Lebesgue Constants (Fourier Series)

N.B. A detailed on-line essay by S. Finch was the starting point for this entry.

Assume a function $f$ is integrable over the interval $[-\pi, \pi]$ and $S_n(f,x)$ is the $n$th partial sum of the Fourier Series of $f$, so that

$$a_k = {1\over\pi}\int_{-\pi}^\pi f(t)\cos(kt)\,dt$$ (1)

$$b_k = {1\over\pi}\int_{-\pi}^\pi f(t)\sin(kt)\,dt$$ (2)

and

$$S_n(f,x)={\textstyle{1\over 2}}a_0+\left\{{\sum_{k=1}^n [a_k\cos(kx)+b_k\sin(kx)]}\right\}.$$ (3)

If

$$\vert f(x)\vert\leq 1$$ (4)

for all $x$, then

$$S_n(f,x)\leq {1\over\pi}\int_0^\pi {\vert\sin[{\textstyle{1\... ...ta]\vert\over\sin({\textstyle{1\over 2}}\theta)}\,d\theta=L_n,$$ (5)

and $L_n$ is the smallest possible constant for which this holds for all continuous $f$. The first few values of $L_n$ are

$$L_0 = 1$$ (6)

$$L_1 = {1\over 3}+{2\sqrt{3}\over\pi}=1.435991124\ldots$$ (7)

$$L_2 = 1.642188435\ldots$$ (8)

$$L_3 = 1.778322862.$$ (9)

Some Formulas for $L_n$ include

$$L_n = {1\over 2n+1}+{2\over\pi}\sum_{k=1}^n {1\over k}\tan\left({\pi k\over 2n+1}\right)$$

$$= {16\over \pi^2}\sum_{k=1}^\infty \sum_{j=1}^{(2n+1)k} {1\over 4k^2-1} {1\over 2j-1}$$ (10)

(Zygmund 1959) and integral Formulas include

$$L_n = 4\int_0^\infty {\tanh[(2n+1)x]\over\tanh x} {dx\over \pi^2+4x^2}$$

$$= {4\over\pi^2} \int_0^\infty {\sinh[(2n+1)x]\over\sinh x} \ln\{\coth[{\textstyle{1\over 2}}(2n+1)x]\}\,dx$$ (11)

(Hardy 1942). For large $n$,

$${4\over\pi^2}\ln n < L_n < 3+{4\over\pi^2}\ln n.$$ (12)

This result can be generalized for an $r$-differentiable function satisfying

$$\left\vert{d^rf\over dx^r}\right\vert\leq 1$$ (13)

for all $x$. In this case,

$$\vert f(x)-S_n(f,x)\vert\leq L_{n,r}={4\over\pi^2}{\ln n\over n^r}+{\mathcal O}\left({1\over n^r}\right),$$ (14)

where

$$L_{n,r}=\cases{ {1\over\pi}\int_{-\pi}^\pi \left\vert{\sum_... ... {\cos(kx)\over k^r}}\right\vert\,dx & for $r\geq 1$\ even\cr}$$ (15)

(Kolmogorov 1935, Zygmund 1959).

Watson (1930) showed that

$$\lim_{n\to\infty} \left[{L_n-{4\over\pi^2}\ln(2n+1)}\right]=c,$$ (16)

where

$$c = {8\over\pi^2}\left({\,\sum_{k=1}^\infty {\ln k\over 4k^2-1}}\righ... ...ver\pi^2} {\Gamma'({\textstyle{1\over 2}})\over \Gamma({\textstyle{1\over 2}})}$$ (17)

$$= {8\over \pi^2}\left[{\,\sum_{j=0}^\infty {\lambda(2j+2)-1\over 2j+1}}\right]+{4\over\pi^2} (2\ln 2+\gamma)$$ (18)

$$= 0.9894312738\ldots,$$ (19)

where $\Gamma(z)$ is the Gamma Function, $\lambda(z)$ is the Dirichlet Lambda Function, and $\gamma$ is the Euler-Mascheroni Constant.

References

Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/lbsg/lbsg.html

Hardy, G. H. ``Note on Lebesgue's Constants in the Theory of Fourier Series.'' J. London Math. Soc. 17, 4-13, 1942.

Kolmogorov, A. N. ``Zur Grössenordnung des Restgliedes Fourierscher reihen differenzierbarer Funktionen.'' Ann. Math. 36, 521-526, 1935.

Watson, G. N. ``The Constants of Landau and Lebesgue.'' Quart. J. Math. Oxford 1, 310-318, 1930.

Zygmund, A. G. Trigonometric Series, 2nd ed., Vols. 1-2. Cambridge, England: Cambridge University Press, 1959.