Robert E. Zink

Greedy approximation with respect to certain subsystems of the Walsh orthonormal system

In an article that appeared in 1967, J.J. Price has shown that there is a vast family of subsystems of the Walsh orthonormal system each of which is complete on sets of large measure. In the present work it is shown that the greedy algorithm, when applied to functions in L 1 [0, 1], is surprisingly effective for these nearly-complete families. Indeed, if Φ is such a subsystem of the Walsh system, then to each positive e, however small, there corresponds a Lebesgue measurable set E such that for every f, Lebesgue integrable on [0,1], the greedy approximants to f, associated with Φ, converge, in the L 1 norm, to an integrable function g that coincides with f on E.

Some ramifications of a theorem of Boas and Pollard concerning the completion of a set of functions in

About fifty years ago, R. P. Boas and Harry Pollard proved that an orthonormal system that is completable by the adjunction of a finite number of functions also can be completed by multiplying the elements of the given system by a fixed, bounded, nonnegative measurable function. In subsequent years, several variations and extensions of this theorem have been given by a number of other investigators, and this program is continued here. A mildly surprising corollary of one of the results is that the trigonometric and Walsh systems can be multiplicatively transformed into quasibases for L1[0, 1]. 1. A curious connection between disparate mathematical ideas occurs in the theory of complete systems of real–valued measurable functions. The first of these notions is due to Boas and Pollard [3], who showed that certain incomplete systems in L may be transformed into complete systems by multiplying the elements of the system by a (fixed) bounded, nonnegative measurable function. Subsequently, Talalyan [17], [18] proved that a system Φ is complete in measure on a set E if and only if, for every positive e, Φ is complete in L(Ee), where Ee is a measurable subset of E that has measure greater than |E| − e. In addition, Talalyan showed that if Φ has these properties, so also does any family obtained from Φ by deleting a finite number of its members. Later, Goffman and Waterman [6] gave a new proof of this theorem and observed that it is always possible to make certain infinite deletions from a system that is complete in measure so as to leave a residual system that also satisfies this condition. The somewhat surprising coincidence of these ideas was noted in [15], where the following result was established. Theorem A. Let E be a measurable set of finite, positive measure, and let Φ = {φn : n = 1, 2, . . . } be a subset of L(E). Then, the following conditions are equivalent: (BP) There exists a bounded, measurable function, m, such that {mφn : n = 1, 2, . . .} is complete in L(E); (M) Φ is complete in measure on E; Received by the editors March 8, 1995 and, in revised form, July 21, 1995. 1991 Mathematics Subject Classification. Primary 42B65, 42C15, 46B15, 41A30, 41A58.

On a theorem of Goffman concerning Schauder series

1. A classical question posed by Lusin asks whether it is possible to find for each measurable function defined on [0, 2ir] a corresponding trigonometric series, with coefficients converging to zero, that converges almost everywhere to the function. This question was answered affirmatively by Mensov [4] for real-valued functions, but the general question remains unanswered. Thus, it is of interest to inquire whether there be any Schauder bases for LP, p> l, with respect to which every measurable function has a pointwise representation. Although Talalyan and Arutyanyan [8] have shown that a prime candidate, the Haar system, does not have this property, Gundy [3 ] has proved that systems of the specified type do exist. The Schauder functions are total in each of the LP spaces, although they in no case constitute a basis. In [1], Goffman solved Lusins problem for this system by way of a sequence of careful estimates culminating in a construction of the required series. An interesting aspect of this work is that not all of the Schauder functions are required for the construction. Indeed, it is clear from a superficial examination of the arguments employed in [1] that any finite number of functions could be deleted from the system and the work carried through with no resulting essential modification of the demonstrations. In analogy with work of Talalyan [6], and Goffman and Waterman [2], it is appropriate to ask whether infinitely many Schauder functions could be discarded in such a way that the above-mentioned result of Goffman would remain in force for the abbreviated system. In the present note it is shown that this is the case, and a characterization is given of those subsystems for which the Goffman theorem holds. The result suggests that there may be lurking in the background some very general form of the Muintz theorem.

Subsystems of the Walsh orthogonal system whose multiplicative completions are quasibases for ^{{}}[0,1], 1≤<+∞

If one discards some of the elements from the Walsh family, an ancient example of a system that serves as a Schauder basis for each of the L P -spaces, with 1 < p < +∞, the residual system will net be a Schauder basis for any of those spaces. Nevertheless, Price has shown that each member of a large class of such subsystems is complete on subsets of [0,1] that have measure arbitrarily close to 1. In the present work, it is shown that subsystems of this kind can be multiplicatively completed in such a way that the resulting systems are quasibases for each space L P [0,1], 1 < p < +00, from which the earlier completeness result follows as a corollary.

On semicontinuous fuctions and Baire functions

1. According to a classical theorem, every semicontinuous real-valued function of a real variable can be obtained as the limit of a sequence of continuous functions. The corresponding proposition need not hold for the semicontinuous functions defined on an arbitrary topological space; indeed, it is known that the theorem holds in the general case if and only if the topology is perfectly normal [6]. For some purposes, it is important to know when a topological measure space has the property that each semicontinuous real-valued function defined thereon is almost everywhere equal to a function of the first Baire class. In the present article, we are able to resolve this question when the topology is completely regular. A most interesting by-product of our consideration of this problem is the discovery that every Lebesgue measurable function is equivalent to the limit of a sequence of approximately continuous ones. This fact enables us to answer a question, posed in an earlier work [7], concerning the existence of topological measure spaces in which every bounded measurable function is equivalent to a Baire function of the first class.

A CONTINUOUS BASIS FOR ORLICZ SPACES

1. In 1910 Haar created the orthonormal system that bears his name in order to show that there are ON systems with respect to which the Fourier series of each continuous function converges uniformly to the function. The Haar functions are themselves discontinuous, however, and this led Franklin to construct a continuous ON set that plays the same role. Now the Haar functions comprise a Schauder basis for each of the spaces LP [0, 1], p _ 1, as Schauder himself has shown [5]. Indeed, this system serves as a Schauder basis for each of the separable Orlicz spaces associated with the unit interval, see [2], [4]. Thus, it is natural to inquire whether the Franklin system also serves as a basis for these spaces. In the present article this question is answered affirmatively. The proof is elementary, requiring only the judicious application of the uniform boundedness principle and the Jensen integral inequality. For the spaces LP, p _ 1, this result has been noted recently by Ciesielski [1]. Nevertheless, even for this special case, the demonstration given below may be of interest by virtue of its utter simplicity.

On a note of Marcus concerning a problem posed by Frink

In his measure-theoretic treatment of Riemann integration, Frink has shown that the set of discontinuities of a Riemann integrable function is a countable union of sets of zero Jordan content [21. Such a set is thus of Lebesgue measure zero and of the first category. By constructing a counterexample in the plane, Frink also established the fact that not every first category null set is the set of discontinuities of some Riemann integrable function. In the course of his consideration of Jordan content and Riemann integration in topological measure spaces, Marcus has proved the direct generalization of the positive assertion above [5] and has addressed himself to the converse proposition in this more general setting [6]. In the latter investigation, restricting his attention to somewhat special topological measure spaces, Marcus was able to establish the following proposition: