Dirichlet kernel

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

$${\displaystyle D_{n}(x)=\sum _{k=-n}^{n}e^{ikx}=\left(1+2\sum _{k=1}^{n}\cos(kx)\right)={\frac {\sin \left(\left(n+1/2\right)x\right)}{\sin(x/2)}},}$$ where n is any nonnegative integer. The kernel functions are periodic with period ${\displaystyle 2\pi }$.

Plot restricted to one period ${\displaystyle [-L,L],~L=\pi ,~}$ of the first few Dirichlet kernels showing their convergence to one of the Dirac delta distributions of the Dirac comb.

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 $${\displaystyle (D_{n}*f)(x)=\int _{-\pi }^{\pi }f(y)D_{n}(x-y)\,dy=2\pi \sum _{k=-n}^{n}{\hat {f}}(k)e^{ikx},}$$ where $${\displaystyle {\widehat {f}}(k)={\frac {1}{2\pi }}\int _{-\pi }^{\pi }f(x)e^{-ikx}\,dx}$$ 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.

Plot restricted to one period of the first few Dirichlet kernels (multiplied by ${\displaystyle 2\pi }$).

L1 norm of the kernel function

Of particular importance is the fact that the L¹ norm of D_n on ${\displaystyle [0,2\pi ]}$ diverges to infinity as n → ∞. One can estimate that $${\displaystyle \|D_{n}\|_{L^{1}}=\Omega (\log n).}$$

By using a Riemann-sum argument to estimate the contribution in the largest neighbourhood of zero in which ${\displaystyle D_{n}}$ is positive, and Jensen's inequality for the remaining part, it is also possible to show that: $${\displaystyle \|D_{n}\|_{L^{1}}\geq 4\operatorname {Si} (\pi )+{\frac {8}{\pi }}\log n}$$ where ${\textstyle \operatorname {Si} (x)}$ is the sine integral ${\textstyle \int _{0}^{x}(\sin t)/t\,dt.}$

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 ${\displaystyle \|D_{n}\|_{L^{1}[0,2\pi ]}=\Omega (\log n)}$ is given by

$${\displaystyle {\begin{aligned}\int _{0}^{2\pi }|D_{n}(x)|\,dx&\geq \int _{0}^{\pi }{\frac {\left|\sin[(2n+1)x]\right|}{x}}\,dx\\[5pt]&\geq \sum _{k=0}^{2n}\int _{k\pi }^{(k+1)\pi }{\frac {\left|\sin s\right|}{s}}\,ds\\[5pt]&\geq \left|\sum _{k=0}^{2n}\int _{0}^{\pi }{\frac {\sin s}{(k+1)\pi }}\,ds\right|\\[5pt]&={\frac {2}{\pi }}H_{2n+1}\\[5pt]&\geq {\frac {2}{\pi }}\log(2n+1),\end{aligned}}}$$

where we have used the Taylor series identity that ${\displaystyle 2/x\leq 1/\left|\sin(x/2)\right|}$ and where ${\displaystyle H_{n}}$ are the first-order harmonic numbers.

Relation to the periodic delta function

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

${\displaystyle \sum _{m=-\infty }^{\infty }e^{\pm i\omega mT}={\frac {2\pi }{T}}\sum _{k=-\infty }^{\infty }\delta (\omega -2\pi k/T)={\frac {1}{T}}\sum _{k=-\infty }^{\infty }\delta (\xi -k/T)~,}$

with the angular frequency ${\displaystyle \omega =2\pi \xi }$.

