Integrability theorems for trigonometric series

Studia Mathematica (1993)

• Volume: 107, Issue: 1, page 33-59
• ISSN: 0039-3223

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Abstract

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We show that, if the coefficients (an) in a series ${a}_{0}/2+{\sum }_{n=1}^{\infty }{a}_{n}cos\left(nt\right)$ tend to 0 as n → ∞ and satisfy the regularity condition that ${\sum }_{m=0}^{\infty }{{\sum }_{j=1}^{\infty }\left[{\sum }_{n=j{2}^{m}}^{\left(j+1\right){2}^{m}-1}|{a}_{n}-{a}_{n+1}|\right]²}^{1/2}<\infty$, then the cosine series represents an integrable function on the interval [-π,π]. We also show that, if the coefficients (bn) in a series ${\sum }_{n=1}^{\infty }{b}_{n}sin\left(nt\right)$ tend to 0 and satisfy the corresponding regularity condition, then the sine series represents an integrable function on [-π,π] if and only if ${\sum }_{n=1}^{\infty }|{b}_{n}|/n<\infty$. These conclusions were previously known to hold under stronger restrictions on the sizes of the differences $\Delta {a}_{n}={a}_{n}-{a}_{n+1}$ and $\Delta {b}_{n}={b}_{n}-{b}_{n+1}$. We were led to the mixed-norm conditions that we use here by our recent discovery that the same combination of conditions implies the integrability of Walsh series with coefficients (an) tending to 0. We also show here that this condition on the differences implies that the cosine series converges in L¹-norm if and only if ${a}_{n}logn\to 0$ as n → ∞. The corresponding statement also holds for sine series for which ${\sum }_{n=1}^{\infty }|{b}_{n}|/n<\infty$. If either type of series is assumed a priori to represent an integrable function, then weaker regularity conditions suffice for the validity of this criterion for norm convergence.

How to cite

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Aubertin, Bruce, and Fournier, John. "Integrability theorems for trigonometric series." Studia Mathematica 107.1 (1993): 33-59. <http://eudml.org/doc/216021>.

@article{Aubertin1993,
abstract = {We show that, if the coefficients (an) in a series $a_0/2+∑_\{n=1\}^∞ a_n cos(nt)$ tend to 0 as n → ∞ and satisfy the regularity condition that $∑_\{m=0\}^∞ \{∑_\{j=1\}^∞ [∑_\{n=j2^m\}^\{(j+1)2^m-1\} |a_n - a_\{n+1\}|]²\}^\{1/2\} < ∞$, then the cosine series represents an integrable function on the interval [-π,π]. We also show that, if the coefficients (bn) in a series $∑_\{n=1\}^∞ b_n sin(nt)$ tend to 0 and satisfy the corresponding regularity condition, then the sine series represents an integrable function on [-π,π] if and only if $∑_\{n=1\}^∞ |b_n|/n < ∞$. These conclusions were previously known to hold under stronger restrictions on the sizes of the differences $Δa_n = a_n - a_\{n+1\}$ and $Δb_n = b_n - b_\{n+1\}$. We were led to the mixed-norm conditions that we use here by our recent discovery that the same combination of conditions implies the integrability of Walsh series with coefficients (an) tending to 0. We also show here that this condition on the differences implies that the cosine series converges in L¹-norm if and only if $a_n log n → 0$ as n → ∞. The corresponding statement also holds for sine series for which $∑_\{n=1\}^∞ |b_n|/n < ∞$. If either type of series is assumed a priori to represent an integrable function, then weaker regularity conditions suffice for the validity of this criterion for norm convergence.},
author = {Aubertin, Bruce, Fournier, John},
journal = {Studia Mathematica},
keywords = {integrability theorems; trigonometric series; regularity condition; cosine series; sine series; Walsh series; norm convergence},
language = {eng},
number = {1},
pages = {33-59},
title = {Integrability theorems for trigonometric series},
url = {http://eudml.org/doc/216021},
volume = {107},
year = {1993},
}

TY - JOUR
AU - Aubertin, Bruce
AU - Fournier, John
TI - Integrability theorems for trigonometric series
JO - Studia Mathematica
PY - 1993
VL - 107
IS - 1
SP - 33
EP - 59
AB - We show that, if the coefficients (an) in a series $a_0/2+∑_{n=1}^∞ a_n cos(nt)$ tend to 0 as n → ∞ and satisfy the regularity condition that $∑_{m=0}^∞ {∑_{j=1}^∞ [∑_{n=j2^m}^{(j+1)2^m-1} |a_n - a_{n+1}|]²}^{1/2} < ∞$, then the cosine series represents an integrable function on the interval [-π,π]. We also show that, if the coefficients (bn) in a series $∑_{n=1}^∞ b_n sin(nt)$ tend to 0 and satisfy the corresponding regularity condition, then the sine series represents an integrable function on [-π,π] if and only if $∑_{n=1}^∞ |b_n|/n < ∞$. These conclusions were previously known to hold under stronger restrictions on the sizes of the differences $Δa_n = a_n - a_{n+1}$ and $Δb_n = b_n - b_{n+1}$. We were led to the mixed-norm conditions that we use here by our recent discovery that the same combination of conditions implies the integrability of Walsh series with coefficients (an) tending to 0. We also show here that this condition on the differences implies that the cosine series converges in L¹-norm if and only if $a_n log n → 0$ as n → ∞. The corresponding statement also holds for sine series for which $∑_{n=1}^∞ |b_n|/n < ∞$. If either type of series is assumed a priori to represent an integrable function, then weaker regularity conditions suffice for the validity of this criterion for norm convergence.
LA - eng
KW - integrability theorems; trigonometric series; regularity condition; cosine series; sine series; Walsh series; norm convergence
UR - http://eudml.org/doc/216021
ER -

References

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