A transplantation theorem for ultraspherical polynomials at critical index

J. J. Guadalupe ¹   ; V. I. Kolyada ²

¹ Departamento de Matemáticas y Computación Universidad de La Rioja Edif. Vives, c. Luis de Ulloa 26004 Logroño, La Rioja, Spain
² Department of Mathematics Odessa National University 2 Dvoryanskaya st. 270000 Odessa, Ukraine

Studia Mathematica, Tome 147 (2001) no. 1, pp. 51-72

Résumé

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 $\| \varphi \| _{\mathop {\rm Re} H^1}\le c\| f\| _{{\cal L}_\lambda }$.

Détail

DOI : 10.4064/sm147-1-5

Keywords: investigate behaviour fourier coefficients respect system ultraspherical polynomials leads study boundary lorentz space cal lambda corresponding endpoint mean convergence interval ultraspherical coefficients lambda cal lambda functions turn out behave fourier coefficients functions real hardy space mathop namely prove cal lambda series sum infty lambda mathop cos nolimits theta fourier series function varphi mathop varphi mathop cal lambda