Trigonometric Series and Theories of Integration

Abstract

This essay was written in response to my experience in teaching the definite integral of a function of one variable. In covering the topic first in freshman calculus, then in advanced calculus, and finally in a course on real variables, these questions would occur to me repeatedly: What is the need for more general theories of integration? To what mathematical use can these theories be put? Is there a way that we can justify the time spent on developing the theories of Riemann and Lebesgue integration, a context in which their development seems natural and answers previously unanswered questions? The usual justification for the development of these theories is the desire for generality, the tendency to see how far concepts can be pushed in an effort to get at their essence. If this motivating principle is acceptable to both students and instructor then the above questions are perhaps less urgent. But my classroom experiences have suggested to me that something more would be helpful, namely an idea of the necessity for the new concepts within a mathematical framework. As an example, consider the transition from the Riemann to the Lebesgue integral. The usual view of the inadequacy of the Riemann integral is that the theory is incomplete because one can have a sequence of functions, each of which is Riemann integrable, but whose pointwise limit function is not. From an advanced viewpoint, say, that of function space theory, that is indeed a drawback, but should it be considered as one by a student of advanced calculus? After all, limit functions do not always share the properties of their defining sequences-continuity is an example of this. A related problem is that of reversing the order of integration and summation, and of convergence theorems in general: Why all the concern over the weakest conditions that assure that lim I = fj blim O fn? Of what use are these theorems? (Some answers will be given later in this paper.) In doing some background preparation for a seminar in Fourier series, I learned about their history, and I believe that the connection with trigonometric series can be useful in putting these integration issues into a context. The three main integration theories that we teach can roughly be attributed to Cauchy, Riemann, and Lebesgue, and each of these men was very interested in trigonometric series. Cauchy devoted a paper to the subject [7], Riemann introduced his integral in a classic on the series, [19], and Lebesgue devoted several papers and a short book to them [15], [16]. (Books [3], [12], and [14] are excellent general references for this material.) This paper is an attempt to show how we may pedagogically relate integration theories and trigonometric series. Let us begin with a simple question seemingly unrelated to integration: Is there a closed-form analytic representation for an arbitrary real-valued function defined on [-w, 7]? That is, can every function defined on this