An essay on some problems of approximation theory

Abstract

Several questions of approximation theory are discussed: 1) can one approximate stably in $L^{\infty }$ norm $f^{\prime }$ given approximation $f_{\delta },\parallel f_{\delta }-f{\parallel }_{L^{\infty }}<\delta$, of an unknown smooth function $f(x)$, such that $\parallel f^{\prime }(x){\parallel }_{L^{\infty }}\le m_1$? 2) can one approximate an arbitrary $f\in L^2(D),D\subset {\mathbb{R}}^n,n\ge 3$, is a bounded domain, by linear combinations of the products $u_1{u}_2$, where $u_m\in N(L_m),m=1,2,$ $L_m$ is a formal linear partial differential operator and $N(L_m)$ is the null-space of $L_m$ in $D$, $3)canoneapproximateanarbitrary$L^2(D)$ function by an entire function of exponential type whose Fourier transform has support in an arbitrary small open set? Is there an analytic formula for such an approximation? N(L_m) := \{w: L_m w=0 \hbox{in\} D\}$?