arXiv subjects

Kirill A. Kopotun

Publications and source records attributed to Kirill A. Kopotun.

9 recordsLinked to original sources

arXiv2023

Onesided, intertwining, positive and copositive polynomial approximation with interpolatory constraints

Given $k\in N$, a nonnegative function $f\in C^r[a,b]$, $r\ge 0$, an arbitrary finite collection of points $\big\{α_i\big\}_{i\in J} \subset [a,b]$, and a corresponding collection of nonnegative integers $\big\{m_i\big\}_{i\in J}$ with $0\le m_i \le r$, $i\in J$, is it true that, for sufficiently large $n\in N$, there exists a polynomial $P_n$ of degree $n$ such that (i) $|f(x)-P_n(x)| \le c ρ_n^r(x) ω_k(f^{(r)}, ρ_n(x); [a,b])$, $x\in [a,b]$, where $ρ_n (x):= n^{-1} \sqrt{1-x^2} +n^{-2}$ and $ω_k$ is the classical $k$-th modulus of smoothness, (ii) $P^{(ν)}(α_i)=f^{(ν)}(α_i)$, for all $0\le ν\le m_i$ and all $i\in J$, and (iii) either $P \ge f$ on $[a,b]$ (\emph{onesided} approximation), or $P \ge 0$ on $[a,b]$ (\emph{positive} approximation)? We provide {\em precise answers} not only to this question, but also to similar questions for more general {\em intertwining} and {\em copositive} polynomial approximation. It turns out that many of these answers are quite unexpected. We also show that, in general, similar questions for $q$-monotone approximation with $q\ge 1$ have negative answers, i.e., $q$-monotone approximation with general interpolatory constraints is impossible if $q\ge 1$.

math.CA↗

arXiv2021

Exact pointwise estimates for polynomial approximation with Hermite interpolation

We establish best possible pointwise (up to a constant multiple) estimates for approximation, on a finite interval, by polynomials that satisfy finitely many (Hermite) interpolation conditions, and show that these estimates cannot be improved. In particular, we show that {\bf any} algebraic polynomial of degree $n$ approximating a function $f\in C^r(I)$, $I=[-1,1]$, at the classical pointwise rate $ρ_n^r(x) ω_k(f^{(r)}, ρ_n(x))$, where $ρ_n(x)=n^{-1}\sqrt{1-x^2}+n^{-2}$, and (Hermite) interpolating $f$ and its derivatives up to the order $r$ at a point $x_0\in I$, has the best possible pointwise rate of (simultaneous) approximation of $f$ near $x_0$. Several applications are given.

math.CA↗

arXiv2018

On moduli of smoothness with Jacobi weights

The main purpose of this paper is to introduce moduli of smoothness with Jacobi weights $(1-x)^α(1+x)^β$ for functions in the Jacobi weighted $L_p[-1,1]$, $0<p\le \infty$, spaces. These moduli are used to characterize the smoothness of (the derivatives of) functions in the weighted $L_p$ spaces. If $1\le p\le\infty$, then these moduli are equivalent to certain weighted $K$-functionals (and so they are equivalent to certain weighted Ditzian-Totik moduli of smoothness for these $p$), while for $0<p<1$ they are equivalent to certain "Realization functionals".

math.CA↗

arXiv2018

Interpolatory estimates for convex piecewise polynomial approximation

In this paper, among other things, we show that, given $r\in N$, there is a constant $c=c(r)$ such that if $f\in C^r[-1,1]$ is convex, then there is a number ${\mathcal N}={\mathcal N}(f,r)$, depending on $f$ and $r$, such that for $n\ge{\mathcal N}$, there are convex piecewise polynomials $S$ of order $r+2$ with knots at the Chebyshev partition, satisfying \[ |f(x)-S(x)|\le c(r)\left( \min\left\{ 1-x^2, n^{-1}\sqrt{1-x^2} \right\} \right)^r ω_2\left(f^{(r)}, n^{-1}\sqrt{1-x^2} \right), \] for all $x\in [-1,1]$. Moreover, ${\mathcal N}$ cannot be made independent of $f$.

math.CA↗

arXiv2017

On weighted approximation with Jacobi weights

We obtain matching direct and inverse theorems for the degree of weighted $L_p$-approximation by polynomials with the Jacobi weights $(1-x)^α(1+x)^β$. Combined, the estimates yield a constructive characterization of various smoothness classes of functions via the degree of their approximation by algebraic polynomials.

