### Article

## A Nonperiodic Spline Analog of the Akhiezer–Krein–Favard Operators

In what follows, $\sigma>0$, $m,r\in\mathbb N$, $m\geqslant r$, ${\mathbf S}_{\sigma, m}$ is the space of splines of order~$m$ and minimal defect with nodes $\frac{j\pi}{\sigma}$ ($j\in\mathbb Z$), $A_{\sigma,m}(f)_{p}$ is the best approximation of a function~$f$ by the set ${\mathbf S}_{\sigma,m}$ in the space $L_p(\mathbb R)$. It is known that for $p=1,+\infty$

$$

\supl_{f\in W^{(r)}_{p}(\mathbb R)}

\frac{A_{\sigma,m}(f)_{p}}{\|f^{(r)}\|_{p}}=

\frac{{\mathcal K}_r}{\sigma^r}.\eqno(1)

$$

In this paper we construct linear operators ${\mathcal X}_{\sigma,r,m}$ with their values in ${\mathbf S}_{\sigma,m}$, such that for all $p\in[1,+\infty]$ and $f\in W_p^{(r)}(\mathbb R)$

$$

\|f-{\mathcal X}_{\sigma,r,m}(f)\|_{p}\leqslant

\frac{{\mathcal K}_r}{\sigma^r}\|f^{(r)}\|_p.

$$

So we establish the possibility to achieve the upper bounds in~(1) by linear methods of approximation, which was unknown before.

We establish several Jackson type inequalities with explicit constants for spline approximation of functions deﬁned on the real ax is. The inequalities for the ﬁrst modulus of continuity of odd derivatives are sharp. We also obtain inequalities for high-order moduli of continuity of a function itself. One of the inequalities for th e second modulus of continuity is sharp. Up to the present paper no estimates for spline approximation on the axis in terms of high-order moduli of continuity, with constants written explicitly, were known.

Suppose that $\sigma>0$, $r,\mu\in{\mathbb N}$, $\mu\geqslant r+1$, $r$ is

odd,

$p\in[1,+\infty]$, $f\in W^{(r)}_{p}(\mathbb R)$.

We construct linear operators ${\mathcal X}_{\sigma,r,\mu}$ whose

values are splines of degree~$\mu$ and of minimal defect with knots

$\frac{k\pi}{\sigma}$ ($k\in\mathbb Z$) such that

{\allowdisplaybreaks

\begin{align*}

&\|f-{\mathcal X}_{\sigma,r,\mu}(f)\|_{p}

\\

&\qquad\leqslant

\left(\frac{\pi}{\sigma}\right)^{\!r}\left\{\frac{A_{r,0}}{2}

\,\omega_{1}\left(f^{(r)},\frac{\pi}{\sigma}\right)_{\!p}+

\sum\limits_{\nu=1}^{\mu-r-1}A_{r,\nu}

\omega_{\nu}\left(f^{(r)},\frac{\pi}{\sigma}\right)_{\!p}\right\}

\\

&\qquad+\left(\frac{\pi}{\sigma}\right)^{\!r}\biggl(

\frac{{\mathcal K}_r}{\pi^r}-

\sum\limits_{\nu=0}^{\mu-r-1}2^{\nu}A_{r,\nu}

\biggr)

2^{r-\mu}\omega_{\mu-r}\left(f^{(r)},\frac{\pi}{\sigma}\right)_{\!p},

\end{align*}}

where for $p=1,\dots, +\infty$ the constants cannot be reduced on the

class $W^{(r)}_{p}(\mathbb R)$.

Here

${\mathcal K}_r=\frac{4}{\pi}\sum\limits_{l=0}^{\infty}

\frac{(-1)^{l(r+1)}}{(2l+1)^{r+1}}$ are the Favard constants,

the constants $A_{r,\nu}$ are constructed explicitly,

$\omega_{\nu}$ is a modulus of continuity of order~$\nu$.

As a corollary, we get the sharp Jackson type inequality

$$

\|f-{\mathcal X}_{\sigma,r,\mu}(f)\|_{p}\leqslant

\frac{{\mathcal K}_r}{2\sigma^r}\,

\omega_1\left(f^{(r)},\frac{\pi}{\sigma}\right)_p.

$$

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.

This proceedings publication is a compilation of selected contributions from the “Third International Conference on the Dynamics of Information Systems” which took place at the University of Florida, Gainesville, February 16–18, 2011. The purpose of this conference was to bring together scientists and engineers from industry, government, and academia in order to exchange new discoveries and results in a broad range of topics relevant to the theory and practice of dynamics of information systems. Dynamics of Information Systems: Mathematical Foundation presents state-of-the art research and is intended for graduate students and researchers interested in some of the most recent discoveries in information theory and dynamical systems. Scientists in other disciplines may also benefit from the applications of new developments to their own area of study.