## Lebesgue Constants (Fourier Series)

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

Assume a function is integrable over the interval and is the th partial sum of the Fourier Series of , so that

 (1) (2)

and
 (3)

If
 (4)

for all , then
 (5)

and is the smallest possible constant for which this holds for all continuous . The first few values of are
 (6) (7) (8) (9)

Some Formulas for include
 (10)

(Zygmund 1959) and integral Formulas include
 (11)

(Hardy 1942). For large ,
 (12)

This result can be generalized for an -differentiable function satisfying

 (13)

for all . In this case,
 (14)

where
 (15)

(Kolmogorov 1935, Zygmund 1959).

Watson (1930) showed that

 (16)

where
 (17) (18) (19)

where is the Gamma Function, is the Dirichlet Lambda Function, and 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.