A transplantation theorem for ultraspherical polynomials at critical index
Volume 147 / 2001
Abstract
We investigate the behaviour of Fourier coefficients with respect to the system of ultraspherical polynomials. This leads us to the study of the “boundary” Lorentz space ${\cal L}_\lambda $ corresponding to the left endpoint of the mean convergence interval. The ultraspherical coefficients $\{ c_n^{(\lambda )}(f)\} $ of ${\cal L}_\lambda $-functions turn out to behave like the Fourier coefficients of functions in the real Hardy space $\mathop {\rm Re} H^1$. Namely, we prove that for any $f\in {\cal L}_\lambda $ the series $\sum _{n=1}^\infty c_n^{(\lambda )}(f)\mathop {\rm cos}\nolimits n\theta $ is the Fourier series of some function $\varphi \in \mathop {\rm Re} H^1$ with $\| \varphi \| _{\mathop {\rm Re} H^1}\le c\| f\| _{{\cal L}_\lambda }$.
