P. L. Chebyshev solved the problem of finding a polynomial of degree n with leading coefficient one that has the least deviation from zero with respect to the maximum norm. In the case of entire functions, a similar problem can be solved for some classes. We find an entire function of exponential type σ such thatf σ ρ m for any nonzero entire function Q of type less than σ of class C. Bibliography: 5 titles.
Results of Chebyshev and Bernstein about polynomials with the smallest deviation from zero in a weighted norm are extended to entire functions of exponential type. Suppose that a function ρ m belongs to the Cartwright class, is of type m, and is positive on the real axis. Let σ ≥ m. Functions that have the smallest deviation from zero among the entire functions of type σ are constructed in the uniform and integral metrics.
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