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A Note on Approximation by Trigonometric Polynomials
Let E = ∪ k = 1 n a k b k ⊂ ℝ ; if n > 1, then we assume that the segments [ a k , b k ] are pairwise disjoint. Assume that the following property holds: E ∩ ( E + 2π ν ) = ∅, ν ∈ ℤ , ν ≠ 0. Denote by H ω + r ( E ) the space of functions f defined on E such that | f ( r ) ( x 2 ) − f ( r...
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| Published in: | Journal of mathematical sciences (New York, N.Y.) N.Y.), 2020-03, Vol.243 (6), p.981-984 |
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| Language: | English |
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| container_title | Journal of mathematical sciences (New York, N.Y.) |
| container_volume | 243 |
| creator | Shirokov, N. A. |
| description | Let
E
=
∪
k
=
1
n
a
k
b
k
⊂
ℝ
; if
n
> 1, then we assume that the segments [
a
k
,
b
k
] are pairwise disjoint. Assume that the following property holds:
E
∩ (
E
+ 2π
ν
) = ∅,
ν
∈
ℤ
,
ν
≠ 0. Denote by
H
ω
+
r
(
E
) the space of functions
f
defined on
E
such that |
f
(
r
)
(
x
2
) −
f
(
r
)
(
x
1
)| ≤
c
f
ω
(|
x
2
−
x
1
|),
x
1
,
x
2
∈
E
,
f
(0)
≡
f
. Assume that a modulus of continuity
ω
satisfies the condition
∫
0
x
ω
t
t
dt
+
x
∫
x
∞
ω
t
t
2
dt
≤
cω
x
.
We find a constructive description of the space
H
ω
+
r
(
E
) in terms of the rate of nonuniform approximation of a function
f
∈
H
ω
+
r
(
E
) by trigonometric polynomials if
E
and
ω
satisfy the above conditions. |
| doi_str_mv | 10.1007/s10958-019-04598-y |
| format | article |
| fullrecord | <record><control><sourceid>gale_proqu</sourceid><recordid>TN_cdi_proquest_journals_2317062660</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><galeid>A615361060</galeid><sourcerecordid>A615361060</sourcerecordid><originalsourceid>FETCH-LOGICAL-c463y-71f7f1d8189d593a8cd8fa8837a5c8927c5cda3287e464d0fed9c59afd1519053</originalsourceid><addsrcrecordid>eNqNkl9LwzAUxYsoOKdfwKeCTz5kJk3TJI9j-GcwVHQ-h5gmJWNrZtLB-u2Nq7ANxpQ83NzwO-cm5CTJNYIDBCG9CwhywgBEHMCccAbak6SHCMWAUU5O4x7SDGBM8_PkIoQZjKKC4V6SD9Nn1-jU1elwufRubReysbH7bNOpt5Wr3UI33qr01c3b2Fg5D5fJmYlFX_3WfvLxcD8dPYHJy-N4NJwAlRe4BRQZalDJEOMl4VgyVTIjGcNUEsV4RhVRpcQZozov8hIaXXJFuDQlIohDgvvJTecbL_a10qERM7fydRwpMowoLLKigFuqknMtbG1c46Va2KDEsGBZfCjBxXEKRQLBjRc4QFW61l7OXa2Njcd7rv_id_wHB_i4Sr2w6uCA_wl2JtzuCSLT6HVTyVUIYvz-tm_-J7vjm3Ws8i4Er41Y-pgV3woExU8GRZdBETMoNhkUbRThThQiXFfabz_wiOob6GLYmw</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>2317062660</pqid></control><display><type>article</type><title>A Note on Approximation by Trigonometric Polynomials</title><source>Springer Link</source><creator>Shirokov, N. A.</creator><creatorcontrib>Shirokov, N. A.</creatorcontrib><description>Let
E
=
∪
k
=
1
n
a
k
b
k
⊂
ℝ
; if
n
> 1, then we assume that the segments [
a
k
,
b
k
] are pairwise disjoint. Assume that the following property holds:
E
∩ (
E
+ 2π
ν
) = ∅,
ν
∈
ℤ
,
ν
≠ 0. Denote by
H
ω
+
r
(
E
) the space of functions
f
defined on
E
such that |
f
(
r
)
(
x
2
) −
f
(
r
)
(
x
1
)| ≤
c
f
ω
(|
x
2
−
x
1
|),
x
1
,
x
2
∈
E
,
f
(0)
≡
f
. Assume that a modulus of continuity
ω
satisfies the condition
∫
0
x
ω
t
t
dt
+
x
∫
x
∞
ω
t
t
2
dt
≤
cω
x
.
