## Favard Constants

N.B. A detailed on-line essay by S. Finch was the starting point for this entry.

Let be an arbitrary trigonometric Polynomial

where the Coefficients are real. Let the th derivative of be bounded in , then there exists a Polynomial for which

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

These can be expressed by

where is the Dirichlet Lambda Function and is the Dirichlet Beta Function. Explicitly,

References

Finch, S. Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/favard/favard.html

Kolmogorov, A. N. Zur Grössenordnung des Restgliedes Fourierscher reihen differenzierbarer Funktionen.'' Ann. Math. 36, 521-526, 1935.

Zygmund, A. G. Trigonometric Series, Vols. 1-2, 2nd ed. New York: Cambridge University Press, 1959.