# Wydawnictwa / Czasopisma IMPAN / Studia Mathematica / Wszystkie zeszyty

## A transplantation theorem for ultraspherical polynomials at critical index

### Tom 147 / 2001

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

#### Streszczenie

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

#### Autorzy

• J. J. GuadalupeDepartamento de Matemáticas y Computación
Edif. Vives, c. Luis de Ulloa
26004 Logroño, La Rioja, Spain