In
mathematics, the Fejér kernel is a
summability kernel used to express the effect of
Cesàro summation
In mathematical analysis, Cesàro summation (also known as the Cesàro mean
) assigns values to some infinite sums that are not necessarily convergent in the usual sense. The Cesàro sum is defined as the limit, as ''n'' tends to infinity, of ...
on
Fourier series. It is a non-negative kernel, giving rise to an
approximate identity
In mathematics, particularly in functional analysis and ring theory, an approximate identity is a net in a Banach algebra or ring (generally without an identity) that acts as a substitute for an identity element.
Definition
A right approximate ...
. It is named after the
Hungarian mathematician
Lipót Fejér (1880–1959).
Definition
The Fejér kernel has many equivalent definitions. We outline three such definitions below:
1) The traditional definition expresses the Fejér kernel
in terms of the Dirichlet kernel:
where
:
is the ''k''th order
Dirichlet kernel In mathematical analysis, the Dirichlet kernel, named after the German mathematician Peter Gustav Lejeune Dirichlet, is the collection of periodic functions defined as
D_n(x)= \sum_^n e^ = \left(1+2\sum_^n\cos(kx)\right)=\frac,
where is any nonneg ...
.
2) The Fejér kernel
may also be written in a closed form expression as follows
This closed form expression may be derived from the definitions used above. The proof of this result goes as follows.
First, we use the fact that the Dirichet kernel may be written as:
:
Hence, using the definition of the Fejér kernel above we get:
:
Using the trigonometric identity:
:
Hence it follows that:
:
3) The Fejér kernel can also be expressed as:
Properties
The Fejér kernel is a positive summability kernel. An important property of the Fejér kernel is
with average value of
.
Convolution
The
convolution
In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution'' ...
''F
n'' is positive: for
of period
it satisfies
:
Since
, we have
, which is
Cesàro summation
In mathematical analysis, Cesàro summation (also known as the Cesàro mean
) assigns values to some infinite sums that are not necessarily convergent in the usual sense. The Cesàro sum is defined as the limit, as ''n'' tends to infinity, of ...
of Fourier series.
By
Young's convolution inequality,
:
Additionally, if
, then
:
a.e.
Since