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A transplantation theorem for ultraspherical polynomials at critical index

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Guadalupe J., Kolyada V.

Studia Mathematica

2001

tom 147

nr 1

51-72

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 ${cₙ^{(λ)}(f)}$ 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 $f ∈ ℒ_{λ}$ the series $∑_{n=1}^{∞} cₙ^{(λ)}(f) cos nθ $ is the Fourier series of some function φ ∈ ReH¹ with $||φ||_{ReH¹} ≤ c||f||_{ℒ_{λ}}$.

Ograniczanie wyników

1 Studia Mathematica

1 Guadalupe J.

1 Kolyada V.

1 2001

Wyniki wyszukiwania

A transplantation theorem for ultraspherical polynomials at critical index