q-Blossoming for analytic functions
We construct a q -analog of the blossom for analytic functions, the analytic q -blossom. This q -analog also extends the notion of q -blossoming from polynomials to analytic functions. We then apply this analytic q -blossom to derive identities for analytic functions represented in terms of the q -P...
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| Published in | Numerical algorithms Vol. 82; no. 1; pp. 107 - 121 |
|---|---|
| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
New York
Springer US
01.09.2019
Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
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| Abstract | We construct a
q
-analog of the blossom for analytic functions, the analytic
q
-blossom. This
q
-analog also extends the notion of
q
-blossoming from polynomials to analytic functions. We then apply this analytic
q
-blossom to derive identities for analytic functions represented in terms of the
q
-Poisson basis, including
q
-versions of the Marsden identity and the de Boor-Fix formula for analytic functions. |
|---|---|
| AbstractList | We construct a
q
-analog of the blossom for analytic functions, the analytic
q
-blossom. This
q
-analog also extends the notion of
q
-blossoming from polynomials to analytic functions. We then apply this analytic
q
-blossom to derive identities for analytic functions represented in terms of the
q
-Poisson basis, including
q
-versions of the Marsden identity and the de Boor-Fix formula for analytic functions. We construct a q-analog of the blossom for analytic functions, the analytic q-blossom. This q-analog also extends the notion of q-blossoming from polynomials to analytic functions. We then apply this analytic q-blossom to derive identities for analytic functions represented in terms of the q-Poisson basis, including q-versions of the Marsden identity and the de Boor-Fix formula for analytic functions. |
| Author | Goldman, Ron Simeonov, Plamen |
| Author_xml | – sequence: 1 givenname: Ron surname: Goldman fullname: Goldman, Ron email: rng@cs.rice.edu organization: Department of Computer Science, Rice University – sequence: 2 givenname: Plamen surname: Simeonov fullname: Simeonov, Plamen organization: Department of Mathematics and Statistics, University of Houston-Downtown |
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| CitedBy_id | crossref_primary_10_1016_j_cagd_2023_102252 |
| Cites_doi | 10.1016/j.cam.2015.04.027 10.1016/j.cam.2014.12.034 10.1051/m2an/1998320607731 10.1007/s003659900096 10.1016/j.cam.2013.01.009 10.1006/jath.1996.0056 10.1016/j.cagd.2016.02.019 10.1023/A:1018995019439 10.1201/9781439803356 10.1016/S0167-8396(99)00005-9 10.1016/j.jat.2011.09.006 10.1016/0167-8396(93)90036-3 10.1016/j.cagd.2011.09.003 10.1006/jath.1998.3306 10.1007/s10543-012-0395-z 10.1006/jath.2000.3529 10.1016/j.cagd.2012.10.002 10.1016/j.jat.2014.12.007 10.1016/S0377-0427(02)00733-1 10.1023/A:1019131529398 10.1023/A:1019114424971 |
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| Keywords | Marsden identity 30Bxx 41Axx 30E10 blossom bracket operator 65Dxx Poisson basis de Boor-Fix formula 65D17 Analytic function derivative |