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

${\displaystyle {\mathcal {F}}\left[D_{n}(2\pi x)\right](\xi )={\mathcal {F}}^{-1}\left[D_{n}(2\pi x)\right](\xi )=\int _{-\infty }^{\infty }D_{n}(2\pi x)e^{\pm i2\pi \xi x}\,dx=\sum _{k=-n}^{+n}\delta (\xi -k)\equiv \operatorname {comb} _{n}(\xi )}$

${\displaystyle {\mathcal {F}}\left[\operatorname {comb} _{n}\right](x)={\mathcal {F}}^{-1}\left[\operatorname {comb} _{n}\right](x)=\int _{-\infty }^{\infty }\operatorname {comb} _{n}(\xi )e^{\pm i2\pi \xi x}\,d\xi =D_{n}(2\pi x),}$

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

In a different vein, consider ∆(x) as the identity element for convolution on functions of period 2π. In other words, we have $${\displaystyle f*(\Delta )=f}$$ for every function f of period 2π. The Fourier series representation of this "function" is $${\displaystyle \Delta (x)\sim \sum _{k=-\infty }^{\infty }e^{ikx}=\left(1+2\sum _{k=1}^{\infty }\cos(kx)\right).}$$

(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 convergence mentioned above).

Proof of the trigonometric identity

The trigonometric identity $${\displaystyle \sum _{k=-n}^{n}e^{ikx}={\frac {\sin((n+1/2)x)}{\sin(x/2)}}}$$ displayed at the top of this article may be established as follows. First recall that the sum of a finite geometric series is $${\displaystyle \sum _{k=0}^{n}ar^{k}=a{\frac {1-r^{n+1}}{1-r}}.}$$

In particular, we have $${\displaystyle \sum _{k=-n}^{n}r^{k}=r^{-n}\cdot {\frac {1-r^{2n+1}}{1-r}}.}$$

Multiply both the numerator and the denominator by ${\displaystyle r^{-1/2}}$, getting $${\displaystyle {\frac {r^{-n-1/2}}{r^{-1/2}}}\cdot {\frac {1-r^{2n+1}}{1-r}}={\frac {r^{-n-1/2}-r^{n+1/2}}{r^{-1/2}-r^{1/2}}}.}$$

In the case ${\displaystyle r=e^{ix}}$ we have $${\displaystyle \sum _{k=-n}^{n}e^{ikx}={\frac {e^{-(n+1/2)ix}-e^{(n+1/2)ix}}{e^{-ix/2}-e^{ix/2}}}={\frac {-2i\sin((n+1/2)x)}{-2i\sin(x/2)}}={\frac {\sin((n+1/2)x)}{\sin(x/2)}}}$$ as required.

Alternative proof of the trigonometric identity

Start with the series $${\displaystyle f(x)=1+2\sum _{k=1}^{n}\cos(kx).}$$ Multiply both sides by ${\textstyle \sin(x/2)}$ and use the trigonometric identity $${\displaystyle \cos(a)\sin(b)={\frac {\sin(a+b)-\sin(a-b)}{2}}}$$ to reduce the terms in the sum. $${\displaystyle \sin(x/2)f(x)=\sin(x/2)+\sum _{k=1}^{n}\left(\sin((k+{\tfrac {1}{2}})x)-\sin((k-{\tfrac {1}{2}})x)\right)}$$ 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 that is not centered), then using similar techniques we can show the following identity: $${\displaystyle \sum _{k=0}^{N-1}e^{ikx}=e^{i(N-1)x/2}{\frac {\sin(N\,x/2)}{\sin(x/2)}}}$$Another variant is ${\displaystyle D_{n}(x)-{\frac {1}{2}}\cos(nx)={\frac {\sin \left(nx\right)}{2\tan({\frac {x}{2}})}}}$ and this can be easily proved by using an identity ${\displaystyle \sin(\alpha +\beta )=\sin(\alpha )\cos(\beta )+\cos(\alpha )\sin(\beta )}$.^[1]

See also

• Fejér kernel

References

1. Fay, Temple H.; Kloppers, P. Hendrik (2001). "The Gibbs' phenomenon". International Journal of Mathematical Education in Science and Technology. 32 (1): 73–89. doi:10.1080/00207390117151. S2CID 120595055.

Sources

• Andrew M. Bruckner, Judith B. Bruckner, Brian S. Thomson: Real Analysis. ClassicalRealAnalysis.com 1996, ISBN 0-13-458886-X, 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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• Dirichlet kernel at PlanetMath