Approximation by Unimodular Functions
The theorems in this paper are all concerned with either pointwise or uniform approximation by functions which have unit modulus or by convex combinations of such functions. The results are related to, and are outgrowths of, the theorems in [4; 5; 10]. In § 1, we show that a function bounded by 1, w...
Uloženo v:
| Vydáno v: | Canadian journal of mathematics Ročník 23; číslo 2; s. 257 - 269 |
|---|---|
| Hlavní autor: | |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
Cambridge, UK
Cambridge University Press
01.04.1971
|
| ISSN: | 0008-414X, 1496-4279 |
| On-line přístup: | Získat plný text |
| Tagy: |
Přidat tag
Žádné tagy, Buďte první, kdo vytvoří štítek k tomuto záznamu!
|
| Abstract | The theorems in this paper are all concerned with either pointwise or uniform approximation by functions which have unit modulus or by convex combinations of such functions. The results are related to, and are outgrowths of, the theorems in [4; 5; 10]. In § 1, we show that a function bounded by 1, which is analytic in the open unit disc Δ and continuous on may be approximated uniformly on the set where it has modulus 1 (subject to certain restrictions; see Theorem 1) by a finite Blaschke product; that is, by a function of the form
*
where |λ| = 1 and |αi
| < 1, i = 1, …, N. In § 1 we also discuss pointwise approximation by Blaschke products with restricted zeros. |
|---|---|
| AbstractList | The theorems in this paper are all concerned with either pointwise or uniform approximation by functions which have unit modulus or by convex combinations of such functions. The results are related to, and are outgrowths of, the theorems in [
4; 5; 10
].
In § 1, we show that a function bounded by 1, which is analytic in the open unit disc Δ and continuous on
may be approximated uniformly on the set where it has modulus 1 (subject to certain restrictions; see Theorem 1) by a finite Blaschke product; that is, by a function of the form
*
where |
λ
| = 1 and |
α
i
| < 1,
i
= 1, …,
N
. In § 1 we also discuss pointwise approximation by Blaschke products with restricted zeros. The theorems in this paper are all concerned with either pointwise or uniform approximation by functions which have unit modulus or by convex combinations of such functions. The results are related to, and are outgrowths of, the theorems in [4; 5; 10]. In § 1, we show that a function bounded by 1, which is analytic in the open unit disc Δ and continuous on may be approximated uniformly on the set where it has modulus 1 (subject to certain restrictions; see Theorem 1) by a finite Blaschke product; that is, by a function of the form * where |λ| = 1 and |αi | < 1, i = 1, …, N. In § 1 we also discuss pointwise approximation by Blaschke products with restricted zeros. |
| Author | Fisher, Stephen |
| Author_xml | – sequence: 1 givenname: Stephen surname: Fisher fullname: Fisher, Stephen organization: Northwestern University, Evanston, Illinois |
| BookMark | eNp1j89LwzAUgINMsJtePffiMTMvTX8dR3FOmXhx4C0kaSoZbVqSFtx_b6o7CTs93oPv8X1LtLC91QjdA1kzSJPH6vUNQ5kDJjTF7ApFwMoMM5qXCxQRQgrMgH3eoKX3x7AmWQoRetgMg-u_TSdG09tYnuKDNV1fT61w8Xayaj77W3TdiNbru_NcocP26aPa4f3780u12WNFUzJiyUDVOdMCiA5KOm9A5wBUskYxJSmtZVEWWUKCCFFKFWUuUlEIDUkthWqSFVr__VWu997phg8uqLkTB8LnSB4i-RzJQyRnAWD_AGXG35TRCdNexsgZE510pv7S_NhPzoa2S8gPF-hnEA |
| CitedBy_id | crossref_primary_10_1016_0022_1236_77_90074_X crossref_primary_10_1080_17476933_2021_1968381 |
| ContentType | Journal Article |
| Copyright | Copyright © Canadian Mathematical Society 1971 |
| Copyright_xml | – notice: Copyright © Canadian Mathematical Society 1971 |
| DBID | AAYXX CITATION |
