Scenes from the History of Real Functions by Fyodor A. Medvedev

Author:Fyodor A. Medvedev
Language: eng
Format: epub
Publisher: Springer-Verlag Wien 2012
Published: 2015-02-08T16:00:00+00:00

Here ρ(f, g) denotes the distance between the functions f and g [1, p. 98]. Besides the fact that essentially different integrals are intended in (1) and (2), these definitions differ in that the square of the difference occurs in the integrand of (1) while the absolute value of the difference occurs in (2). In their book Elements of the Theory of Functions and Functional Analysis Kolmogorov and Fomin call convergence of type (1) with a more general integral than Natanson uses square-mean convergence [1, Vol. 2, p. 84].

In the present section and the one following the term convergence in mean will be understood in the sense of (1) with the Lebesgue integral in its original form.

Convergence in mean, both in the sense of (1) and in the sense of (2), is more general than convergence in measure; historically the former was introduced before the latter. Nevertheless we have chosen to consider them in the opposite order. Our grounds for doing so are that convergence in mean was the source of the concepts of strong and weak convergence, which play such a large role in functional analysis, and in this one of the main epochs in the growth of the theory of functions into functional analysis shows up very clearly.

We shall begin to study the question of the introduction of the concept of convergence in mean at a slightly earlier period than in the preceding section, on the still stronger grounds that, while mathematicians worked implicitly with convergence in measure for a long time—from 1885 to 1909—the attempts to introduce convergence in mean in explicit form, though not completed for objective reasons, was undertaken as early as 1880. It is extremely interesting, and we have ventured to devote an entire section of the present essays to it.

It was noted in the second section of this essay that the mastery of uniform convergence by mathematicians led to a re-examination of many questions of the theory of functions, especially the theory of trigonometric series. This re-examination took place in the 1870’s and 1880’s, and Harnack joined the effort starting in 1880.52 The history of the concept of convergence in mean should evidently begin with his memoir [1].

From the outset Harnack takes as the basic premise of his research the assumption that the functions considered are square-integrable53 [1, p. 124]. He remarks that if f(x) is bounded,54 the integrability of [f(x)]2 follows from the integrability of f(x); in the case of unbounded functions the situation is different, so that the class of integrable functions is larger than the class of square-integrable functions.

The main theorem of this paper of Harnack’s is the following: If [f(x)]2 is integrable on (−π, π), then the Fourier series (3)

possesses the property that for any arbitrarily small positive number S there exists a value of n such that not only are all the coefficients an and bn from that value on less than δ in absolute value, but

for every natural number l [1, pp. 124–125].55

Rephrasing this statement, one can

Download

Copyright Disclaimer:
This site does not store any files on its server. We only index and link to content provided by other sites. Please contact the content providers to delete copyright contents if any and email us, we'll remove relevant links or contents immediately.