Fejér kernel

# Fejér kernel

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Encyclopedia
In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the Fejér kernel is used to express the effect of Cesàro summation
Cesàro summation
In mathematical analysis, Cesàro summation is an alternative means of assigning a sum to an infinite series. If the series converges in the usual sense to a sum A, then the series is also Cesàro summable and has Cesàro sum A...

on Fourier series
Fourier series
In mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a set of simple oscillating functions, namely sines and cosines...

. It is a non-negative kernel, giving rise to an approximate identity
Approximate identity
In functional analysis and ring theory, an approximate identity is a net in a Banach algebra or ring that acts as a substitute for an identity element....

.

The Fejér kernel is defined as

where
is the kth order Dirichlet kernel. It can also be written in a closed form as
,

where this expression is defined. It is named after the Hungarian
Hungary
Hungary , officially the Republic of Hungary , is a landlocked country in Central Europe. It is situated in the Carpathian Basin and is bordered by Slovakia to the north, Ukraine and Romania to the east, Serbia and Croatia to the south, Slovenia to the southwest and Austria to the west. The...

mathematician Lipót Fejér
Lipót Fejér
Lipót Fejér , was a Hungarian mathematician. Fejér was born Leopold Weiss, and changed to the Hungarian name Fejér around 1900....

(1880–1959).

The important property of the Fejér kernel is . The convolution
Convolution
In mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions. Convolution is similar to cross-correlation...

Fn is positive: for of period it satisfies

and, by the Young's inequality
Young's inequality
In mathematics, the term Young's inequality is used for two inequalities: one about the product of two numbers, and one about the convolution of two functions. They are named for William Henry Young....

, for every
or continuous function ; moreover, for every ()
or continuous function . Indeed, if is continuous, then the convergence is uniform.

• Fejér's theorem
Fejér's theorem
In mathematics, Fejér's theorem, named for Hungarian mathematician Lipót Fejér, states that if f:R → C is a continuous function with period 2π, then the sequence of Cesàro means of the sequence of partial sums of the Fourier series of f converges uniformly to f on...

• Dirichlet kernel
• Gibbs phenomenon
Gibbs phenomenon
In mathematics, the Gibbs phenomenon, named after the American physicist J. Willard Gibbs, is the peculiar manner in which the Fourier series of a piecewise continuously differentiable periodic function behaves at a jump discontinuity: the nth partial sum of the Fourier series has large...

• Charles Jean de la Vallée-Poussin
Charles Jean de la Vallée-Poussin
Charles-Jean Étienne Gustave Nicolas de la Vallée Poussin was a Belgian mathematician. He is most well known for proving the Prime number theorem.The king of Belgium ennobled him with the title of baron.-Biography:...