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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 **Ge ...
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, structure, space, models, and change. History On ...
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 In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''cardinal n ...
. 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.html" "title="Dirac_delta_function.html" "title="L,L~L=\pi,~ of the first few Dirichlet kernels showing their convergence to one of the Dirac delta function">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 the ...
. 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.


''L''1 norm of the kernel function

Of particular importance is the fact that the ''L''1 norm of ''Dn'' 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 cornerst ...
, it can be used to show that the Fourier series of a
continuous function In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...
may fail to converge pointwise, in rather dramatic fashion. See
convergence of Fourier series In mathematics, the question of whether the Fourier series of a periodic function converges to a given function is researched by a field known as classical harmonic analysis, a branch of pure mathematics. Convergence is not necessarily given in th ...
for further details. A precise proof of the first result that \, D_n \, _ = \Omega(\log n) is given by \begin \int_0^ , D_n(x), \, dx & \geq \int_0^\pi \frac \, dx \\ pt & \geq \sum_^ \int_^ \frac \, ds \\ pt & \geq \left, \sum_^ \int_0^ \frac \, ds\ \\ pt & = \frac H_ \\ pt & \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 number In mathematics, the -th harmonic number is the sum of the reciprocals of the first natural numbers: H_n= 1+\frac+\frac+\cdots+\frac =\sum_^n \frac. Starting from , the sequence of harmonic numbers begins: 1, \frac, \frac, \frac, \frac, \dot ...
s.


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 the ...
, 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, ...
: :\mathcal\left D_n(2 \pi x) \right\xi) = \mathcal^\left D_n(2 \pi x) \right\xi) = \int_^ D_n(2 \pi x) e^ dx = \sum_^ \delta(\xi-k) \equiv comb_n(\xi) :\mathcal\left comb_n \rightx) = \mathcal^\left comb_n \rightx) = \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 the ...
\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, ...
: \mathcal operatorname \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 su ...
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 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 ...
''. 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 Equality (mathematics), equalities that involve trigonometric functions and are true for every value of the occurring Variable (mathematics), variables for which both sides of the equality are defined. ...
\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 succ ...
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 samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a complex- ...
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. It is a non-negative kernel, giving rise to an approximate identity. It is named after the Hungarian mathematician Lipà ...


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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{{dead link, date=December 2016 , bot=InternetArchiveBot , fix-attempted=yes Mathematical analysis Fourier series Approximation theory Articles containing proofs