Favard Constants -- from Wolfram MathWorld

Favard Constants

The th Favard constant is the smallest constant for which, whenever is a function that is integrable over the interval and the th derivative of is bounded in , there exists a trigonometric polynomial of the form

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

(1)

The polynomial has real coefficients and satisfies

(2)

for all .

can be given explicitly by the sum

(3)

which can be written in terms of the Lerch transcendent as

(4)

These can be expressed by

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

(5)

where is the Dirichlet lambda function and is the Dirichlet beta function. Explicitly,

			(6)
			(7)
			(8)
			(9)
			(10)
			(11)

(OEIS A050970 and A050971).

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.

Favard Constants

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