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This paper deals with approximation of smooth convex functions $f$ on an interval by convex algebraic polynomials which interpolate $f$ at the endpoints of this interval. We call such estimates interpolatory. One important corollary of our main theorem is the following result on approximation of $fin Delta^{(2)}$, the set of convex functions, from $W^r$, the space of functions on $[-1,1]$ for which $f^{(r-1)}$ is absolutely continuous and $|f^{(r)}|_{infty} := ess,sup_{xin[-1,1]} |f^{(r)}(x)| < infty$: For any $fin W^r capDelta^{(2)}$, $rin {mathbb N}$, there exists a number ${mathcal N}={mathcal N}(f,r)$, such that for every $nge {mathcal N}$, there is an algebraic polynomial of degree $le n$ which is in $Delta^{(2)}$ and such that [ left| frac{f-P_n}{varphi^r} right|_{infty} leq frac{c(r)}{n^r} left| f^{(r)}right|_{infty} , ] where $varphi(x):= sqrt{1-x^2}$. For $r=1$ and $r=2$, the above result holds with ${mathcal N}=1$ and is well known. For $rge 3$, it is not true, in general, with ${mathcal N}$ independent of $f$.
Given a nondecreasing function $f$ on $[-1,1]$, we investigate how well it can be approximated by nondecreasing algebraic polynomials that interpolate it at $pm 1$. We establish pointwise estimates of the approximation error by such polynomials that
For a polynomial $P$ mapping the integers into the integers, define an averaging operator $A_{N} f(x):=frac{1}{N}sum_{k=1}^N f(x+P(k))$ acting on functions on the integers. We prove sufficient conditions for the $ell^{p}$-improving inequality begin{e
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Limits on the number of satisfying assignments for CNS instances with n variables and m clauses are derived from various inequalities. Some bounds can be calculated in polynomial time, sharper bounds demand information about the distribution of the n
We investigate the dispersive properties of solutions to the Schrodinger equation with a weakly decaying radial potential on cones. If the potential has sufficient polynomial decay at infinity, then we show that the Schrodinger flow on each eigenspac