Dirichlet kernel

 ( Mathematical Analysis ) 

In mathematical analysis, the Dirichlet kernel, named after the Germany mathematician Peter Gustav Lejeune Dirichlet, is the collection of periodic functions defined as

$$D\_n(x)=\; \backslash sum\_\{k=-n\}^n\; e^\{ikx\}\; =\; \backslash left(1+2\backslash sum\_\{k=1\}^n\backslash cos(kx)\backslash right)=\backslash frac\{\backslash sin\backslash left(\backslash left(n\; +1/2\backslash right)\; x\; \backslash right)\}\{\backslash sin(x/2)\},$$

where is any nonnegative integer. The kernel functions are periodic with period $2\backslash pi$.

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 $2\backslash pi$ is the $n$th-degree Fourier series approximation to $f$, i.e., we have

$$(D\_n*f)(x)=\backslash int\_\{-\backslash pi\}^\backslash pi\; f(y)D\_n(x-y)\backslash ,dy=2\backslash pi\backslash sum\_\{k=-n\}^n\; \backslash hat\{f\}(k)e^\{ikx\},$$

where

$$\backslash widehat\{f\}(k)=\backslash frac\; 1\; \{2\backslash pi\}\backslash int\_\{-\backslash pi\}^\backslash pi\; f(x)e^\{-ikx\}\backslash ,dx$$

is the $k$th 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.

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Applications In signal processing, the Dirichlet kernel is often called the periodic sinc function:

$P(\backslash omega)\; =\; D\_n(x)|\_\{x\; =\; 2\; \backslash pi\; \backslash omega/\backslash omega\_0\}\; =\; \{\; \backslash sin(\backslash pi\; M\; \backslash omega\; /\; \backslash omega\_0\; )\; \backslash over\; \backslash sin(\backslash pi\; \backslash omega\; /\; \backslash omega\_0\; )\; \}$

where $M\; =\; 2n\; +\; 1\; \backslash ge\; 3$ is an odd integer. In this form, $\backslash omega$ is the angular frequency, and $\backslash omega\_0$ is half of the periodicity in frequency. In this case, the periodic sinc function in the frequency domain can be thought of as the Fourier transform of a time bounded impulse train in the time domain:

$p(t)\; =\; \backslash sum\_\{k=-n\}^n\; \backslash delta(t\; -\; kT)$

where $T\; =\; \backslash frac\{\backslash pi\}\{\backslash omega\_0\}$ is the time increment between each impulse and $M\; =\; 2n+1$ represents the number of impulses in the impulse train.

In optics, the Dirichlet kernel is part of the mathematical description of the diffraction pattern formed when monochromatic light passes through an aperture with multiple narrow slits of equal width and equally spaced along an axis perpendicular to the optical axis. In this case, $M$ is the number of slits.

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norm of the kernel function Of particular importance is the fact that the Lp space norm of $D\_n$ on $0,$ diverges to infinity as $n\backslash to\backslash infty$. One can estimate that

$$\backslash |\; D\_n\; \backslash |\; \_\{L^1\}\; =\; \backslash Omega(\backslash log\; n).$$

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:

$$\backslash |\; D\_n\; \backslash |\_\{L^1\}\; \backslash geq\; 4\backslash operatorname\{Si\}(\backslash pi)+\backslash frac\; 8\; \backslash pi\; \backslash log\; n$$

where $\backslash operatorname\{Si\}(x)$ is the sine integral $\backslash int\_0^x\; (\backslash sin\; t)/t\backslash ,\; dt.$

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. See convergence of Fourier series for further details.

A precise proof of the first result that $\backslash |\; D\_n\; \backslash |\; \_\{L^10,2\backslash pi\}\; =\; \backslash Omega(\backslash log\; n)$ is given by

    & \geq \sum_{k=0}^{2n} \int_{k\pi}^{(k+1)\pi} \frac{\left|\sin s\right|}{s} \, ds \\[5pt]
    & \geq \left|\sum_{k=0}^{2n} \int_0^\pi \frac{\sin s}{(k+1)\pi} \, ds\right| \\[5pt]
    & = \frac{2}{\pi} H_{2n+1} \\[5pt]
    & \geq \frac{2}{\pi} \log(2n+1),
     

\end{align}

where we have used the Taylor series identity that $2/x\; \backslash leq\; 1\; /\; \backslash left|\backslash sin(x/2)\backslash right|$ and where $H\_n$ are the first-order .

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Relation to the periodic delta function The Dirichlet kernel is a periodic function which becomes the Dirac comb, i.e. the periodic delta function, in the limit

$\backslash lim\_\{n\; \backslash to\; \backslash infty\}\; D\_n(\backslash omega)\; =\; \backslash lim\_\{n\; \backslash to\; \backslash infty\}\; \backslash sum\_\{k=-n\}^\{n\}\; e^\{\backslash pm\; i\; \backslash omega\; k\}\; =\; \backslash sum\_\{k=-\backslash infty\}^\{\backslash infty\}\; e^\{\backslash pm\; i\; 2\; \backslash pi\; k\; x\}\; =\; \backslash sum\_\{k=-\backslash infty\}^\{\backslash infty\}\; \backslash delta(x-\; k)=2\backslash pi\; \backslash sum\_\{k=-\backslash infty\}^\{\backslash infty\}\; \backslash delta(\backslash omega\; -\; 2\backslash pi\; k)\; ~,$

with the angular frequency $\backslash omega=2\; \backslash pi\; x$.