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| References | Ait-Haddou, Sakane, Nomura (CR1) 2013; 30 Ismail (CR11) 2009 Laurent, Maxim, Mazure (CR12) 1997; 15 Gonsor, Neamtu (CR9) 1996; 86 Mazure (CR19) 1999; 15 Budakçi, Disibüyük, Goldman, Oruç (CR3) 2015; 282 Goldman, Simeonov (CR8) 2016; 47 CR18 CR15 Simeonov, Zafiris, Goldman (CR27) 2012; 164 Laurent, Mazure (CR13) 1998; 32 Oruc, Phillips (CR23) 2003; 151 Ait-Haddou, Sakane, Nomura (CR2) 2013; 247 Laurent, Mazure (CR14) 1999; 16 Mazure (CR16) 1999; 10 Goldman (CR6) 2009 Morin (CR21) 2001 Goldman, Simeonov (CR7) 2015; 288 CR5 Mazure (CR17) 1999; 22 Disibuyuk, Goldman (CR4) 2015; 192 Goodman, Mazure (CR10) 2001; 109 CR22 Simeonov, Zafiris, Goldman (CR26) 2011; 28 Mazure, Laurent (CR20) 1999; 98 Simeonov, Goldman (CR25) 2013; 53 Pottmann (CR24) 1993; 10 P Simeonov (595_CR27) 2012; 164 P-J Laurent (595_CR14) 1999; 16 M-L Mazure (595_CR19) 1999; 15 G Gonsor (595_CR9) 1996; 86 P Simeonov (595_CR26) 2011; 28 TNT Goodman (595_CR10) 2001; 109 M-L Mazure (595_CR16) 1999; 10 R Ait-Haddou (595_CR2) 2013; 247 M-L Mazure (595_CR17) 1999; 22 R Goldman (595_CR7) 2015; 288 H Oruc (595_CR23) 2003; 151 MEH Ismail (595_CR11) 2009 P-J Laurent (595_CR13) 1998; 32 R Goldman (595_CR6) 2009 H Pottmann (595_CR24) 1993; 10 C Disibuyuk (595_CR4) 2015; 192 P-J Laurent (595_CR12) 1997; 15 595_CR18 R Ait-Haddou (595_CR1) 2013; 30 595_CR15 G Morin (595_CR21) 2001 M-L Mazure (595_CR20) 1999; 98 595_CR22 P Simeonov (595_CR25) 2013; 53 R Goldman (595_CR8) 2016; 47 G Budakçi (595_CR3) 2015; 282 595_CR5 |
| References_xml | – ident: CR22 – volume: 288 start-page: 284 year: 2015 end-page: 303 ident: CR7 article-title: Quantum Bernstein bases and quantum Bézier curves publication-title: J. Comput. Appl. Math. doi: 10.1016/j.cam.2015.04.027 contributor: fullname: Simeonov – ident: CR18 – volume: 282 start-page: 17 year: 2015 end-page: 33 ident: CR3 article-title: Extending fundamental formulas from classical B-splines to quantum B-splines publication-title: J. Comput. Appl. Math. doi: 10.1016/j.cam.2014.12.034 contributor: fullname: Oruç – volume: 32 start-page: 773 year: 1998 end-page: 788 ident: CR13 article-title: Nested sequences of Chebyshev spaces publication-title: Math. Mod. Numer. Anal. doi: 10.1051/m2an/1998320607731 contributor: fullname: Mazure – volume: 15 start-page: 33 year: 1999 end-page: 68 ident: CR19 article-title: Blossoming: a geometric approach, Constr publication-title: Approx doi: 10.1007/s003659900096 contributor: fullname: Mazure – volume: 247 start-page: 172 year: 2013 end-page: 208 ident: CR2 article-title: Chebyshev blossoming in Müntz spaces: toward shaping with Young diagrams publication-title: J. Comput. Appl. Math. doi: 10.1016/j.cam.2013.01.009 contributor: fullname: Nomura – volume: 86 start-page: 81 year: 1996 end-page: 107 ident: CR9 article-title: Null spaces of differential operators, polar forms, and splines publication-title: J. Approx. Theory. doi: 10.1006/jath.1996.0056 contributor: fullname: Neamtu – volume: 47 start-page: 29 year: 2016 end-page: 54 ident: CR8 article-title: Generalized quantum splines publication-title: Comput. Aided Geom. Design doi: 10.1016/j.cagd.2016.02.019 contributor: fullname: Simeonov – volume: 10 start-page: 219 year: 1999 end-page: 238 ident: CR16 article-title: Chebyshev spaces with polynomial blossoms publication-title: Adv. Comput. Math. doi: 10.1023/A:1018995019439 contributor: fullname: Mazure – year: 2009 ident: CR6 publication-title: An integrated introduction to computer graphics and geometric modeling doi: 10.1201/9781439803356 contributor: fullname: Goldman – volume: 16 start-page: 317 year: 1999 end-page: 343 ident: CR14 article-title: Polynomial Chebyshev splines publication-title: Comput. Aided Geom. Design doi: 10.1016/S0167-8396(99)00005-9 contributor: fullname: Mazure – ident: