Lebesgue Constants

The term "Lebesgue constants" refers to either of two commonly used sets. The first is related to approximation of functions via Fourier series, while 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 functions . The first few values of are

			(6)
			(7)
			(8)
			(9)
			(10)

			(11)
			(12)

Some sum formulas for include

			(13)
			(14)

(Zygmund 1959) and integral formulas include

			(15)
		$4/(pi^2)int_0^infty(sinh[(2n+1)x])/(sinhx)ln{coth[1/2(2n+1)x]}dx$	(16)

(Hardy 1942). For large ,

(17)

This result can be generalized for a -periodic function with continuous derivatives, where is an integer, satisfying

(18)

for all . In this case,

(19)

where

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

(20)

(Kolmogorov 1935, Zygmund 1959).

Watson (1930) showed that

(21)

where

			(22)
			(23)
			(24)

(OEIS A243277), where is the gamma function, is the Dirichlet lambda function, and is the Euler-Mascheroni constant.

For a positive integer and distinct nodes $X={x_1,...,x_n}$ with , define the th Lebesgue constant for the Lagrange interpolating polynomial by

(25)

It is then true that

(26)

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

(27)

for all . For the nodes given by the roots of the Chebyshev polynomial of the first kind , Erdős (1961) notes that

(28)

where is a positive constant independent of . Thus the leading coefficient of in (27) cannot be increased.

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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 A243277 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.

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Lebesgue Constants

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Weisstein, Eric W. "Lebesgue Constants." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/LebesgueConstants.html

Lebesgue Constants

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