Homework 1 was a difficult one. Many of the problems were given at Qualifying Examinations from universities around the US. Others appeared as part of the Putnam Competition.

The problem that sparked the most attention was the following:

**Problem 7.** *If are continuous functions on of period , then*

Although elementary, this problem has deep connections with Fourier analysis and it is often also regarded as the *Riemann-Lebesgue* lemma. The classical *Riemann-Lebesgue* lemma states that for a Riemann integrable function ,

.

This implies one of the most important facts in classical Fourier analysis, if is a periodic function of period , then under suitable conditions,

converges to as , where are the Fourier coefficients,

, , and .

The following result generalizes the classical Riemann-Lebesgue lemma:

**Theorem 1.** * Let be a continuous function, . Suppose that is a periodic function of period . Then we have*

However, in 1962, Luxemburg proved the following more general result:

**Theorem 2.** ([1]) *Let be a bounded real or complex valued measurable function defined on , and assume that is periodic with period . If is an arbitrary interval and a real or complex-valued measurable function defined on and integrable over , then*

.

The reference is given below

[1] W. A. J. Luxemburg, A property of the Fourier coefficients of an integrable function, *The American Mathematical Monthly*, **69** (1962), 94-98.

The solution of the above theorem as well as many other homework problems can be found here. Also, homework 3 is here.