Let  be an arbitrary trigonometric polynomial

{sum_(k=1)^n[a_kcos(kx)+b_ksin(kx)]} " src="https://mathworld.wolfram.com/images/equations/FavardConstants/NumberedEquation1.gif" style="height:45px; width:274px" />

(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

{4/pilambda(r+1) for r even; 4/pibeta(r+1) for r odd, " src="https://mathworld.wolfram.com/images/equations/FavardConstants/NumberedEquation5.gif" style="height:82px; width:177px" />

(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.