Properties of the Fejer kernel

Question

The Fejer kernel $k_m : \mathbb{R} \to \mathbb{C}$ is defined by $k_m (t) = \frac{1}{2\pi (m+1)} \sum^m_{n=0} \sum^n _{k=-n} e^{ikt}$

One of the properties of the Fejer kernel is

For any $\delta \in (0, \pi)$

$\int_{t \in [- \pi , \pi] \backslash (-\delta , \delta)} k_m (t) dt \to 0$ as $m \to \infty$

I do not understand this what this is saying. What is it saying and why is it important?

Answer 1

$[-\pi,\pi]\backslash(-\delta,\delta)$ is the interval $[-\pi,\pi]$, excluding the (small) interval $(-\delta,\delta)$. The fact that the integral over this region goes to zero means that most of the "mass" of the Fejer kernels clusters near the origin as $m\rightarrow\infty$, in particular moving away from the regions that are farther away from the origin. See a picture of the graphs of the Fejer kernels to see this phenomenon: https://en.wikipedia.org/wiki/Fej%C3%A9r_kernel

This property is important to show that the Fejer kernels form a family of functions known as an "approximation to the identity". What that means is the following: For a given function $f$, consider the function $f\ast k_m$, where $$ (f\ast k_m)(x) = \int\limits_{-\pi}^{\pi}{f(x-t)k_m(t)\text{ d}t}. $$ Using the fact that $\int\limits_{-\pi}^{\pi}{k_m(t)\text{ d}t} = 1$ and the fact that $k_m\ge 0$, along with the property in question, it can be shown that if $f$ is nice enough, then $f\ast k_m\rightarrow f$ as $m\rightarrow\infty$ in some way (e.g. uniformly if $f$ is continuous, and in $L^p$ if $f$ is in $L^p$). In other words, the sequence of functions $\{f\ast k_m\}$ is an approximation to $f$. Intuitively, this is because all of the "mass" of the Fejer kernels is shifted to the origin, which means that the only values of $f$ that contribute to the integral above are those where $t$ is near zero, i.e. the values of $f$ near $x$. Now, we also have $$ (f\ast k_m)(x) = \frac{1}{m+1}\sum\limits_{n=0}^{m}{S_nf(x)} $$ where $S_nf$ is the $n$th partial Fourier series of $f$, i.e. $f\ast k_m$ is the Cesaro average of the partial Fourier series of $f$. This has two important consequences:

1. Since $f\ast k_m$ is a trigonometric polynomial, this shows that trig polynomials are dense in $L^p([-\pi,\pi])$.
2. If $f$ is continuous, and for some $x$ we have $S_nf(x)$ converges to some value as $n\rightarrow\infty$, then for that $x$ we have that $S_nf(x)$ converges to $f(x)$. This is because $(f\ast k_m)(x)$ converges to $f(x)$ as $m\rightarrow\infty$, and $f\ast k_m$ is the Cesaro average of the partial Fourier series.