| DOI | 10.4153/CJM-1971-025-4 |
| DatabaseName | CrossRef |
| DatabaseTitle | CrossRef |
| DatabaseTitleList | CrossRef |
| DeliveryMethod | fulltext_linktorsrc |
| Discipline | Mathematics |
| EISSN | 1496-4279 |
| EndPage | 269 |
| ExternalDocumentID | 10_4153_CJM_1971_025_4 |
| GroupedDBID | --Z -~X 09C 09E 0R~ 186 3O- 69Q 6TJ 8FQ AABWE AAGFV AAIKC AAMNW AANRG AASVR AAUKB AAYEQ AAYJJ ABBZL ABCQX ABEFU ABGDZ ABJNI ABUWG ABVKB ABVZP ABXAU ABZCX ACDLN ACGFO ACIPV ACKIV ACNCT ACYZP ACZWT ADDNB ADIYS ADKIL ADOVH ADVJH AEBAK AECCQ AENCP AFKQG AFKRA AFLVW AFZFC AGABE AGBYD AGJUD AHRGI AI. AIOIP AJCYY AJPFC ALEEW ALMA_UNASSIGNED_HOLDINGS AQJOH ARAPS ARZZG ATUCA AYIQA BBLKV BCGOX BENPR BESQT BGLVJ BLZWO CCPQU CCQAD CCUQV CFBFF CGQII CHEAL CJCSC DOHLZ DWQXO EBS EGQIC EJD FRP HCIFZ HF~ IH6 IOO JHPGK K7- KCGVB KFECR L7B LW7 MVM NHB NZEOI OHT OK1 P2P RCA RCD ROL S10 UHB VH1 WFFJZ XJT XOL XSW YYP ZKB ZMEZD AAXMD AAYXX ABUFD ABXHF ADXHL AFFHD AKMAY AMVHM CITATION PHGZM PHGZT PQGLB |
| ID | FETCH-LOGICAL-c250t-b41cd74ea10e415e7f1e7112b4fc4cb22db8986304140ccc897a5a8ae13dbacf3 |
| ISICitedReferencesCount | 4 |
| ISICitedReferencesURI | http://www.webofscience.com/api/gateway?GWVersion=2&SrcApp=Summon&SrcAuth=ProQuest&DestLinkType=CitingArticles&DestApp=WOS_CPL&KeyUT=10.4153/cjm-1971-025-4&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D |
| ISSN | 0008-414X |
| IngestDate | Tue Nov 18 22:30:17 EST 2025 Sat Nov 29 05:00:06 EST 2025 Tue Jan 21 06:17:57 EST 2025 |
| IsDoiOpenAccess | false |
| IsOpenAccess | true |
| IsPeerReviewed | true |
| IsScholarly | true |
| Issue | 2 |
| Language | English |
| License | https://www.cambridge.org/core/terms |
| LinkModel | OpenURL |
| MergedId | FETCHMERGED-LOGICAL-c250t-b41cd74ea10e415e7f1e7112b4fc4cb22db8986304140ccc897a5a8ae13dbacf3 |
| OpenAccessLink | https://www.cambridge.org/core/services/aop-cambridge-core/content/view/BF3B77C30E09219C30739F1FED87564F/S0008414X00053165a.pdf/div-class-title-approximation-by-unimodular-functions-div.pdf |
| PageCount | 13 |
| ParticipantIDs | crossref_primary_10_4153_CJM_1971_025_4 crossref_citationtrail_10_4153_CJM_1971_025_4 cambridge_journals_10_4153_CJM_1971_025_4 |
| PublicationCentury | 1900 |
| PublicationDate | 1971-04-00 |
| PublicationDateYYYYMMDD | 1971-04-01 |
| PublicationDate_xml | – month: 04 year: 1971 text: 1971-04-00 |
| PublicationDecade | 1970 |
| PublicationPlace | Cambridge, UK |
| PublicationPlace_xml | – name: Cambridge, UK |
| PublicationTitle | Canadian journal of mathematics |
| PublicationTitleAlternate | Can. j. math |
| PublicationYear | 1971 |
| Publisher | Cambridge University Press |
| Publisher_xml | – name: Cambridge University Press |
| References | Rudin (S0008414X00053165_ref010) 1969; 76 Fisher (S0008414X00053165_ref005) 1969; 75 Fisher (S0008414X00053165_ref004) 1968; 74 Phelps (S0008414X00053165_ref009) 1965; 32 Rudin (S0008414X00053165_ref011) 1969 Carleson (S0008414X00053165_ref003) 1960; 73 Frostman (S0008414X00053165_ref006) 1935; 3 Hoffman (S0008414X00053165_ref008) 1962 Carathéodory (S0008414X00053165_ref002) 1954; 2 Helson (S0008414X00053165_ref007) 1967; 21 Ahern (S0008414X00053165_ref001) 1970; 92 |
