Lebesgue Constants

There are two sets of constants that are commonly known as Lebesgue constants. The first is related to approximation of function via Fourier series, which the other arises in the computation of Lagrange interpolating polynomials.

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

			(1)
			(2)

and

$S_n(f,x)=1/2a_0+{sum_(k=1)^n[a_kcos(kx)+b_ksin(kx)]}.$

(3)

(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)
			(10)
			(11)
			(12)
			(13)

Some sum formulas for include

			(14)
			(15)

(Zygmund 1959) and integral formulas include

			(16)
		$4/(pi^2)int_0^infty(sinh[(2n+1)x])/(sinhx)ln{coth[1/2(2n+1)x]}dx$	(17)

(Hardy 1942). For large ,

(18)

This result can be generalized for an -differentiable function satisfying

(19)

for all . In this case,

(20)

where

$L_(n,r)={1/piint_(-pi)^pi|sum_(k=n+1)^(infty)(sin(kx))/(k^r)|dx for r>=1 odd; 1/piint_(-pi)^pi|sum_(k=n+1)^(infty)(cos(kx))/(k^r)|dx for r>=1 even$

(21)

(Kolmogorov 1935, Zygmund 1959).

Watson (1930) showed that

(22)

where

			(23)
			(24)
			(25)

(OEIS A086052), where is the gamma function, is the Dirichlet lambda function, and is the Euler-Mascheroni constant.

Define the th Lebesgue constant for the Lagrange interpolating polynomial by

(26)

It is then true that

(27)

The efficiency of a Lagrange interpolation is related to the rate at which increases. Erdős (1961) proved that there exists a positive constant such that

(28)

for all . Erdős (1961) further showed that

(29)

so (◇) cannot be improved upon.

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References

Erdős, P. "Problems and Results on the Theory of Interpolation, II." Acta Math. Acad. Sci. Hungary 12, 235-244, 1961.Finch, S. R. "Lebesgue Constants." §4.2 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 250-255, 2003.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.Sloane, N. J. A. Sequence A086052 in "The On-Line Encyclopedia of Integer Sequences."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.