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On the exact constant in the Jackson-Stechkin inequality for the uniform metric. (English) Zbl 1181.41017

In the well-known Jackson-Stechkin inequality the value of the best approximation E n-1 (f) of a 2π-periodic function by trigonometric polynomials of degree n-1 is estimated by r-th modolus of smoothness ω r of f. This inequality has the form

E n-1 (f)c r w r (f;2π n),

where c r is some constant that depends only on r. The main result is that

(1-1 r+1)γ r * c r <5γ r * ,whereγ r * =1 r [r 2]r 1/2 2 r ·

Moreover, the same upper bound is valid for the constant c r,p in the Stechkin inequality for L p -metrics with p[1,)·


MSC:
41A17Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities)
41A44Best constants (approximations and expansions)
42A10Trigonometric approximation
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