| References_xml | – volume: 2 volume-title: Theory of functions of a complex variable year: 1954 ident: S0008414X00053165_ref002 – volume: 73 start-page: 190 year: 1960 ident: S0008414X00053165_ref003 article-title: Math. Z. publication-title: A representation formula for the Dirichlet integral – volume: 75 start-page: 1037 year: 1969 ident: S0008414X00053165_ref005 article-title: Bull. Amer. Math. Soc. publication-title: Another theorem on convex combinations of unimodular functions – volume: 3 start-page: 1 year: 1935 ident: S0008414X00053165_ref006 article-title: Medd. Lunds Univ. Mat. Sem. publication-title: Potentiel d'équilibre et capacité des ensembles avec quelque applications à la théorie des fonctions – volume-title: Function theory in poly discs year: 1969 ident: S0008414X00053165_ref011 – volume-title: Banach spaces of analytic functions year: 1962 ident: S0008414X00053165_ref008 – volume: 92 start-page: 332 year: 1970 ident: S0008414X00053165_ref001 article-title: Amer. J. Math. publication-title: Radial limits and invariant subspaces – volume: 74 start-page: 1128 year: 1968 ident: S0008414X00053165_ref004 article-title: Bull Amer. Math. Soc. publication-title: The convex hull of the finite Blaschke products – volume: 21 start-page: 5 year: 1967 ident: S0008414X00053165_ref007 article-title: Math. Scand. publication-title: Past and future – volume: 32 start-page: 267 year: 1965 ident: S0008414X00053165_ref009 article-title: Duke Math. J. publication-title: Extreme points in function algebras – volume: 76 start-page: 795 year: 1969 ident: S0008414X00053165_ref010 article-title: Bull. Amer. Math. Soc. publication-title: Convex combinations of unimodular functions |
| SSID | ssj0003651 |
| Score | 1.1954093 |
| Snippet | The theorems in this paper are all concerned with either pointwise or uniform approximation by functions which have unit modulus or by convex combinations of... |
| SourceID | crossref cambridge |
| SourceType | Enrichment Source Index Database Publisher |
| StartPage | 257 |
| Title | Approximation by Unimodular Functions |
| URI | https://www.cambridge.org/core/product/identifier/S0008414X00053165/type/journal_article |
| Volume | 23 |
| WOSCitedRecordID | wos10.4153/cjm-1971-025-4&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D |
| hasFullText | 1 |
| inHoldings | 1 |
| isFullTextHit | |
| isPrint | |
| journalDatabaseRights | – providerCode: PRVAJQ databaseName: Google Magazines customDbUrl: eissn: 1496-4279 dateEnd: 19961231 omitProxy: false ssIdentifier: ssj0003651 issn: 0008-414X databaseCode: 69Q dateStart: 19490101 isFulltext: true titleUrlDefault: http://www.google.com/books/magazines providerName: Google.com |
| link | http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwtV3PS8MwFA46PehB_InzFz0o4iHapmmbHMdQRNhQmLBbadIUBq4b25T53_uyZG03HMyDl1JKkib50pcv5b3vIXQNS8RV3PMxkNUQUxEmmHGSYkFFBmwjpNIVs2QTUbvNul3-ajNkj2fpBKI8Z9MpH_4r1PAMwNahs3-Au2gUHsA9gA5XgB2uawHf0Crh054JSdTsEmhlf5DO3E2fYBcr_9AtCxRUVCT6hZprGR9SJEm3nmHlDwOPR17Fz2RuBBmmnnGMvFfG7lEOSBGT12VuGE0gsF0ApGrljKa03TCJybWybIuBGWhNiOZLC5tekMDG8yyIXi9tRoWLIBxOdAsx1I91_Rjqx3QTbZEo4NoCh_yt2HL9MLCpEc3IjDqnrv-w-P6qgkaFiVQoRWcf7dmzgNMwGB6gDZUfot1WOfVH6GYBTUd8OyWaToHmMXp_euw0n7HNbIElfBsT-BI8mUZUJZ6roJcqyjwVAfMVNJNUCkJSwTgLfRcG40opGY-SIGGJ8vxUJDLzT1AtH-TqFDk-HJlJ6ofAazPdoBCuT1KX8QC4GZDPOrorRhzbdTSOf5_dOsLzGYmllYjXmUo-Vpa_LcoPjTjKipJna5c8Rzvlor1AtcnoU12ibfk16Y1HVzPYfwAL91S2 |
| linkProvider | Google.com |
| openUrl | ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Approximation+by+Unimodular+Functions&rft.jtitle=Canadian+journal+of+mathematics&rft.au=Fisher%2C+Stephen&rft.date=1971-04-01&rft.issn=0008-414X&rft.eissn=1496-4279&rft.volume=23&rft.issue=2&rft.spage=257&rft.epage=269&rft_id=info:doi/10.4153%2FCJM-1971-025-4&rft.externalDBID=n%2Fa&rft.externalDocID=10_4153_CJM_1971_025_4 |
| thumbnail_l | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=0008-414X&client=summon |
| thumbnail_m | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=0008-414X&client=summon |
| thumbnail_s | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=0008-414X&client=summon |