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The Jackson-Stechkin inequality in \(L^2\) with trigonometric modulus of continuity. (English. Russian original) Zbl 0960.42001
Math. Notes 65, No. 6, 777-781 (1999); translation from Mat. Zametki 65, No. 6, 928-932 (1999).
In the paper a new Jackson-Stechkin type inequality between the best \(L^2\)-approximation of an arbitrary complex \(2\pi\)-periodic function and its trigonometric modulus of continuity is obtained. Let \(L^2\) be the Hilbert space of complex \(2\pi\)-periodic functions with the norm \[ \|f\|=\left({1\over 2\pi}\int_{0}^{2\pi}|f(x)|^{2} dx\right)^{1/2} \] and \(T_{n}\) its subspace consisting of trigonometric polynomials of order \(n\). We define the best approximation of a function \(f\in L^2\) by trigonometric polynomials of order \(n\): \(E_n(f)=\min\{\|f-g\|:g\in T_{n}\}\). For a natural number \(r\) and a real number \(t\) we set \[ \Delta_t^r f(x)=(f(x+t)-f(x))\prod_{j=1}^r \{f(x+2t)-2f(x+t)\cos jt+f(x)\}. \] The trigonometric modulus of continuity of the function \(f\in L^2\) is defined by the equality \(\omega_{L}(f;\delta)=\sup\{\|\Delta_t^r f\|:|t|\leq\delta\}\).
The main result of the paper is the following theorem.
Theorem 2. For any function \(f\in L^2\) and any natural number \(n\geq 2r+1\) the inequality \[ E_{n-1}(f)\leq K\omega_L\left(f,{\pi \over n}\right) \tag{1} \] is valid with the constant \(K< 2^{-r-{1\over 2}}/\prod_{j=1}^r \cos {j\pi\over 2r+1}\). Besides, in the case \(n\geq r+1\) the constant \(K\) in \((1)\) cannot be less than \(2^{-2r-1}\).

42A10 Trigonometric approximation
41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiń≠-type inequalities)
41A50 Best approximation, Chebyshev systems
Full Text: DOI
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