A transplantation theorem for ultraspherical polynomials at critical index

Volume 147 / 2001

J. J. Guadalupe, V. I. Kolyada Studia Mathematica 147 (2001), 51-72 MSC: 42C10, 33C45, 42A50. DOI: 10.4064/sm147-1-5


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


  • J. J. GuadalupeDepartamento de Matemáticas y Computación
    Universidad de La Rioja
    Edif. Vives, c. Luis de Ulloa
    26004 Logroño, La Rioja, Spain
  • V. I. KolyadaDepartment of Mathematics
    Odessa National University
    2 Dvoryanskaya st.
    270000 Odessa, Ukraine

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