Sharp Estimates of Linear Approximations by Nonperiodic Splines in Terms of Linear Combinations of Moduli of Continuity

Abstract

Assume that σ > 0, r, μ ???? ℕ, μ ≥ r + 1, r is odd, p ???? [1,+∞], and \( f\kern0.5em \in \kern0.5em {W}_p^{(r)}\left(\mathrm{\mathbb{R}}\right) \). We construct linear operators X_σ,r,μ whose values are splines of degree μ and of minimal defect with knots \( \frac{k\pi}{\sigma },k\in \mathrm{\mathbb{Z}} \), such that

\( {\displaystyle \begin{array}{l}{\left\Vert f-{X}_{\sigma, r,u}(f)\right\Vert}_p\le {\left(\frac{\pi }{\sigma}\right)}^r\left\{\frac{A_r,0}{2}\left.{\upomega}_1\right|{\left({f}^{(r)},\frac{\pi }{\sigma}\right)}_p+\sum \limits_{v=1}^{u-r-1}{A}_{r,v}{\omega}_v{\left({f}^{(r)},\frac{\pi }{\sigma}\right)}_p\right\}\\ {}\kern9em +{\left(\frac{\pi }{\sigma}\right)}^r\left(\frac{{\mathcal{K}}_r}{\pi^r}-\sum \limits_{v=0}^{u-r-1}{2}^v{A}_{r,v}\right){2}^{r-\mu }{\omega}_{\mu -r}{\left({f}^{(r)},\frac{\pi }{\sigma}\right)}_p,\end{array}} \) where for p = 1, . . . ,+∞, the constants cannot be reduced on the class \( {W}_p^{(r)}\left(\mathrm{\mathbb{R}}\right) \). Here \( {\mathcal{K}}_r=\frac{4}{\pi}\sum \limits_{l=0}^{\infty}\frac{{\left(-1\right)}^{l\left(r+1\right)}}{{\left(2l+1\right)}^{r+1}} \) are the Favard constants, the constants A_r,ν are constructed explicitly, and ω_v is a modulus of continuity of order ν. As a corollary, we get the sharp Jackson type inequality

\( {\left\Vert f-{X}_{\sigma, r,\mu }(f)\right\Vert}_p\le \frac{{\mathcal{K}}_r}{2{\sigma}^r}{\omega}_1{\left({f}^{(r)},\frac{\uppi}{\sigma}\right)}_p. \)