math.CA↗

arXiv2015

Uniform polynomial approximation with $A^*$ weights having finitely many zeros

We prove matching direct and inverse theorems for uniform polynomial approximation with $A^*$ weights (a subclass of doubling weights suitable for approximation in the $L_\infty$ norm) having finitely many zeros and not too "rapidly changing" away from these zeros. This class of weights is rather wide and, in particular, includes the classical Jacobi weights, generalized Jacobi weights and generalized Ditzian-Totik weights. Main part and complete weighted moduli of smoothness are introduced, their properties are investigated, and equivalence type results involving related realization functionals are discussed.

math.CA↗

arXiv2014

Weighted moduli of smoothness of $k$-monotone functions and applications

Let $ω_φ^k(f,δ)_{w,L_q}$ be the Ditzian-Totik modulus with weight $w$, $M^k$ be the cone of $k$-monotone functions on $(-1,1)$, i.e., those functions whose $k$th divided differences are nonnegative for all selections of $k+1$ distinct points in $(-1,1)$, and denote $E (X, Π_n)_{w,q} := \sup_{f\in X} \inf_{P\inΠ_n}\|w(f-P)\|_{L_q}$, where $Π_n$ is the set of algebraic polynomials of degree at most $n$. Additionally, let $w_{α,β}(x) := (1+x)^α(1-x)^β$ be the classical Jacobi weight, and denote by $S_p^{α,β}$ the class of all functions such that $\| w_{α,β}f\|_{L_p}=1$. In this paper, we determine the exact behavior (in terms of $δ$) of $\sup_{f\in S_p^{α,β}\cap M^k} ω_φ^k(f,δ)_{w_{α,β},L_q}$ for $1\leq p, q\leq \infty$ (the interesting case being $q -1/p$ (if $p<\infty$) or $α,β\geq 0$ (if $p=\infty$). It is interesting to note that, in one case, the behavior is different for $α=β=0$ and for $(α,β)\neq (0,0)$. Several applications are given. For example, we determine the exact (in some sense) behavior of $E (M^k\cap S_p^{α,β}, Π_n)_{w_{α,β},L_q}$ for $α,β\geq 0$.

math.CA↗

arXiv2014

Polynomial approximation with doubling weights

Among other things, we prove that, for a doubling weight $w$, $0< p\leq\infty$, $r\in{\mathbb N}_0$, and $0<α<r+1 - 1/λ_p$, we have \[ E_n(f)_{p, w_n} = O(n^{-α}) \iff ω_φ^{r+1}(f, n^{-1})_{p, w_n} = O(n^{-α}), \] where $λ_p := p$ if $0 < p < \infty$, $λ_p :=1$ if $p=\infty$, $\|f\|_{p,w} := \left( \int_{-1}^1 |f(u)|^p w(u) du \right)^{1/p}$, $\|f\|_{\infty,w} := {\mathop{\rm ess\: sup}\nolimits}_{u\in [-1,1]} \left(|f(u)| w(u)\right)$, $ω_φ^r(f, t)_{p, w} := \sup_{0<h\leq t} \| Δ_{hφ(\cdot)}^r(f,\cdot)\|_{p, w}$, $E_n(f)_{p, w} := \inf_{P_n\inΠ_n} \|f-P_n\|_{p,w}$, and $Π_n$ is the set of all algebraic polynomials of degree $\leq n-1$. Equivalence type results involving related $K$-functionals and realization type results (obtained as corollaries of our estimates) are also discussed. Finally, we mention that (*) closes a gap left in the paper by G. Mastroianni and V. Totik "Best Approximation and moduli of smoothness for doubling weights", J. Approx. Theory {\bf 110} (2001), 180-199, where ($\ast$) was established for $p=\infty$ and $ω_φ^{r+2}$ instead of $ω_φ^{r+1}$ (it was shown there that, in general, ($\ast$) is not valid for $p=\infty$ if $ω_φ^{r+1}$ is replaced by $ω_φ^{r}$).

math.CA↗

arXiv2014

Polynomial approximation with doubling weights having finitely many zeros and singularities

We prove matching direct and inverse theorems for (algebraic) polynomial approximation with doubling weights $w$ having finitely many zeros and singularities (i.e., points where $w$ becomes infinite) on an interval and not too ``rapidly changing'' away from these zeros and singularities. This class of doubling weights is rather wide and, in particular, includes the classical Jacobi weights, generalized Jacobi weights and generalized Ditzian-Totik weights. We approximate in the weighted $L_p$ (quasi) norm $\|f\|_{p, w}$ with $0<p<\infty$, where $\|f\|_{p, w} := \left(\int_{-1}^1 |f(u)|^p w(u) du \right)^{1/p}$. Equivalence type results involving related realization functionals are also discussed.

math.CA↗