This can be inferred from the autoconjugation property of the Dirichlet kernel under forward and inverse Fourier transform:

$\backslash mathcal\{F\}\backslash left(\backslash xi)\; =\; \backslash mathcal\{F\}^\{-1\}\backslash left(\backslash xi)\; =\; \backslash int\_\{-\backslash infty\}^\{\backslash infty\}\; D\_n(2\; \backslash pi\; x)\; e^\{\backslash pm\; i\; 2\backslash pi\; \backslash xi\; x\}\; \backslash ,\; dx\; =\; \backslash sum\_\{k=-n\}^\{+n\}\; \backslash delta(\backslash xi-k)\; \backslash equiv\; \backslash operatorname\{comb\}\_n(\backslash xi)$

$\backslash mathcal\{F\}\backslash left(x)\; =\; \backslash mathcal\{F\}^\{-1\}\backslash left(x)\; =\; \backslash int\_\{-\backslash infty\}^\backslash infty\; \backslash operatorname\{comb\}\_n(\backslash xi)\; e^\{\; \backslash pm\; i\; 2\; \backslash pi\; \backslash xi\; x\; \}\; \backslash ,\; d\backslash xi\; =\; D\_n(2\; \backslash pi\; x),$

and $\backslash operatorname\{comb\}\_n(x)$ goes to the Dirac comb $\backslash operatorname\{\backslash text\{Ш\}\}$ of period $T=1$ as $n\; \backslash rightarrow\; \backslash infty$, which remains invariant under Fourier transform: $\backslash mathcal\{F\}\backslash operatorname\{\backslash text\{Ш\}\}=\; \backslash operatorname\{\backslash text\{Ш\}\}$. Thus $D\_n(2\; \backslash pi\; x)$ must also have converged to $\backslash operatorname\{\backslash text\{Ш\}\}$ as $n\; \backslash rightarrow\; \backslash infty$.

In a different vein, consider $\backslash Delta(x)$ as the identity element for convolution on functions of period $2\backslash pi$. In other words, we have

$$f*(\; \backslash Delta)=f$$

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

$$\backslash Delta(x)\backslash sim\backslash sum\_\{k=-\backslash infty\}^\backslash infty\; e^\{ikx\}=\; \backslash left(1\; +\; 2\backslash sum\_\{k=1\}^\backslash infty\; \backslash cos(kx)\backslash right).$$

(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 convergence mentioned above).

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Proof of the trigonometric identity The trigonometric identity

$$\backslash sum\_\{k=-n\}^n\; e^\{ikx\}\; =\; \backslash frac\{\backslash sin((n+1/2)x)\}\{\backslash sin(x/2)\}$$

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

$$\backslash sum\_\{k=0\}^n\; a\; r^k=a\backslash frac\{1-r^\{n+1\}\}\{1-r\}.$$

In particular, we have

$$\backslash sum\_\{k=-n\}^n\; r^k=r^\{-n\}\backslash cdot\backslash frac\{1-r^\{2n+1\}\}\{1-r\}.$$

Multiply both the numerator and the denominator by $r^\{-1/2\}$, getting

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

In the case $r\; =\; e^\{ix\}$ we have

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

as required.

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Alternative proof of the trigonometric identity Start with the series

$$f(x)\; =\; 1\; +\; 2\; \backslash sum\_\{k=1\}^n\backslash cos(kx).$$

Multiply both sides by $\backslash sin(x/2)$ and use the trigonometric identity

$$\backslash cos(a)\backslash sin(b)\; =\; \backslash frac\{\backslash sin(a\; +\; b)\; -\; \backslash sin(a\; -\; b)\}\; 2$$

to reduce the terms in the sum.

$$\backslash sin(x/2)f(x)\; =\; \backslash sin(x/2)+\; \backslash sum\_\{k=1\}^n\; \backslash left(\; \backslash sin((k\; +\; \backslash tfrac\; 1\; 2\; )x)-\; \backslash sin((k-\backslash tfrac\; 1\; 2\; )x)\; \backslash right)$$

which telescopes down to the result.

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Variant of identity If the sum is only over non-negative integers (which may arise when computing a discrete Fourier transform that is not centered), then using similar techniques we can show the following identity:

$$\backslash sum\_\{k=0\}^\{N-1\}\; e^\{ikx\}\; =\; e^\{i(N-1)x/2\}\backslash frac\{\backslash sin(N\; \backslash ,\; x/2)\}\{\backslash sin(x/2)\}$$

Another variant is

$D\_n(x)\; -\; \backslash frac\{1\}\{2\}\backslash cos\; (nx)\; =\; \backslash frac\{\backslash sin\backslash left(nx\; \backslash right)\}\{2\; \backslash tan(\backslash frac\{x\}\{2\})\}$

and this can be easily proved by using an identity $\backslash sin\; (\backslash alpha\; +\; \backslash beta)\; =\; \backslash sin\; (\backslash alpha)\; \backslash cos\; (\backslash beta)\; +\; \backslash cos\; (\backslash alpha)\; \backslash sin(\backslash beta)$.

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See also

• Fejér kernel

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Sources

• (1997). 013458886X, Prentice-Hall. 013458886X
• Dirichlet kernel at PlanetMath

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