Favard Constants
N.B. A detailed on-line essay by S. Finchwas 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 
, thenthere 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, |  |  | |
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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.
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