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Summability of double Fourier series on quantum tori
In this paper, we study two general summability methods generated by a function θ for noncommutative Fourier series on quantum tori T q 2 . For the rectangular θ -summation, we establish the noncommutative weak type maximal inequality ‖ ( σ m , n θ ( f ) ) ( m , n ) ∈ Σ β ‖ Λ 1 , ∞ ( T q 2 , ℓ ∞ ) ≤...
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| Published in: | Mathematische Zeitschrift 2024-11, Vol.308 (3), Article 47 |
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| creator | Jiao, Yong Zhou, Dejian |
| description | In this paper, we study two general summability methods generated by a function
θ
for noncommutative Fourier series on quantum tori
T
q
2
. For the rectangular
θ
-summation, we establish the noncommutative weak type maximal inequality
‖
(
σ
m
,
n
θ
(
f
)
)
(
m
,
n
)
∈
Σ
β
‖
Λ
1
,
∞
(
T
q
2
,
ℓ
∞
)
≤
c
β
,
θ
‖
f
‖
L
1
(
T
q
2
)
,
which generalizes the result due to Marcinkiewicz and Zygmund (Fundam Math 32:122–132, 1939). For the Marcinkiewicz
θ
-summation, we prove that
‖
(
F
n
θ
(
f
)
)
n
≥
1
‖
Λ
1
,
∞
(
T
q
2
,
ℓ
∞
)
≤
c
θ
‖
f
‖
L
1
(
T
q
2
)
.
Both noncommutative weak type maximal inequalities imply the bilateral almost uniform convergence. The
θ
-summation contains almost all well known summability methods, such as the Fejér, Weierstrass, Riesz, Picard, Bessel, Riemann, Rogosinski and de La Vallée–Poussin summations. |
| doi_str_mv | 10.1007/s00209-024-03604-7 |
| format | article |
| fullrecord | <record><control><sourceid>proquest_cross</sourceid><recordid>TN_cdi_proquest_journals_3115961207</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>3115961207</sourcerecordid><originalsourceid>FETCH-LOGICAL-c200t-ed0cd06c7924e924d34fadeab240610c7bbc6173966af7b7afc5b24cbaa816d13</originalsourceid><addsrcrecordid>eNp9kE9LxDAQxYMoWFe_gKeA5-jkT5P2KIurwoIH9RySNJUubbObtIf99mat4M3D8A7vvZnhh9AthXsKoB4SAIOaABMEuARB1BkqqOCM0Irxc1RkvyRlpcQlukppB5BNJQok3udhMLbru-mIQ4ubMNve402YY-cjTj5LwmHEh9mM0zzgKcTuGl20pk_-5ldX6HPz9LF-Idu359f145Y4BjAR34BrQDpVM-HzNFy0pvHGMgGSglPWOkkVr6U0rbLKtK7MnrPGVFQ2lK_Q3bJ3H8Nh9mnSu_zXmE9qTmlZS8pA5RRbUi6GlKJv9T52g4lHTUGf6OiFjs509A8dfSrxpZRyePzy8W_1P61vb8hnRw</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>3115961207</pqid></control><display><type>article</type><title>Summability of double Fourier series on quantum tori</title><source>Springer Link</source><creator>Jiao, Yong ; Zhou, Dejian</creator><creatorcontrib>Jiao, Yong ; Zhou, Dejian</creatorcontrib><description>In this paper, we study two general summability methods generated by a function
θ
for noncommutative Fourier series on quantum tori
T
q
2
. For the rectangular
θ
-summation, we establish the noncommutative weak type maximal inequality
‖
(
σ
m
,
n
θ
(
f
)
)
(
m
,
n
)
∈
Σ
β
‖
Λ
1
,
∞
(
T
q
2
,
ℓ
∞
)
≤
c
β
,
θ
‖
f
‖
L
1
(
T
q
2
)
,
which generalizes the result due to Marcinkiewicz and Zygmund (Fundam Math 32:122–132, 1939). For the Marcinkiewicz
θ
-summation, we prove that
‖
(
F
n
θ
(
f
)
)
n
≥
1
‖
Λ
1
,
∞
(
T
q
2
,
ℓ
∞
)
≤
c
θ
‖
f
‖
L
1
(
T
q
2
)
.
