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Favard Constants

Let 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 be a function that is integrable over the interval , and let the th derivative of be bounded in . Then there exists a polynomial for which

(2)

for all , where is the smallest constant possible, known as the th Favard constant.

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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