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An Interpolation Problem for Coefficients of H∞Functions
H∞denotes the space of all bounded functions g on the unit circle whose Fourier coefficients$\hat g(n)$are zero for all negative n. It is known that, if {nk}∞ k = 0is a sequence of nonnegative integers with$n_{k + 1} > (1 + \delta)n_k$for all k, and if$\sum^\infty_{k = 0} |v_k|^2 < \infty$, th...
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| Published in: | Proceedings of the American Mathematical Society 1974-02, Vol.42 (2), p.402-408 |
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| container_end_page | 408 |
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| container_title | Proceedings of the American Mathematical Society |
| container_volume | 42 |
| creator | Fournier, John J. F. |
| description | H∞denotes the space of all bounded functions g on the unit circle whose Fourier coefficients$\hat g(n)$are zero for all negative n. It is known that, if {nk}∞
k = 0is a sequence of nonnegative integers with$n_{k + 1} > (1 + \delta)n_k$for all k, and if$\sum^\infty_{k = 0} |v_k|^2 < \infty$, then there is a function g in H∞with$\hat g(n_k) = v_k$for all k. Previous proofs of this fact have not indicated how to construct such H∞functions. This paper contains a simple, direct construction of such functions. The construction depends on properties of some polynomials similar to those introduced by Shapiro and Rudin. There is also a connection with a type of Riesz product studied by Salem and Zygmund. |
| doi_str_mv | 10.2307/2039516 |
| format | article |
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k = 0is a sequence of nonnegative integers with$n_{k + 1} > (1 + \delta)n_k$for all k, and if$\sum^\infty_{k = 0} |v_k|^2 < \infty$, then there is a function g in H∞with$\hat g(n_k) = v_k$for all k. Previous proofs of this fact have not indicated how to construct such H∞functions. This paper contains a simple, direct construction of such functions. The construction depends on properties of some polynomials similar to those introduced by Shapiro and Rudin. There is also a connection with a type of Riesz product studied by Salem and Zygmund.</description><identifier>ISSN: 0002-9939</identifier><identifier>EISSN: 1088-6826</identifier><identifier>DOI: 10.2307/2039516</identifier><language>eng</language><publisher>American Mathematical Society</publisher><subject>Coefficients ; Fourier coefficients ; Fourier series ; Integers ; Interpolation ; Mathematical functions ; Mathematical theorems ; Polynomials ; Series convergence</subject><ispartof>Proceedings of the American Mathematical Society, 1974-02, Vol.42 (2), p.402-408</ispartof><rights>Copyright 1974 American Mathematical Society</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c996-c4b88de015c9c47bba2d2b5cfe43a6b5df6365592a61fe85e0294a627843e9013</citedby></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://www.jstor.org/stable/pdf/2039516$$EPDF$$P50$$Gjstor$$H</linktopdf><linktohtml>$$Uhttps://www.jstor.org/stable/2039516$$EHTML$$P50$$Gjstor$$H</linktohtml><link.rule.ids>314,776,780,27901,27902,58213,58446</link.rule.ids></links><search><creatorcontrib>Fournier, John J. F.</creatorcontrib><title>An Interpolation Problem for Coefficients of H∞Functions</title><title>Proceedings of the American Mathematical Society</title><description>H∞denotes the space of all bounded functions g on the unit circle whose Fourier coefficients$\hat g(n)$are zero for all negative n. It is known that, if {nk}∞
k = 0is a sequence of nonnegative integers with$n_{k + 1} > (1 + \delta)n_k$for all k, and if$\sum^\infty_{k = 0} |v_k|^2 < \infty$, then there is a function g in H∞with$\hat g(n_k) = v_k$for all k. Previous proofs of this fact have not indicated how to construct such H∞functions. This paper contains a simple, direct construction of such functions. The construction depends on properties of some polynomials similar to those introduced by Shapiro and Rudin. There is also a connection with a type of Riesz product studied by Salem and Zygmund.</description><subject>Coefficients</subject><subject>Fourier coefficients</subject><subject>Fourier series</subject><subject>Integers</subject><subject>Interpolation</subject><subject>Mathematical functions</subject><subject>Mathematical theorems</subject><subject>Polynomials</subject><subject>Series convergence</subject><issn>0002-9939</issn><issn>1088-6826</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>1974</creationdate><recordtype>article</recordtype><recordid>eNp1z01KAzEcBfAgCo5VvEIWgqvRfE_irgzWFgq66H5IMv_AlGlSknHhDTyFh_MktrRbV48HPx48hO4peWKcNM-McCOpukAVJVrXSjN1iSpCCKuN4eYa3ZSyPVRqRFOhl3nEqzhB3qfRTkOK-CMnN8IOh5RxmyCEwQ8Qp4JTwMvf75_FZ_RHWG7RVbBjgbtzztBm8bppl_X6_W3Vzte1N0bVXjiteyBUeuNF45xlPXPSBxDcKif7oLiS0jCraAAtgTAjrGKNFhwMoXyGHk-zPqdSMoRun4edzV8dJd3xcXd-fJAPJ7ktU8r_sj8milNt</recordid><startdate>19740201</startdate><enddate>19740201</enddate><creator>Fournier, John J. F.</creator><general>American Mathematical Society</general><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>19740201</creationdate><title>An Interpolation Problem for Coefficients of H∞Functions</title><author>Fournier, John J. F.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c996-c4b88de015c9c47bba2d2b5cfe43a6b5df6365592a61fe85e0294a627843e9013</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>1974</creationdate><topic>Coefficients</topic><topic>Fourier coefficients</topic><topic>Fourier series</topic><topic>Integers</topic><topic>Interpolation</topic><topic>Mathematical functions</topic><topic>Mathematical theorems</topic><topic>Polynomials</topic><topic>Series convergence</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Fournier, John J. F.</creatorcontrib><collection>CrossRef</collection><jtitle>Proceedings of the American Mathematical Society</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Fournier, John J. F.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>An Interpolation Problem for Coefficients of H∞Functions</atitle><jtitle>Proceedings of the American Mathematical Society</jtitle><date>1974-02-01</date><risdate>1974</risdate><volume>42</volume><issue>2</issue><spage>402</spage><epage>408</epage><pages>402-408</pages><issn>0002-9939</issn><eissn>1088-6826</eissn><abstract>H∞denotes the space of all bounded functions g on the unit circle whose Fourier coefficients$\hat g(n)$are zero for all negative n. It is known that, if {nk}∞
k = 0is a sequence of nonnegative integers with$n_{k + 1} > (1 + \delta)n_k$for all k, and if$\sum^\infty_{k = 0} |v_k|^2 < \infty$, then there is a function g in H∞with$\hat g(n_k) = v_k$for all k. Previous proofs of this fact have not indicated how to construct such H∞functions. This paper contains a simple, direct construction of such functions. The construction depends on properties of some polynomials similar to those introduced by Shapiro and Rudin. There is also a connection with a type of Riesz product studied by Salem and Zygmund.</abstract><pub>American Mathematical Society</pub><doi>10.2307/2039516</doi><tpages>7</tpages></addata></record> |
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| source | JSTOR Archival Journals and Primary Sources Collection; American Mathematical Society Publications (Freely Accessible) |
| subjects | Coefficients Fourier coefficients Fourier series Integers Interpolation Mathematical functions Mathematical theorems Polynomials Series convergence |
| title | An Interpolation Problem for Coefficients of H∞Functions |
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