Homework: Fejér kernel

Trigonometric polynomials and the Fejér kernel #

Background #

The approach used in solving differential equations by mathematicians centuries ago is very similar in spirit to today’s machine learning. Suppose that you are given a differential equation and would like to find a continuous function \(f\) that is a solution. If you have no idea where to look for a function that might work, the approach was to try:

\[ \begin{align} f(x) \approx a_0 & + a_1 \cos(\pi x) + a_2 \cos(2\pi x) + \ldots + \cos(n\pi x) \\ & + b_1 \sin(\pi x) + b_2 \sin(2\pi x) + \ldots + \sin(n\pi x) \\ \end{align} \]

Such a function is called a trigonometric polynomial of degree \(n\). The question is now a matter of finding values of the coefficients that yield a solution, or at least an approximation to the solution. Here is a fundamental question mathematicians faced:

Question: Suppose no adequate values for the coefficents could be found. Does this mean that there is no solution to the differential equation? Or does it mean that not all solutions can be approximated by trigonometric polynomials and another technique must be found?

The purpose of this exercise is to show that in fact every continuous function on an interval can be uniformly approximated by a sequence of trigonometric polynomials.

The Fejer kernel #

Consider a family of functions defined by \[F_n(x) = \tfrac{1}{n} \sum_{k=0}^{n-1} \Big( \sum_{m= -k}^k \cos(m \pi x) \Big)\] and further define

\[\phi_n(x) = \begin{cases} F_n(x) & x \in [-1,1] \\ 0 & \text{otherwise} \end{cases} \]

The original functions \(F_n(x) \) are a bit of a disaster, but you can basically think of them as a very long sum of terms each of which is a dilated cosine. The family of functions specified by \(\phi_n\) are known as the Fejér kernels.

The proof of the following theorem is a feat of trigometric identities and a little analysis. We will omit it.

Theorem: The collection \(\{\phi_n\}_{n \in \mathbb{N}}\) is a family of good kernels.

Your homework exercise is to show that the approach used by mathematicians to solve differential equations was on solid footing: every continuous function can be approximated by a trigonometric polynomial. More precisely:

Homework exercise:

Show that if \(f \in \mathcal{C}[0,1]\), then there exist a sequence of trigonometric polynomials \(\{t_n\}_{n \in \mathbb{N}}\) such that \(\lim t_n = f\) uniformly.

Hints: Imitate our proof of the Weierstrass Theorem via Landau kernels but using the Fejér kernel instead. Use the following outline:

Explain why the sequence \(\{f *\phi_n\}_{n \in \mathbb{N}}\) converges uniformly to \(f\). You can use the fact that \(\{\phi_n\}_{n \in \mathbb{N}}\) is a family of good kernels without proof.
Show that \(\int_0^1 f(y) \cos(m \pi (x-y)) \; dy \) is a trigonometric polynomial in the variable \(x\). You may need the following trigonometric idenitity: \[ \cos(a-b) = \cos(a)\cos(b) + \sin(a) \sin(b).\]
To formally finish the proof, explain why \(f *\phi_n\) is always a trigonometric polynomial.

Historical note: The problem in differential equations introduced above was the impetus for the field of Fourier analysis, see p. 14 of Stein and Shakarchi. Joseph Fourier was the first to believe such an approximation was always possible. After almost a century of work, his suspicions finally verified in 1900 by Lipót Fejér as part of his doctoral thesis. J. J. O’Connor and E. F. Robertson write

Fejér’s theorem is a simple, beautiful theorem, and, in the opinion of Jean-Pierre Kahane, its discovery restored to Fourier series a fundamental role in analysis for at least fifty years. Also, at that time the theory of functions of one real variable had produced a number of strange anomalies (like continuous, nowhere differentiable functions), and Fejér’s theorem, by its simplicity, reassured mathematicians who were disturbed by these pathological results.