Density of bivariate homogeneous polynomials on non-convex curves

Abstract:

The density of bivariate homogeneous polynomials is studied in the space of continuous functions on the $L_{\alpha }$ sphere given by $K_{\alpha }:=\{(x,y)\in \mathbb R^2: |x|^{\alpha }+|y|^{\alpha }= 1\}, \; \alpha >0.$ The goal is to approximate functions $f\in C(K_{\alpha })$ by sums of the form $h_{2n}+h_{2n+1}$, where $h_{2n},h_{2n+1}$ are bivariate homogeneous polynomials of degree $2n$ and $2n+1$, respectively. It is known that whenever $\alpha \geq 1$, i.e., when $K_{\alpha }$ is convex, a Weierstrass-type approximation result holds, namely for every $f\in C(K_{\alpha })$ there are homogeneous polynomials $h_{2n},h_{2n+1}$ for which $f=\lim _{n\rightarrow \infty }(h_{2n}+h_{2n+1})$ uniformly on $K_{\alpha }$. In this note the problem is solved in the non-convex case $0<\alpha <1$. It is verified that $f(x,y)$ is a uniform limit on $K_{\alpha }$ of sums $h_{2n}+h_{2n+1}$ of homogeneous polynomials if and only if $f(\pm 1,0)=f(0,\pm 1)=0.$ The theorem is proven in an equivalent form: $g\in C(\mathbb R)$ is a uniform limit as $n\rightarrow \infty$ of weighted polynomials $(1+|t|^{\alpha })^{-n/\alpha }p_{n}(t)$ (degree $p_{n}\leq n$) if and only if $g(0)=g(\infty )=g(-\infty )=0.$