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Keywords:
Bernstein theorems; interpolating operators; Bernstein's inequality; function of exponential type; uniform norm; space of uniformly continuous functions
Bernstein theorems; interpolating operators; Bernstein's inequality; function of exponential type; uniform norm; space of uniformly continuous functions
Summary:
One of the Bernstein theorems that the class of bounded functions of the exponential type is dense in the space of bounded and uniformly continuous functions. This theorem follows from a convergence theorem for some interpolating operators on the real axis.
References:
[1] С. H. Бернштейн: Экстремальные свойства полиномов и наилучшее приближение непрерывных функций одной вещственной переменной. (Extremal properties of polynomials and the best approximations of continuous functions of one real variable.), Гонти, 1937. Zbl 0131.10103
[2] С. H. Бернштейн: О наилучшем приближении непрерывных функций на всей вещественной оси при помощи целых функций данной степени I. (On the best approximation of continuous functions on the whole real axis in terms of entire functions of a given degree I.) Сочинения, т. II, 1946. Zbl 0074.10805
[3] А. Ф. Тиман: Теория приближения функций действительного переменного. (Theory of approximation of functions of real variable.), Госиздат физмат лит, Moskva, 1960. Zbl 1004.90500
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