A transplantation theorem for ultraspherical polynomials at critical index (Q2717719)

In this noteworthy paper the authors prove some interesting theorems. A precise characterization of the results is given in the abstract as follows: ``We investigate the behaviour of Fourier coefficients with respect to the system of the ``boundary'' Lorentz space \({\mathcal L}_\lambda\) corresponding to the left endpoint of the mean convergence interval. The ultraspherical coefficients \(\{c^{(\lambda)}_n(f)\}\) of \({\mathcal L}_\lambda\)-functions turn out to behave like the Fourier coefficients of functions in the real Hardy space \(\text{Re }H^1\). Namely, we prove that for any \(f\in{\mathcal L}_\lambda\) the series \(\sum^\infty_{n=1} c^{(\lambda)}_n(f)\cos n\theta\) is the Fourier series of some function \(\varphi\in \text{Re }H^1\) with \(\|\varphi\|_{\text{Re }H^1}\leq c\|f\|_{{\mathcal L}_\lambda}\)''.