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Lebesgue Constants

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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 f is integrable over the interval [-pi,pi] and S_n(f,x) is the nth partial sum of the Fourier series of f, so that

a_k=1/piint_(-pi)^pif(t)cos(kt)dt
(1)
b_k=1/piint_(-pi)^pif(t)sin(kt)dt
(2)

and

S_n(f,x)=1/2a_0+{sum_(k=1)^n[a_kcos(kx)+b_ksin(kx)]}.
(3)

If

|f(x)|<=1
(4)

for all x, then

S_n(f,x)<=1/piint_0^pi(|sin[1/2(2n+1)theta]|)/(sin(1/2theta))dtheta=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=1/3+(2sqrt(3))/pi
(7)
=1.435991124...
(8)
L_2=1/5+(sqrt(25-2sqrt(5)))/pi=1.642188435...
(9)
L_3=1/7+1/(3pi)[22sin(pi/7)-2cos(pi/(14))+10cos((3pi)/(14))]
(10)
=1.778322861....
(11)
L_4=(13)/(2sqrt(3)pi)+1/9+1/pi[7sin((2pi)/9)-5sin(pi/9)-cos(pi/(18))]
(12)
=1.880080599....
(13)

Some sum formulas for L_n include

L_n=1/(2n+1)+2/pisum_(k=1)^(n)1/ktan((pik)/(2n+1))
(14)
=(16)/(pi^2)sum_(k=1)^(infty)sum_(j=1)^((2n+1)k)1/(4k^2-1)1/(2j-1)
(15)

(Zygmund 1959) and integral formulas include

L_n=4int_0^infty(tanh[(2n+1)x])/(tanhx)(dx)/(pi^2+4x^2)
(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 n,

4/(pi^2)lnn<L_n<3+4/(pi^2)lnn.
(18)

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

|(d^rf)/(dx^r)|<=1
(19)

for all x. In this case,

|f(x)-S_n(f,x)|<=L_(n,r)=4/(pi^2)(lnn)/(n^r)+O(1/(n^r)),
(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

lim_(n->infty)[L_n-4/(pi^2)ln(2n+1)]=c,
(22)

where

c=8/(pi^2)(sum_(k=1)^(infty)(lnk)/(4k^2-1))-4/(pi^2)(Gamma^'(1/2))/(Gamma(1/2))
(23)
=8/(pi^2)[sum_(j=0)^(infty)(lambda(2j+2)-1)/(2j+1)]+4/(pi^2)(2ln2+gamma)
(24)
=0.9894312738...
(25)

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

Define the nth Lebesgue constant for the Lagrange interpolating polynomial by

Lambda_n(X)=max_(-1<=x<=1)sum_(k=1)^n|product_(j!=k)(x-x_j)/(x_k-x_j)|.
(26)

It is then true that

Lambda_n>4/(pi^2)lnn-1.
(27)

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

Lambda_n>2/pilnn-C
(28)

for all n. Erdős (1961) further showed that

Lambda_n<2/pilnn+4,
(29)

so (◇) cannot be improved upon.

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References

Finch, S. R. "Lebesgue Constants." §4.2 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 250-255, 2003.Erdős, P. "Problems and Results on the Theory of Interpolation, II." Acta Math. Acad. Sci. Hungary 12, 235-244, 1961.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.

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Lebesgue Constants

Cite this as:

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

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