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Favard Constants

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

Let $T_n(x)$ be an arbitrary trigonometric Polynomial

T_n(x)={\textstyle{1\over 2}}a_0 +\left\{{\sum_{k=1}^n [a_k\cos(kx)+b_k\sin(kx)]}\right\},

where the Coefficients are real. Let the $r$th derivative of $T_n(x)$ be bounded in $[-1,1]$, then there exists a Polynomial $T_n(x)$ for which

\vert f(x)-T_n(x)\vert\leq {K_r\over (n+1)^r},

for all $x$, where $K_r$ is the $r$th Favard constant, which is the smallest constant possible.

K_r={4\over\pi} \sum_{k=0}^\infty \left[{(-1)^k\over 2k+1}\right]^{r+1}.

These can be expressed by

{4\over\pi} \lambda(r+1) & for $r$\ odd\cr
{4\over\pi} \beta(r+1) & for $r$\ even,\cr}

where $\lambda$ is the Dirichlet Lambda Function and $\beta$ is the Dirichlet Beta Function. Explicitly,
$\displaystyle K_0$ $\textstyle =$ $\displaystyle 1$  
$\displaystyle K_1$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}\pi$  
$\displaystyle K_2$ $\textstyle =$ $\displaystyle {\textstyle{1\over 8}}\pi^2$  
$\displaystyle K_3$ $\textstyle =$ $\displaystyle {\textstyle{1\over 24}}\pi^3.$  


Finch, S. ``Favorite Mathematical Constants.''

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.

© 1996-9 Eric W. Weisstein