We find a constructive description of the space
H
ω
+
r
(
E
) in terms of the rate of nonuniform approximation of a function
f
∈
H
ω
+
r
(
E
) by trigonometric polynomials if
E
and
ω
satisfy the above conditions.</description><identifier>ISSN: 1072-3374</identifier><identifier>EISSN: 1573-8795</identifier><identifier>DOI: 10.1007/s10958-019-04598-y</identifier><language>eng</language><publisher>New York: Springer US</publisher><subject>Approximation ; Mathematical analysis ; Mathematics ; Mathematics and Statistics ; Polynomials</subject><ispartof>Journal of mathematical sciences (New York, N.Y.), 2020-03, Vol.243 (6), p.981-984</ispartof><rights>Springer Science+Business Media, LLC, part of Springer Nature 2019</rights><rights>COPYRIGHT 2019 Springer</rights><rights>COPYRIGHT 2020 Springer</rights><rights>2019© Springer Science+Business Media, LLC, part of Springer Nature 2019</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><cites>FETCH-LOGICAL-c463y-71f7f1d8189d593a8cd8fa8837a5c8927c5cda3287e464d0fed9c59afd1519053</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>314,780,784,27924,27925</link.rule.ids></links><search><creatorcontrib>Shirokov, N. A.</creatorcontrib><title>A Note on Approximation by Trigonometric Polynomials</title><title>Journal of mathematical sciences (New York, N.Y.)</title><addtitle>J Math Sci</addtitle><description>Let
E
=
∪
k
=
1
n
a
k
b
k
⊂
ℝ
; if
n
> 1, then we assume that the segments [
a
k
,
b
k
] are pairwise disjoint. Assume that the following property holds:
E
∩ (
E
+ 2π
ν
) = ∅,
ν
∈
ℤ
,
ν
≠ 0. Denote by
H
ω
+
r
(
E
) the space of functions
f
defined on
E
such that |
f
(
r
)
(
x
2
) −
f
(
r
)
(
x
1
)| ≤
c
f
ω
(|
x
2
−
x
1
|),
x
1
,
x
2
∈
E
,
f
(0)
≡
f
. Assume that a modulus of continuity
ω
satisfies the condition
∫
0
x
ω
t
t
dt
+
x
∫
x
∞
ω
t
t
2
dt
≤
cω
x
.
We find a constructive description of the space
H
ω
+
r
(
E
) in terms of the rate of nonuniform approximation of a function
f
∈
H
ω
+
r
(
E
) by trigonometric polynomials if
E
and
ω
satisfy the above conditions.</description><subject>Approximation</subject><subject>Mathematical analysis</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Polynomials</subject><issn>1072-3374</issn><issn>1573-8795</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2020</creationdate><recordtype>article</recordtype><recordid>eNqNkl9LwzAUxYsoOKdfwKeCTz5kJk3TJI9j-GcwVHQ-h5gmJWNrZtLB-u2Nq7ANxpQ83NzwO-cm5CTJNYIDBCG9CwhywgBEHMCccAbak6SHCMWAUU5O4x7SDGBM8_PkIoQZjKKC4V6SD9Nn1-jU1elwufRubReysbH7bNOpt5Wr3UI33qr01c3b2Fg5D5fJmYlFX_3WfvLxcD8dPYHJy-N4NJwAlRe4BRQZalDJEOMl4VgyVTIjGcNUEsV4RhVRpcQZozov8hIaXXJFuDQlIohDgvvJTecbL_a10qERM7fydRwpMowoLLKigFuqknMtbG1c46Va2KDEsGBZfCjBxXEKRQLBjRc4QFW61l7OXa2Njcd7rv_id_wHB_i4Sr2w6uCA_wl2JtzuCSLT6HVTyVUIYvz-tm_-J7vjm3Ws8i4Er41Y-pgV3woExU8GRZdBETMoNhkUbRThThQiXFfabz_wiOob6GLYmw</recordid><startdate>20200323</startdate><enddate>20200323</enddate><creator>Shirokov, N. A.</creator><general>Springer US</general><general>Springer</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope><scope>ISR</scope></search><sort><creationdate>20200323</creationdate><title>A Note on Approximation by Trigonometric Polynomials</title><author>Shirokov, N. A.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c463y-71f7f1d8189d593a8cd8fa8837a5c8927c5cda3287e464d0fed9c59afd1519053</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2020</creationdate><topic>Approximation</topic><topic>Mathematical analysis</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Polynomials</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Shirokov, N. A.</creatorcontrib><collection>CrossRef</collection><collection>Gale In Context: Science</collection><jtitle>Journal of mathematical sciences (New York, N.Y.)</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Shirokov, N. A.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>A Note on Approximation by Trigonometric Polynomials</atitle><jtitle>Journal of mathematical sciences (New York, N.Y.)</jtitle><stitle>J Math Sci</stitle><date>2020-03-23</date><risdate>2020</risdate><volume>243</volume><issue>6</issue><spage>981</spage><epage>984</epage><pages>981-984</pages><issn>1072-3374</issn><eissn>1573-8795</eissn><abstract>Let
E
=
∪
k
=
1
n
a
k
b
k
⊂
ℝ
; if
n
> 1, then we assume that the segments [
a
k
,
b
k
] are pairwise disjoint. Assume that the following property holds:
E
∩ (
E
+ 2π
ν
) = ∅,
ν
∈
ℤ
,
ν
≠ 0. Denote by
H
ω
+
r
(
E
) the space of functions
f
defined on
E
such that |
f
(
r
)
(
x
2
) −
f
(
r
)
(
x
1
)| ≤
c
f
ω
(|
x
2
−
x
1
|),
x
1
,
x
2
∈
E
,
f
(0)
≡
f
. Assume that a modulus of continuity
ω
satisfies the condition
∫
0
x
ω
t
t
dt
+
x
∫
x
∞
ω
t
t
2
dt
≤
cω
x
.
We find a constructive description of the space
H
ω
+
r
(
E
) in terms of the rate of nonuniform approximation of a function
f
∈
H
ω
+
r
(
E
) by trigonometric polynomials if
E
and
ω
satisfy the above conditions.</abstract><cop>New York</cop><pub>Springer US</pub><doi>10.1007/s10958-019-04598-y</doi><tpages>4</tpages></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 1072-3374 |
| ispartof | Journal of mathematical sciences (New York, N.Y.), 2020-03, Vol.243 (6), p.981-984 |
| issn | 1072-3374 1573-8795 |
| language | eng |
| recordid | cdi_proquest_journals_2317062660 |
| source | Springer Link |
| subjects | Approximation Mathematical analysis Mathematics Mathematics and Statistics Polynomials |
| title | A Note on Approximation by Trigonometric Polynomials |
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