Fejér kernel

In mathematics, the Fejér kernel is a summability kernel used to express the effect of Cesàro summation on Fourier series. It is a non-negative kernel, giving rise to an approximate identity. It is named after the Hungarian mathematician Lipót Fejér (1880–1959).

Definition

The Fejér kernel has many equivalent definitions. Three such definitions are outlined below:

1) The traditional definition expresses the Fejér kernel $F_{n} (x)$ in terms of the Dirichlet kernel

F_{n} (x) = \frac{1}{n} \sum_{k = 0}^{n - 1} D_{k} (x)

where

D_{k} (x) = \sum_{s = - k}^{k} e^{i s x}

is the $k$ th order Dirichlet kernel.

2) The Fejér kernel $F_{n} (x)$ may also be written in a closed form expression as follows:^[1]

$F_{n} (x) = \frac{1}{n} {(\frac{\sin (\frac{n x}{2})}{\sin (\frac{x}{2})})}^{2} = \frac{1}{n} \frac{1 - \cos (n x)}{1 - \cos (x)}$

This closed form expression may be derived from Definition 1; a proof of this result goes as follows.

Using the fact that the Dirichlet kernel may be written as:^[2]

D_{k} (x) = \frac{\sin ((k + \frac{1}{2}) x)}{\sin \frac{x}{2}}

,

one obtains from the definition of the Fejér kernel above:

F_{n} (x) = \frac{1}{n} \sum_{k = 0}^{n - 1} D_{k} (x) = \frac{1}{n} \sum_{k = 0}^{n - 1} \frac{\sin ((k + \frac{1}{2}) x)}{\sin (\frac{x}{2})} = \frac{1}{n} \frac{1}{\sin (\frac{x}{2})} \sum_{k = 0}^{n - 1} \sin ((k + \frac{1}{2}) x) = \frac{1}{n} \frac{1}{\sin^{2} (\frac{x}{2})} \sum_{k = 0}^{n - 1} [\sin ((k + \frac{1}{2}) x) \cdot \sin (\frac{x}{2})]

By the trigonometric identity: $\sin (α) \cdot \sin (β) = \frac{1}{2} (\cos (α - β) - \cos (α + β))$ , one has

F_{n} (x) = \frac{1}{n} \frac{1}{\sin^{2} (\frac{x}{2})} \sum_{k = 0}^{n - 1} [\sin ((k + \frac{1}{2}) x) \cdot \sin (\frac{x}{2})] = \frac{1}{n} \frac{1}{2 \sin^{2} (\frac{x}{2})} \sum_{k = 0}^{n - 1} [\cos (k x) - \cos ((k + 1) x)],

which allows evaluation of $F_{n} (x)$ as a telescoping sum:

F_{n} (x) = \frac{1}{n} \frac{1}{\sin^{2} (\frac{x}{2})} \frac{1 - \cos (n x)}{2} = \frac{1}{n} \frac{1}{\sin^{2} (\frac{x}{2})} \sin^{2} (\frac{n x}{2}) = \frac{1}{n} {(\frac{\sin (\frac{n x}{2})}{\sin (\frac{x}{2})})}^{2} .

3) The Fejér kernel can also be expressed as:

F_{n} (x) = \sum_{| k | \leq n - 1} (1 - \frac{| k |}{n}) e^{i k x}

Properties

The Fejér kernel is a positive summability kernel. An important property of the Fejér kernel is $F_{n} (x) \geq 0$ with average value of $1$ .

Convolution

The convolution $F_{n}$ is positive: for $f \geq 0$ of period $2 π$ it satisfies

0 \leq (f * F_{n}) (x) = \frac{1}{2 π} \int_{- π}^{π} f (y) F_{n} (x - y) d y .

Since

f * D_{n} = S_{n} (f) = \sum_{| j | \leq n} {\hat{f}}_{j} e^{i j x},

we have

f * F_{n} = \frac{1}{n} \sum_{k = 0}^{n - 1} S_{k} (f),

which is Cesàro summation of Fourier series. By Young's convolution inequality,

‖ F_{n} * f ‖_{L^{p} ([- π, π])} \leq ‖ f ‖_{L^{p} ([- π, π])} for every 1 \leq p \leq \infty for f \in L^{p} .

Additionally, if $f \in L^{1} ([- π, π])$ , then

f * F_{n} \to f

a.e.

Since $[- π, π]$ has finite volume, $L^{1} ([- π, π]) \supset L^{2} ([- π, π]) \supset \dots \supset L^{\infty} ([- π, π])$ , so the result holds for other $L^{p}$ spaces, $p \geq 1$ , as well.

If $f$ is continuous, then the convergence is uniform, yielding a proof of the Weierstrass theorem.

One consequence of the pointwise a.e. convergence is the uniqueness of Fourier coefficients: If $f, g \in L^{1}$ with $\hat{f} = \hat{g}$ , then $f = g$ a.e. This follows from writing

f * F_{n} = \sum_{| j | \leq n} (1 - \frac{| j |}{n}) {\hat{f}}_{j} e^{i j t},

which depends only on the Fourier coefficients.

A second consequence is that if $lim_{n \to \infty} S_{n} (f)$ exists a.e., then $lim_{n \to \infty} F_{n} (f) = f$ a.e., since the Cesàro means of $F_{n} * f$ converge to the original sequence limit if it exists.

Applications

The Fejér kernel is used in signal processing and Fourier analysis.

References

↑ Hoffman, Kenneth (1988). Banach Spaces of Analytic Functions. Dover. p. 17. ISBN 0-486-45874-1.
↑ Königsberger, Konrad. Analysis 1 (in German) (6th ed.). Springer. p. 322.

[1] Hoffman, Kenneth (1988). Banach Spaces of Analytic Functions. Dover. p. 17. ISBN 0-486-45874-1.

[2] Königsberger, Konrad. Analysis 1 (in German) (6th ed.). Springer. p. 322.

[1]

[2]

Definition

Properties

Convolution

Applications

See also

References