Entire functions of least deviation from zero in generalized Orlicz classes

Abstract:

Some results of S. N. Bernstein about polynomials of least deviation from zero in some weighted spaces $L_p$ are generalized to entire functions of exponential type.

Suppose that a function $\rho _m$ belongs to the Cartwright class, is of type $m$, is positive on the real axis, and let $\sigma \geq m$. Earlier, the authors constructed functions least deviating from zero among entire functions of type $\sigma$ in the uniform and integral metrics on $\mathbb {R}$ with the weights $\omega =1/\rho _m$ and $\omega =| \cdot |/\rho _m$.

In this paper it is shown that these functions deviate least from zero in some other classes related to the function $\rho _m$ and generalizing the Orlicz classes. In particular, the results are obtained for the spaces $L_p(\mathbb {R})$, $p <\infty$, with the weight $\omega ^p$ for the same $\omega$.