Dirichlet kernel - Citizendia

In mathematical analysis, the Dirichlet kernel is the collection of functions

$D_n(x)=\sum_{k=-n}^n e^{ikx}=1+2\sum_{k=1}^n\cos(kx)=\frac{\sin\left(\left(n +\frac{1}{2}\right)x\right)}{\sin(x/2)}.$

It is named after Johann Peter Gustav Lejeune Dirichlet. Analysis has its beginnings in the rigorous formulation of Calculus. Johann Peter Gustav Lejeune Dirichlet (ləʒœn diʀiçle February 13, 1805 &ndash May 5, 1859) was a German Mathematician

The importance of the Dirichlet kernel comes from its relation to Fourier series. In Mathematics, a Fourier series decomposes a periodic function into a sum of simple oscillating functions The convolution of D_n(x) with any function f of period 2π is the nth-degree Fourier series approximation to f, i. In Mathematics and in particular Functional analysis, convolution is a mathematical operation on two functions f and e. , we have

$(D_n*f)(x)=\frac{1}{2\pi}\int_{-\pi}^\pi f(y)D_n(x-y)\,dy=\sum_{k=-n}^n \hat{f}(k)e^{ikx},$

where

$\hat{f}(k)=\frac{1}{2\pi}\int_{-\pi}^\pi f(x)e^{-ikx}\,dx$

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 n → ∞. In Mathematics, the Lp and ℓp spaces are spaces of p-power integrable functions, and corresponding One can estimate that

$\| D_n \| _{L^1} \approx \log n$

where $\approx$ denotes "is of the order. " This lack of uniform integrability is behind many divergence phenomena for the Fourier series. For example, together with the uniform boundedness principle, it can be used to show that the Fourier series of a continuous function may fail to converge pointwise, in rather dramatic fashion. In Mathematics, the uniform boundedness principle or Banach-Steinhaus theorem is one of the fundamental results in Functional analysis. In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output See convergence of Fourier series for more. In Mathematics, the question of whether the Fourier series of a Periodic function converges to the given function is researched by

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. 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 or Dirac's delta is a mathematical construct introduced by the British theoretical physicist Paul Dirac. In Mathematics, generalized functions are objects generalizing the notion of functions There is more than one recognised theory We get the identity element for convolution on functions of period 2π. 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 In other words, we have

$f*(2\pi \delta)=f \,$

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

$2\pi \delta(x)\sim\sum_{k=-\infty}^\infty e^{ikx}=\left(1 +2\sum_{k=1}^\infty\cos(kx)\right).$

Therefore the Dirichlet kernel, which is just the sequence of partial sums of this series, can be thought of as an approximate identity. In Functional analysis, a right approximate identity in a Banach algebra, A, is a net (or a Sequence) \{\e_\lambda Abstractly speaking it is not however an approximate identity of positive elements (hence the failures mentioned above).

Proof of the trigonometric identity

The trigonometric identity

$\sum_{k=-n}^n e^{ikx} =\frac{\sin\left(\left(n+\frac{1}{2}\right)x\right)}{\sin(x/2)}$

displayed at the top of this article may be established as follows. In Mathematics, trigonometric identities are equalities that involve Trigonometric functions that are true for every single value of the occurring variables First recall that the sum of a finite geometric series is

$\sum_{k=0}^n a r^k=a\frac{1-r^{n+1}}{1-r}.$

In particular, we have

$\sum_{k=-n}^n r^k=r^{-n}\cdot\frac{1-r^{2n+1}}{1-r}.$

Multiply both the numerator and the denominator by r^−1/2, getting

$\frac{r^{-n-1/2}}{r^{-1/2}}\cdot\frac{1-r^{2n+1}}{1-r} =\frac{r^{-n-1/2}-r^{n+1/2}}{r^{-1/2}-r^{1/2}}.$

In case r = e^ix we have

$\sum_{k=-n}^n e^{ikx}=\frac{e^{-(n+1/2)ix}-e^{(n+1/2)ix}}{e^{-ix/2}-e^{ix/2}} =\frac{-2i\sin((n+1/2)x)}{-2i\sin(x/2)} = \frac{\sin((n+1/2)x)}{\sin(x/2)}$

as required. In Mathematics, a geometric series is a series with a constant ratio between successive terms.

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