## Some extremal functions in Fourier analysis, III

##### Authors
Carneiro, Emanuel
Vaaler, Jeffrey D.
##### Description
We obtain the best approximation in $L^1(\R)$, by entire functions of exponential type, for a class of even functions that includes $e^{-\lambda|x|}$, where $\lambda >0$, $\log |x|$ and $|x|^{\alpha}$, where $-1 < \alpha < 1$. We also give periodic versions of these results where the approximating functions are trigonometric polynomials of bounded degree.
Comment: 26 pages. Submitted
##### Keywords
Mathematics - Classical Analysis and ODEs, Mathematics - Complex Variables, 41A30, 41A52, 42A05