Georgia Institute of Technology College of Sciences

This is a joint work with F.~Nazarov and A.~Volberg.Let $s\in(1,2)$, and let $\mu$ be a finite positive Borel measure in $\mathbb R^2$ with $\mathcal H^s(\supp\mu)<+\infty$. We prove that if the lower $s$-density of $\mu$ is+equal to zero $\mu$-a.~e. in $\mathbb R^2$, then$\|R\mu\|_{L^\infty(m_2)}=\infty$, where $R\mu=\mu\ast\frac{x}{|x|^{s+1}}$ and $m_2$ is the Lebesque measure in $\mathbb R^2$. Combined with known results of Prat and+Vihtil\"a, this shows that for any noninteger $s\in(0,2)$ and any finite positive Borel measure in $\mathbb R^2$ with $\mathcal H^s(\supp\mu)<+\infty$, we have+$\|R\mu\|_{L^\infty(m_2)}=\infty$.Also I will tell about the resent result of Ben Jaye, as well as about open problems.

In the talk some problems related with the famous Chernoff square root of n - lemma in the theory of approximation of some semi-groups of operators will be discussed. We present some optimal bounds in these approximations (one of them is Euler approximation) and two new classes of operators, generalizing sectorial and quasi-sectorial operators will be introduced. The talk is based on two papers [V. Bentkus and V. Paulauskas, Letters in Math. Physics, 68, (2004), 131-138] and [V. Paulauskas, J. Functional Anal., 262, (2012), 2074-2099]

We discuss some recent generalizations of Euler--Maclaurin expansions for the trapezoidal rule and of analogous asymptotic expansions for Gauss--Legendre quadrature, in the presence of arbitrary algebraic-logarithmic endpoint singularities. In addition of being of interest by themselves, these asymptotic expansions enable us to design appropriate variable transformations to improve the accuracies of these quadrature formulas arbitrarily. In general, these transformations are singular, and their singularities can be adjusted easily to achieve this improvement. We illustrate this issue with a numerical example involving Gauss--Legendre quadrature. We also discuss some recent asymptotic expansions of the coefficients of Legendre polynomial expansions of functions over a finite interval, assuming that the functions may have arbitrary algebraic-logarithmic interior and endpoint suingularities. These asymptotic expansions can be used to make definitive statements on the convergence acceleration rates of extrapolation methods as these are applied to the Legendre polynomial expansions.