Moduli of smoothness and growth of the Laguerre transform: two-sided estimates

We study weighted coefficient estimates for the Laguerre transform in terms of moduli of smoothness generated by the Laguerre translation. For every integer $r\ge 1$ and $1\le p\le 2$, a weighted ${\mathrm{\ell }}^{p^{\mathrm{\prime }}}$ norm of $\min \{1,nt{\}}^r{\widehat{f}}_{\alpha }(n)$ is bounded by the modulus of order $r$; for $2\le p\le \mathrm{\infty }$ the reverse direction holds on the same weighted sequence scale. At $p=2$ these estimates yield a two-sided equivalence, and testing individual Laguerre polynomials proves optimality of the multiplier weight up to constants. The proof combines Hausdorff–Young inequalities with an averaged lower bound for the translation multiplier. As consequences we recover Jackson- and Bernstein-type estimates and the known $L^2$ approximation characterisation of the Lipschitz classes for $0<\beta <r$. We also give the corresponding one-sided coefficient-tail criteria for general $p$, together with saturation and endpoint statements.