A transplantation theorem for ultraspherical polynomials at critical index

J. J. Guadalupe¹ ; V. I. Kolyada²

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

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