2014 CMS Summer Meeting
University of Manitoba, June 6 - 9, 2014
We will show how to use the Euclidean representation of strongly regular graphs to construct a two-distance set consisting of $416$ points on the unit sphere in the dimension $65$ which cannot be partitioned into $83$ parts of smaller diameter.
Here I am introducing a new different measure of smoothness of functions on the unit ball $B$. This measure beside being ``correct'' has the added advantages of being rotation invariant (independent of the basis of $R^d$), being applicable to more norms, and being amenable to more applications. Computability of the new measure was demonstrated and some of the computations show optimality of results.
In $1926,$ S. Bernstein showed that \[ \sup_{P_n\in\triangle_ n}\frac{\|P'_n\|}{\|P_n\|}=\left\{ \begin{array}{ll} \frac{(n+1)^2}{4} , & \mbox{ if } n=2k+1 ,\ \frac{n(n+2)}{4} , & \mbox{ if } n=2k. \end{array} \right. \] where the supremum is taken over all monotone polynomials of degree $\le n.$
Motivated by this result, T. Erdelyi considered analogous problem for $M^{(l)}_{q,p}(n,m),$ where the supremum is taken over all absolutely monotone polynomials of order $l$ and determined the exact asymptotic in the case $p\le q.$
At the time, A. Kroo and J. Szabados found the exact Markov factors for monotone polynomials of order $k$ in $L_1$ and $L_{\infty}$ norms.
In this talk, I will discuss the sharp order of $M^{(l)}_{q,p}(n,m)$ for all values of $p,q.$ I will also discuss the weighted analog of Berstein result for monotone polynomials.
We will discuss some open problems and results in the the approximation with Jacobi weights $$q_{\alpha,\beta}(x):=(1-x)^\alpha(1+x)^\beta$$ and their generalizations. Say, for the error $$E_n^*(f):=\inf_{P_n}\|(f-P_n)q_{-1/2,-1/2}\|_{C[-1,1]}$$ of the best uniform weighted approximation by algebraic polynomials $P_n$ of degree $<n$ of a continuous on $[-1,1]$ function $f\in \text{Lip} 1$ (that is $|f(x')-f(x'')|\le|x'-x''|, \forall x',x''\in[-1,1]$) the estimate $$ nE_n^*(f)\le c $$ is well known. However the exact value of the absolute constant $c$ is not known.
Recently the asymptotically exact estimates for the generalized Lebesgue constants of the Fourier-Jacobi sums in the weighted integral norm are obtained.
For a domain $D\subset \mathbf{C}$ with piecewise smooth boundary an analog of the Jacobi weight is $$ q_{\alpha_1,\dots,\alpha_s}(z):=|z_1-z|^{\alpha_1} \dots|z_s-z|^{\alpha_s}, $$ where $s$ is the number of the angular points $z_j$. For such domains and weights complete analogs of the estimates, taking place for the interval, are proved.
Finally, we will discuss the shape preserving approximation with
Jacobi weights.
We can restore higher-order convergence, e.g., by reprojecting the slowly convergent Fourier series onto a suitable basis of orthogonal algebraic polynomials, however, all exponentially convergent methods appear to suffer from some sort of ill-conditioning, whereas methods that recover $f$ in a stable manner have a much slower approximation rate.
We give to these observations a definite explanation in terms of the following
fundamental stability barrier: the best possible convergence rate for a stable
reconstruction from the first $m$ Fourier coefficients is root-exponential in $m$.
We will discuss some open problems and results in the the approximation with Jacobi weights $$q_{\alpha,\beta}(x):=(1-x)^\alpha(1+x)^\beta$$ and their generalizations. Say, for the error $$E_n^*(f):=\inf_{P_n}\|(f-P_n)q_{-1/2,-1/2}\|_{C[-1,1]}$$ of the best uniform weighted approximation by algebraic polynomials $P_n$ of degree $<n$ of a continuous on $[-1,1]$ function $f\in \text{Lip} 1$ (that is $|f(x')-f(x'')|\le|x'-x''|, \forall x',x''\in[-1,1]$) the estimate $$ nE_n^*(f)\le c $$ is well known. However the exact value of the absolute constant $c$ is not known.
Recently the asymptotically exact estimates for the generalized Lebesgue constants of the Fourier-Jacobi sums in the weighted integral norm are obtained.
For a domain $D\subset \mathbf{C}$ with piecewise smooth boundary an analog of the Jacobi weight is $$ q_{\alpha_1,\dots,\alpha_s}(z):=|z_1-z|^{\alpha_1} \dots|z_s-z|^{\alpha_s}, $$ where $s$ is the number of the angular points $z_j$. For such domains and weights complete analogs of the estimates, taking place for the interval, are proved.
Finally, we will discuss the shape preserving approximation with Jacobi weights.
This is joint work with O. V. Motorna.