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

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

where $\lambda$ is the Dirichlet Lambda Function and $\beta$ is the Dirichlet Beta Function. Explicitly,

$$K_0 = 1$$

$$K_1 = {\textstyle{1\over 2}}\pi$$

$$K_2 = {\textstyle{1\over 8}}\pi^2$$

$$K_3 = {\textstyle{1\over 24}}\pi^3.$$

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.