In mathematical analysis, the Dirichlet kernel is the collection of functions Analysis is the generic name given to any branch of mathematics which depends upon the concepts of limits and convergence, and studies closely related topics such as continuity, integration, differentiability and transcendental functions. ...

It is named after Johann Peter Gustav Lejeune Dirichlet. Peter Gustav Lejeune Dirichlet. ...

The importance of the Dirichlet kernel comes from its relation to Fourier series. The convolution of D_n(x) with any function f of period 2π is the nth-degree Fourier series approximation to f, i.e., we have The Fourier series, named in honor of Joseph Fourier (1768-1830), is an extremely useful mathematical tool. ... For the computer science usage see convolution (computer science) . In mathematics and in particular, functional analysis, convolution is a mathematical operator which takes two functions f and g and produces a third function that in a sense represents the amount of overlap between f and a reversed and translated version...

where

is the kth Fourier coefficient of f. This implies that in order to study convergence of Fourier series it is enough to study properties of the Dirichlet kernel. Of particular importance is the fact that the L¹ norm of D_n diverges to infinity as . This fact is behind many divergence phenomena. For example, it is the direct reason that the Fourier series of a continuous function may diverge. See convergence of Fourier series for more. In mathematics, the Lp and spaces are spaces of p-power integrable functions, and corresponding sequence spaces. ... In mathematics, a continuous function is a function in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ... In mathematics, the question whether the Fourier series of a periodic function converges to the given function and in what sense is a rich field of research, sometimes called classic harmonic analysis, a branch of pure mathematics. ...

Relation to the delta function

Take the periodic Dirac delta function, which is not really a function, in the sense of mapping one set into another, but is rather a "generalized function", also called a "distribution", and multiply by 2π. We get the identity element for convolution on functions of period 2π. In other words, we have In mathematics, a periodic function is a function that repeats its values after some definite period has been added to its independent variable. ... The Dirac delta function, sometimes referred to as the unit impulse function and introduced by the British theoretical physicist Paul Dirac, can usually be informally thought of as a function δ(x) that has the value of infinity for x = 0, the value zero elsewhere such that the total integral... In mathematics, generalized functions are objects generalizing the notion of functions. ... In mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a binary operation on that set. ...

for every function f of period 2π. The Fourier series representation of this "function" is

Therefore the Dirichlet kernel, which are just the partial sums of this series, can be thought of as an approximate identity.

Proof of the trigonometric identity

The trigonometric identity In mathematics, trigonometric identities are equations involving trigonometric functions that are true for all values of the occurring variables. ...

displayed at the top of this article may be established as follows. First recall that the sum of a finite geometric series is In mathematics, a geometric progression is a sequence of numbers such that the quotient of any two successive members of the sequence is a constant called the common ratio of the sequence. ...

The first term is a; the common ratio by which each term is multiplied to get the next is r; the number of terms is n + 1. In particular, we have

The expression to the left of "=" should make us expect the sum to be a symmetric function of r and 1/r, but the expression to the right of "=" is perhaps less-than-obviously symmetric in those two quantities. The remedy is to multiply both the numerator and the denominator by r^−1/2, getting

In case r = e^ix we have

and then "−2i" cancels.