FoCM 2014 conference

Workshop B1 - Approximation Theory

December 16, 17:05 ~ 17:55 - Room B21

## Weighted D-T moduli revisited and applied

### Dany Leviatan

### Tel Aviv University, Israel - leviatan@math.tau.ac.il

We introduce weighted moduli of smoothness for functions $f\in L_p[-1,1]\cap C^{r-1}(-1,1)$ $r\ge 1$, that have an $(r-1)$st absolutely continuous derivative in $(-1,1)$ and such that $\varphi^rf^r\in L_p[-1,1]$, where $\varphi(x)=(1-x^2)^{1/2}$. These moduli are equivalent to certain weighted D-T moduli, but our definition is more transparent and simpler. In addition, instead of applying these weighted moduli to weighted approximation, which was the purpose of the original D-T moduli, we apply these moduli to obtain Jackson-type estimates on the approximation of functions in $L_p[-1,1]$ (no weight), by means of algebraic polynomials. We also have inverse theorems that yield characterization of the behavior of the derivatives of the function by means of its degree of approximation.

Joint work with K. Kopotun (University of Manitoba, Canada) and I. A. Shevchuk (Ukrainian National Academy of Sciences, Kiev).