In this paper, we prove that for $\ell \,=\,1$ or 2 the rate of best $\ell $ - monotone polynomial approximation in the ${{L}_{p}}$ norm $\left( 1\,\le \,p\,\le \,\infty \right)$ weighted by the Jacobi weight ${{w}_{\alpha ,\,\beta }}\left( x \right)\,:=\,{{\left( 1\,+\,x \right)}^{\alpha }}{{\left( 1\,-\,x \right)}^{\beta }}$ with $\alpha ,\,\beta \,>\,-1/p$ if $p\,<\,\infty $ , or $\alpha ,\,\beta \,\ge \,0$ if $p\,=\,\infty $ , is bounded by an appropriate $\left( \ell \,+\,1 \right)$ -st modulus of smoothness with the same weight, and that this rate cannot be bounded by the $\left( \ell \,+\,2 \right)$ -nd modulus. Related results on constrained weighted spline approximation and applications of our estimates are also given.

Let ${{B}_{p}}$ be the unit ball in ${{\mathbb{L}}_{p}}$ , $0\,<\,p\,<\,1$, and let $\Delta _{+}^{s}$ , $s\,\in \,\mathbb{N}$, be the set of all $s$-monotone functions on a finite interval $I$, i.e., $\Delta _{+}^{s}$ consists of all functions $x\,:\,I\,\mapsto \,\mathbb{R}$ such that the divided differences $[x;\,{{t}_{0}},\,...\,,\,{{t}_{s}}]$ of order $s$ are nonnegative for all choices of $\left( s\,+\,1 \right)$ distinct points ${{t}_{0}},\,.\,.\,.\,,{{t}_{s}}\,\in \,I.$ For the classes $\Delta _{+}^{s}{{B}_{P}}\,:=\,\Delta _{+}^{s}\,\cap \,{{B}_{P}},$ we obtain exact orders of Kolmogorov, linear and pseudo-dimensional widths in the spaces ${{\mathbb{L}}_{q}},$$0\,<\,q\,<\,p\,<\,1$:

$${{d}_{n}}(\Delta _{+}^{s}{{B}_{P}})_{{{\mathbb{L}}_{q}}}^{\text{psd}}\asymp {{d}_{n}}(\Delta _{+}^{s}{{B}_{P}})_{{{\mathbb{L}}_{q}}}^{\text{kol}}\asymp {{d}_{n}}(\Delta _{+}^{s}{{B}_{P}})_{{{\mathbb{L}}_{q}}}^{\text{lin}}\asymp {{n}^{-s}}.$$

Estimating the degree of approximation in the uniform norm, of a convex function on a finite interval, by convex algebraic polynomials, has received wide attention over the last twenty years. However, while much progress has been made especially in recent years by, among others, the authors of this article, separately and jointly, there have been left some interesting open questions. In this paper we give final answers to all those open problems. We are able to say, for each $r$-th differentiable convex function, whether or not its degree of convex polynomial approximation in the uniform norm may be estimated by a Jackson-type estimate involving the weighted Ditzian–Totik $k$th modulus of smoothness, and how the constants in this estimate behave. It turns out that for some pairs $(k,\,r)$ we have such estimate with constants depending only on these parameters. For other pairs the estimate is valid, but only with constants that depend on the function being approximated, while there are pairs for which the Jackson-type estimate is, in general, invalid.

Positive, copositive, onesided and intertwining (co-onesided) polynomial and spline approximations of functions are considered. Both uniform and pointwise estimates, which are exact in some sense, are obtained.

Let A be a (k + 1) × (k + 1) nonzero matrix. For polynomials p ∈ Pn, set and . Let E ⊂ C be a compact set that does not separate the plane and f be a function continuous on E and analytic in the interior of E. Set and . Our goal is to study approximation to f on E by polynomials from Bn(A). We obtain necessary and sufficient conditions on the matrix A for the convergence En(A,f) → 0 to take place. These results depend on whether zero lies inside, on the boundary or outside E and yield generalizations of theorems of Clunie, Hasson and Saff for approximation by polynomials that omit a power of z. Let be such that . We also study the asymptotic behavior of the zeros of and the asymptotic relation between En(f) and En(A,f).

Jackson type theorems are obtained for the comonotone approximation of piecewise monotone functions by polynomials.

Under appropriate conditions it is shown that the best interpolating approximation to a given function in the uniform norm is a limit of best unconstrained approximations with respect to a certain sequence of discontinuous weight functions.