Dirichlet Kernel

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mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...

, the Dirichlet kernel, named after the

German German(s) may refer to: * Germany (of or related to) **Germania (historical use) * Germans, citizens of Germany, people of German ancestry, or native speakers of the German language ** For citizens of Germany, see also German nationality law **Ger ...

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...

Peter Gustav Lejeune Dirichlet Johann Peter Gustav Lejeune Dirichlet (; 13 February 1805 – 5 May 1859) was a German mathematician who made deep contributions to number theory (including creating the field of analytic number theory), and to the theory of Fourier series and ...

, 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 nonnegative integer. The kernel functions are periodic with period

2\pi

. 300px, Plot restricted to one period

Dirac delta In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire ...

distributions of the Dirac comb">Dirac comb In mathematics, a Dirac comb (also known as shah function, impulse train or sampling function) is a periodic function with the formula \operatorname_(t) \ := \sum_^ \delta(t - k T) for some given period T. Here ''t'' is a real variable and t ...

. The importance of the Dirichlet kernel comes from its relation to Fourier series. The convolution of with any function of period 2 is the ''n''th-degree Fourier series approximation to , i.e., we have

(D_n*f)(x)=\int_^\pi f(y)D_n(x-y)\,dy=\sum_^n \hat(k)e^,

where

\widehat(k)=\frac 1 \int_^\pi f(x)e^\,dx

is the th Fourier coefficient of . This implies that in order to study convergence of Fourier series it is enough to study properties of the Dirichlet kernel. Dirichlet kernels

Dirichlet kernels

''L''¹ norm of the kernel function

Of particular importance is the fact that the ''L''¹ norm of ''D_n'' on

, 2\pi /math> diverges to infinity as . One can estimate that \,  D_n \,  _ = \Omega(\log n). By using a Riemann-sum argument to estimate the contribution in the largest neighbourhood of zero in which D_n is positive, and Jensen's inequality for the remaining part, it is also possible to show that: \,  D_n \, _ \geq 4\operatorname(\pi)+\frac 8 \pi \log n. This lack of uniform integrability is behind many divergence phenomena for the Fourier series. For example, together with the

uniform boundedness principle In mathematics, the uniform boundedness principle or Banach–Steinhaus theorem is one of the fundamental results in functional analysis. Together with the Hahn–Banach theorem and the open mapping theorem, it is considered one of the corners ...

, it can be used to show that the Fourier series of a continuous function may fail to converge pointwise, in rather dramatic fashion. See convergence of Fourier series for further details. A precise proof of the first result that

\,  D_n \,  _ = \Omega(\log n)

is given by

& \geq \frac \log(2n+1), \end

where we have used the Taylor series identity that

2/x \leq 1 / , \sin(x/2),

and where

H_n

are the first-order harmonic numbers.

Relation to the periodic delta function

The Dirichlet kernel is a periodic function which becomes the

Dirac comb In mathematics, a Dirac comb (also known as shah function, impulse train or sampling function) is a periodic function with the formula \operatorname_(t) \ := \sum_^ \delta(t - k T) for some given period T. Here ''t'' is a real variable and t ...

, i.e. the periodic delta function, in the limit :

\sum_^ e^ = \frac \sum_^ \delta(\omega-2\pi k/T) = \frac \sum_^ \delta(\xi- k/T)  ~,

with the angular frequency

\omega=2 \pi \xi

. This can be inferred from the autoconjugation property of the Dirichlet kernel under forward and inverse

Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...

: :

\xi) = \int_^ D_n(2 \pi x) e^ dx = \sum_^ \delta(\xi-k) \equiv comb_n(\xi)

:

x) = \int_^ comb_n(\xi) e^ d\xi = D_n(2 \pi x),

and

comb_n(x)

goes to the

Dirac comb In mathematics, a Dirac comb (also known as shah function, impulse train or sampling function) is a periodic function with the formula \operatorname_(t) \ := \sum_^ \delta(t - k T) for some given period T. Here ''t'' is a real variable and t ...

\operatorname

of period

T=1

as

n \rightarrow \infty

, which remains invariant under

Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...

:

\operatorname

. Thus

D_n(2 \pi x)

must also have converged to

\operatorname

as

n \rightarrow \infty

. In a different vein, consider ∆(x) as the

identity element In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures s ...

for convolution on functions of period 2. In other words, we have

f*( \Delta)=f

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

\Delta(x)\sim\sum_^\infty e^= \left(1 + 2\sum_^\infty \cos(kx)\right).

(This Fourier series converges to the function almost nowhere.) Therefore the Dirichlet kernel, which is just the sequence of partial sums of this series, can be thought of as an '' approximate identity''. Abstractly speaking it is not however an approximate identity of ''positive'' elements (hence the failures in pointwise covergence mentioned above).

Proof of the trigonometric identity

The

trigonometric identity In trigonometry, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involvi ...

\sum_^n e^ = \frac

displayed at the top of this article may be established as follows. First recall that the sum of a finite

geometric series In mathematics, a geometric series is the sum of an infinite number of terms that have a constant ratio between successive terms. For example, the series :\frac \,+\, \frac \,+\, \frac \,+\, \frac \,+\, \cdots is geometric, because each su ...

is

\sum_^n a r^k=a\frac.

In particular, we have

\sum_^n r^k=r^\cdot\frac.

Multiply both the numerator and the denominator by

r^

, getting

\frac\cdot\frac =\frac.

In the case

r = e^

we have

\sum_^n e^ = \frac = \frac = \frac

as required.

Alternative proof of the trigonometric identity

Start with the series

f(x) = 1 + 2 \sum_^n\cos(kx).

Multiply both sides by

\sin(x/2)

and use the trigonometric identity

\cos(a)\sin(b) = \frac 2

to reduce the terms in the sum.

\sin(x/2)f(x) = \sin(x/2)+ \sum_^n \sin((k + \tfrac 1 2 )x)- \sin((k-\tfrac 1 2 )x)

which telescopes down to the result.

Variant of identity

If the sum is only over non negative integers (which may arise when computing a

discrete Fourier transform In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced Sampling (signal processing), samples of a function (mathematics), function into a same-length sequence of equally-spaced samples of the discre ...

that is not centered), then using similar techniques we can show the following identity:

\sum_^ e^ = e^\frac

See also

*

Fejér kernel In mathematics, the Fejér kernel is a summability kernel used to express the effect of Cesàro summation on Fourier series A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. ...

References

* Andrew M. Bruckner, Judith B. Bruckner, Brian S. Thomson: ''Real Analysis''. ClassicalRealAnalysis.com 1996, , S.620
vollständige Online-Version (Google Books)
* Podkorytov, A. N. (1988), "Asymptotic behavior of the Dirichlet kernel of Fourier sums with respect to a polygon". ''Journal of Soviet Mathematics'', 42(2): 1640–1646. doi: 10.1007/BF01665052 * Levi, H. (1974), "A geometric construction of the Dirichlet kernel". ''Transactions of the New York Academy of Sciences'', 36: 640–643. doi: 10.1111/j.2164-0947.1974.tb03023.x *
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