A Complete Asymptotic Expansion for the Quasi-interpolants of Gauß–Weierstraß Operators

Abstract

We derive the complete asymptotic expansion for the quasi-interpolants of Gauß–Weierstraß operators $W_{n}$ and their left quasi-interpolants $W_{n}^{\left[ r\right] }$ with explicit representation of the coefficients. The results apply to all locally integrable real functions f on $\mathbb {R}$ satisfying the growth condition ${f}\left( t\right) =O\left( \mathrm{{e}}^{ct^{2}}\right)$ as $\left| t\right| \rightarrow +\infty$, for some $c>0$. All expansions are shown to be valid also for simultaneous approximation.