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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 $$2^{-2r-1}$$.

##### MSC:
 42A10 Trigonometric approximation 41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities) 41A50 Best approximation, Chebyshev systems
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