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Favard Constants


Let T_n(x) be an arbitrary trigonometric polynomial

 T_n(x)=1/2a_0+{sum_(k=1)^n[a_kcos(kx)+b_ksin(kx)]}
(1)

with real coefficients, let f be a function that is integrable over the interval [-pi,pi], and let the rth derivative of f be bounded in [-1,1]. Then there exists a polynomial T_n(x) for which

 |f(x)-T_n(x)|<=(K_r)/((n+1)^r),
(2)

for all x in [-pi,pi], where K_r is the smallest constant possible, known as the rth Favard constant.

K_r can be given explicitly by the sum

 K_r=4/pisum_(k=0)^infty[((-1)^k)/(2k+1)]^(r+1),
(3)

which can be written in terms of the Lerch transcendent as

 K_r=(2^(1-r))/piPhi((-1)^(r+1),r+1,1/2).
(4)

These can be expressed by

 K_r={4/pilambda(r+1)   for r even; 4/pibeta(r+1)   for r odd,
(5)

where lambda(x) is the Dirichlet lambda function and beta(x) is the Dirichlet beta function. Explicitly,

K_0=1
(6)
K_1=1/2pi
(7)
K_2=1/8pi^2
(8)
K_3=1/(24)pi^3
(9)
K_4=5/(384)pi^4
(10)
K_5=1/(240)pi^5
(11)

(OEIS A050970 and A050971).


See also

Dirichlet Beta Function, Dirichlet Lambda Function

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References

Finch, S. R. "Achieser-Krein-Favard Constants." §4. 2 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 255-257, 2003.Kolmogorov, A. N. "Zur Grössenordnung des Restgliedes Fourierscher reihen differenzierbarer Funktionen." Ann. Math. 36, 521-526, 1935.Sloane, N. J. A. Sequences A050970 and A050970 in "The On-Line Encyclopedia of Integer Sequences."Zygmund, A. G. Trigonometric Series, Vols. 1-2, 2nd ed. New York: Cambridge University Press, 1959.

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Favard Constants

Cite this as:

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

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