, X+tk]. If ,n Vm(f) =sSUP{j E If(Ikx) I dx} < oo, Tk=1 where the supremum is taken over all partitions, then f is said to be of bounded variation in the mean (or of bounded variation in the L1 norm). We denote the class of all functions which are of bounded variation in the mean by BVM. This concept was introduced by Moricz and Siddiqi [MS], who investigated the convergence in the mean of the partial sums of S[f], the Fourier series of f. If f is of bounded variation (f E BV) with variation V(f, T), then

On some high indices theorems III

Abstract We extend the class of double null sequences of complex numbers that are of bounded variation and prove the almost everywhere pointwise convergence as well as the convergence in the Lr(T2)-metric for 0