A transplantation theorem for ultraspherical polynomials at critical index

J. J. Guadalupe; V. I. Kolyada

Studia Mathematica, Tome 147 (2001), p. 51-72 / Harvested from The Polish Digital Mathematics Library

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 $ℒ_{λ}$ corresponding to the left endpoint of the mean convergence interval. The ultraspherical coefficients $c ₙ^{(λ)} (f)$ of $ℒ_{λ}$ -functions turn out to behave like the Fourier coefficients of functions in the real Hardy space ReH¹. Namely, we prove that for any $f \in ℒ_{λ}$ the series $\sum_{n = 1}^{\infty} c ₙ^{(λ)} (f) c o s n θ$ is the Fourier series of some function φ ∈ ReH¹ with ${| | φ | |}_{R e H ¹} {\leq c | | f | |}_{ℒ_{λ}}$ .

Publié le : 2001-01-01

Zbl 0979.42012

EUDML-ID : urn:eudml:doc:284884

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     title = {A transplantation theorem for ultraspherical polynomials at critical index},
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     volume = {147},
     year = {2001},
     pages = {51-72},
     zbl = {0979.42012},
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J. J. Guadalupe; V. I. Kolyada. A transplantation theorem for ultraspherical polynomials at critical index. Studia Mathematica, Tome 147 (2001) pp. 51-72. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm147-1-5/