Favard problem

The problem of calculating the least upper bound

$$ \tag{* } \sup _ {\begin{array}{c} {} \\ f \in W ^ {r} MX \end{array} } \ \inf _ {t _ {n} } \ \| f ( x) - t _ {n} ( x) \| _ {X} , $$

where the $ t _ {n} ( x) $ are trigonometric polynomials of order not exceeding $ n $, $ W ^ {r} MX $ is the class of periodic functions whose $ r $- th derivative in the sense of Weyl (see Fractional integration and differentiation) satisfies the inequality $ \| f ^ { ( r) } \| _ {X} \leq M $, and $ X = C [ 0, 2 \pi ] $. The Favard problem was posed by J. Favard [1]. Subsequently, broader classes of functions have been considered and a complete solution of the Favard problem for $ X = C, L $ and arbitrary $ r > 0 $ has been obtained as a corollary of more general results (see [2], [3]).

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