Both noncommutative weak type maximal inequalities imply the bilateral almost uniform convergence. The
θ
-summation contains almost all well known summability methods, such as the Fejér, Weierstrass, Riesz, Picard, Bessel, Riemann, Rogosinski and de La Vallée–Poussin summations.</description><identifier>ISSN: 0025-5874</identifier><identifier>EISSN: 1432-1823</identifier><identifier>DOI: 10.1007/s00209-024-03604-7</identifier><language>eng</language><publisher>Berlin/Heidelberg: Springer Berlin Heidelberg</publisher><subject>Fourier series ; Mathematics ; Mathematics and Statistics ; Toruses</subject><ispartof>Mathematische Zeitschrift, 2024-11, Vol.308 (3), Article 47</ispartof><rights>The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2024. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><cites>FETCH-LOGICAL-c200t-ed0cd06c7924e924d34fadeab240610c7bbc6173966af7b7afc5b24cbaa816d13</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>314,776,780,27900,27901</link.rule.ids></links><search><creatorcontrib>Jiao, Yong</creatorcontrib><creatorcontrib>Zhou, Dejian</creatorcontrib><title>Summability of double Fourier series on quantum tori</title><title>Mathematische Zeitschrift</title><addtitle>Math. Z</addtitle><description>In this paper, we study two general summability methods generated by a function
θ
for noncommutative Fourier series on quantum tori
T
q
2
. For the rectangular
θ
-summation, we establish the noncommutative weak type maximal inequality
‖
(
σ
m
,
n
θ
(
f
)
)
(
m
,
n
)
∈
Σ
β
‖
Λ
1
,
∞
(
T
q
2
,
ℓ
∞
)
≤
c
β
,
θ
‖
f
‖
L
1
(
T
q
2
)
,
which generalizes the result due to Marcinkiewicz and Zygmund (Fundam Math 32:122–132, 1939). For the Marcinkiewicz
θ
-summation, we prove that
‖
(
F
n
θ
(
f
)
)
n
≥
1
‖
Λ
1
,
∞
(
T
q
2
,
ℓ
∞
)
≤
c
θ
‖
f
‖
L
1
(
T
q
2
)
.
Both noncommutative weak type maximal inequalities imply the bilateral almost uniform convergence. The
θ
-summation contains almost all well known summability methods, such as the Fejér, Weierstrass, Riesz, Picard, Bessel, Riemann, Rogosinski and de La Vallée–Poussin summations.</description><subject>Fourier series</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Toruses</subject><issn>0025-5874</issn><issn>1432-1823</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2024</creationdate><recordtype>article</recordtype><recordid>eNp9kE9LxDAQxYMoWFe_gKeA5-jkT5P2KIurwoIH9RySNJUubbObtIf99mat4M3D8A7vvZnhh9AthXsKoB4SAIOaABMEuARB1BkqqOCM0Irxc1RkvyRlpcQlukppB5BNJQok3udhMLbru-mIQ4ubMNve402YY-cjTj5LwmHEh9mM0zzgKcTuGl20pk_-5ldX6HPz9LF-Idu359f145Y4BjAR34BrQDpVM-HzNFy0pvHGMgGSglPWOkkVr6U0rbLKtK7MnrPGVFQ2lK_Q3bJ3H8Nh9mnSu_zXmE9qTmlZS8pA5RRbUi6GlKJv9T52g4lHTUGf6OiFjs509A8dfSrxpZRyePzy8W_1P61vb8hnRw</recordid><startdate>20241101</startdate><enddate>20241101</enddate><creator>Jiao, Yong</creator><creator>Zhou, Dejian</creator><general>Springer Berlin Heidelberg</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>20241101</creationdate><title>Summability of double Fourier series on quantum tori</title><author>Jiao, Yong ; Zhou, Dejian</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c200t-ed0cd06c7924e924d34fadeab240610c7bbc6173966af7b7afc5b24cbaa816d13</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2024</creationdate><topic>Fourier series</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Toruses</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Jiao, Yong</creatorcontrib><creatorcontrib>Zhou, Dejian</creatorcontrib><collection>CrossRef</collection><jtitle>Mathematische Zeitschrift</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Jiao, Yong</au><au>Zhou, Dejian</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Summability of double Fourier series on quantum tori</atitle><jtitle>Mathematische Zeitschrift</jtitle><stitle>Math. Z</stitle><date>2024-11-01</date><risdate>2024</risdate><volume>308</volume><issue>3</issue><artnum>47</artnum><issn>0025-5874</issn><eissn>1432-1823</eissn><abstract>In this paper, we study two general summability methods generated by a function
θ
for noncommutative Fourier series on quantum tori
T
q
2
. For the rectangular
θ
-summation, we establish the noncommutative weak type maximal inequality
‖
(
σ
m
,
n
θ
(
f
)
)
(
m
,
n
)
∈
Σ
β
‖
Λ
1
,
∞
(
T
q
2
,
ℓ
∞
)
≤
c
β
,
θ
‖
f
‖
L
1
(
T
q
2
)
,
which generalizes the result due to Marcinkiewicz and Zygmund (Fundam Math 32:122–132, 1939). For the Marcinkiewicz
θ
-summation, we prove that
‖
(
F
n
θ
(
f
)
)
n
≥
1
‖
Λ
1
,
∞
(
T
q
2
,
ℓ
∞
)
≤
c
θ
‖
f
‖
L
1
(
T
q
2
)
.
Both noncommutative weak type maximal inequalities imply the bilateral almost uniform convergence. The
θ
-summation contains almost all well known summability methods, such as the Fejér, Weierstrass, Riesz, Picard, Bessel, Riemann, Rogosinski and de La Vallée–Poussin summations.</abstract><cop>Berlin/Heidelberg</cop><pub>Springer Berlin Heidelberg</pub><doi>10.1007/s00209-024-03604-7</doi></addata></record> |
| fulltext | fulltext |
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| ispartof | Mathematische Zeitschrift, 2024-11, Vol.308 (3), Article 47 |
| issn | 0025-5874 1432-1823 |
| language | eng |
| recordid | cdi_proquest_journals_3115961207 |
| source | Springer Link |
| subjects | Fourier series Mathematics Mathematics and Statistics Toruses |
| title | Summability of double Fourier series on quantum tori |
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