A transplantation theorem for ultraspherical polynomials at critical index

J. J. Guadalupe, and V. I. Kolyada. "A transplantation theorem for ultraspherical polynomials at critical index." Studia Mathematica 147.1 (2001): 51-72. <http://eudml.org/doc/284884>.

@article{J2001,
abstract = {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 $||φ||_\{ReH¹\} ≤ c||f||_\{ℒ_\{λ\}\}$.},
author = {J. J. Guadalupe, V. I. Kolyada},
journal = {Studia Mathematica},
keywords = {transplantation; Fourier coefficients; orthogonal polynomials; Hardy spaces; Lorentz spaces; Fourier series},
language = {eng},
number = {1},
pages = {51-72},
title = {A transplantation theorem for ultraspherical polynomials at critical index},
url = {http://eudml.org/doc/284884},
volume = {147},
year = {2001},
}

TY - JOUR
AU - J. J. Guadalupe
AU - V. I. Kolyada
TI - A transplantation theorem for ultraspherical polynomials at critical index
JO - Studia Mathematica
PY - 2001
VL - 147
IS - 1
SP - 51
EP - 72
AB - 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 $||φ||_{ReH¹} ≤ c||f||_{ℒ_{λ}}$.
LA - eng
KW - transplantation; Fourier coefficients; orthogonal polynomials; Hardy spaces; Lorentz spaces; Fourier series
UR - http://eudml.org/doc/284884
ER -