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Sharp Kolmogorov-type inequalities for moduli of continuity and best approximations by trigonometric polynomials and splines. (English. Russian original) Zbl 1078.42001
J. Math. Sci., New York 124, No. 2, 4845-4857 (2004); translation from Zap. Nauchn. Semin. POMI 290, 5-26 (2002).
Summary: In what follows, $$C$$ is the space of $$2\pi$$-periodic continuous functions; $$P$$ is a seminorm defined on $$C$$, shift-invariant, and majorized by the uniform norm; $$\omega_m(f,h)_P$$ is the $$m$$th modulus of continuity of a function $$f$$ with step $$h$$ and calculated with respect to $$P$$; $${\mathcal K}_r= \frac 4\pi \sum_{l=0}^\infty \frac {(-1)^{l(r+1)}} {(2l+1)^{r+1}}$$, $$B_r(x)= -\frac{r!}{2^{r-1}\pi^r} \sum_{k=1}^\infty \frac {\cos(2k\pi x-r\pi/2)}{k^r}$$ $$(r\in \mathbb N)$$, $$B_0(x)=1$$, $$\gamma_r= \frac {B_r(\frac12)}{r!}$$; $$(k)= k_1+\cdots+ k_m$$, $K_{r,m}= \{k\in \mathbb Z_+^m: 0\leq k_\nu\leq r+\nu-2-k_1-\cdots- k_{\nu-1}\},$ $A_{r,0}= \frac {2}{r!} \int_0^{1/2} \biggl|B_r(t)-B_r \biggl( \frac12\biggr)\biggr|\,dt,$ $A_{r,m}= \sum_{k\in K_{r,m}} \Biggl( \prod_{j=1}^m |\gamma_{k_j}| \Biggr) A_{r+m-(k),0}, \quad \Sigma_{r,m}= \sum_{\nu=0}^{m-1} 2^\nu A_{r,\nu},$ $M_{r,m}(f,h)_P= \begin{cases} \Sigma_{r,m}^{-1} \sum_{\nu=0}^{m-1} A_{r,\nu} \omega_\nu(f,h)_P, &r\text{ is even},\\ \Sigma_{r,m}^{-1} \left( \frac{A_{r,0}}{2} \omega_1(f,h)_P+ \sum_{\nu=1}^{m-1} A_{r,\nu} \omega_\nu(f,h)_P \right), &r\text{ is odd}. \end{cases}$ Theorem 1. Let $$r,m\in\mathbb N$$, $$n,\lambda> 0$$, $$f\in C^{(r+m)}$$. Then $\begin{split} P(f^{(m)})\leq \lambda^r \Biggl\{ \Sigma_{r,m}+ 2^m \sum_{k\in K_{r,m}} \biggl( \prod_{j=1}^m |\gamma_{k_j}| \biggr) \frac {{\mathcal K}_{r+m-(k)}} {\lambda^{r+m-(k)}} \Biggr\}\\ \times\max \Biggl\{\biggl( \frac {\omega_m \bigl(f, \frac \lambda n\bigr)_P} {{\mathcal K}_{r+m} 2^m} \biggr)^{\frac{r} {r+m}} M_{r,m}^{\frac{m}{r+m}} \biggl(f^{(r+m)}, \frac \lambda n\biggr)_P, \frac {n^m\omega_m \bigl(f, \frac \lambda n\bigr)_P} {{\mathcal K}_{r+m} 2^m}\Biggr\}. \end{split}$ For some values of $$\lambda$$ and seminorms related to best approximations by trigonometric polynomials and splines in the uniform and integral metrics, the inequalities are sharp.

##### MSC:
 42A10 Trigonometric approximation 41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities) 41A15 Spline approximation 26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable 26D10 Inequalities involving derivatives and differential and integral operators 41A50 Best approximation, Chebyshev systems
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