Expansion by orthogonal systems with respect to Freud weights related to Hardy spaces

Abstract:

For the basic class of Freud weights $w_\alpha (x) = \exp (-\vert x\vert ^\alpha /2), \alpha >1$ the coefficients of the expansion of $w_\alpha f\in H_p(R)$ by the Freud orthogonal system $\{w_\alpha p_{n,\alpha }\}^\infty _{n=0} ,$ where $p_{n,\alpha }$ are polynomials of degree $n,$ are related to the quasi-norm (or norm) of $w_\alpha f$ in $H_p(R).$ Relations are achieved for all $\alpha >1$ and $\frac 12 <p<1,$ and for some $\alpha$ for a larger range of $p.$ As a result, estimates for $1<p\le 2$ are also improved.