CR15 – volume: 164 start-page: 77 year: 2012 end-page: 104 ident: CR27 article-title: q-Blossoming: A new approach to algorithms and identities for q-Bernstein bases and q-Bézier curves publication-title: J. Approx. Theory doi: 10.1016/j.jat.2011.09.006 contributor: fullname: Goldman – volume: 10 start-page: 181 year: 1993 end-page: 210 ident: CR24 article-title: The geometry of Tchebycheffian splines publication-title: Comput. Aided Geom. Design doi: 10.1016/0167-8396(93)90036-3 contributor: fullname: Pottmann – volume: 28 start-page: 549 year: 2011 end-page: 565 ident: CR26 article-title: h-Blossoming: A new approach to algorithms and identities for h-Bernstein bases and h-Bézier curves publication-title: Comput. Aided Geom. Design doi: 10.1016/j.cagd.2011.09.003 contributor: fullname: Goldman – volume: 98 start-page: 316 year: 1999 end-page: 353 ident: CR20 article-title: Piecewise smooth spaces in duality: applications to blossoming publication-title: J. Approx. Theory. doi: 10.1006/jath.1998.3306 contributor: fullname: Laurent – volume: 53 start-page: 193 year: 2013 end-page: 223 ident: CR25 article-title: Quantum B-splines publication-title: BIT. doi: 10.1007/s10543-012-0395-z contributor: fullname: Goldman – year: 2009 ident: CR11 publication-title: Classical and quantum orthogonal polynomials in one variable, vol. 98 contributor: fullname: Ismail – year: 2001 ident: CR21 publication-title: Analytic functions in computer aided design. Ph.D. Thesis contributor: fullname: Morin – volume: 109 start-page: 48 year: 2001 end-page: 81 ident: CR10 article-title: Blossoming beyond extended Chebyshev spaces publication-title: J. Approx. Theory. doi: 10.1006/jath.2000.3529 contributor: fullname: Mazure – ident: CR5 – volume: 30 start-page: 199 year: 2013 end-page: 225 ident: CR1 article-title: Gelfond-Bézier curves publication-title: Comput. Aided Geom. Design doi: 10.1016/j.cagd.2012.10.002 contributor: fullname: Nomura – volume: 192 start-page: 234 year: 2015 end-page: 249 ident: CR4 article-title: A unifying functional structure for polar forms and for functional Bernstein Bézier curves publication-title: J. Approx. Theory. doi: 10.1016/j.jat.2014.12.007 contributor: fullname: Goldman – volume: 151 start-page: 1 year: 2003 end-page: 12 ident: CR23 article-title: q-Bernstein polynomials and Bézier curves publication-title: J. Comput. Appl. Math. doi: 10.1016/S0377-0427(02)00733-1 contributor: fullname: Phillips – volume: 22 start-page: 285 year: 1999 end-page: 304 ident: CR17 article-title: Bernstein bases in Müntz spaces publication-title: Numer. Algorithms doi: 10.1023/A:1019131529398 contributor: fullname: Mazure – volume: 15 start-page: 373 year: 1997 end-page: 383 ident: CR12 article-title: Chebyshev splines and shape parameters publication-title: Numer. Algorithms doi: 10.1023/A:1019114424971 contributor: fullname: Mazure – volume: 164 start-page: 77 year: 2012 ident: 595_CR27 publication-title: J. Approx. Theory doi: 10.1016/j.jat.2011.09.006 contributor: fullname: P Simeonov – volume: 192 start-page: 234 year: 2015 ident: 595_CR4 publication-title: J. Approx. Theory. doi: 10.1016/j.jat.2014.12.007 contributor: fullname: C Disibuyuk – volume-title: Analytic functions in computer aided design. Ph.D. Thesis year: 2001 ident: 595_CR21 contributor: fullname: G Morin – volume: 16 start-page: 317 year: 1999 ident: 595_CR14 publication-title: Comput. Aided Geom. Design doi: 10.1016/S0167-8396(99)00005-9 contributor: fullname: P-J Laurent – ident: 595_CR22 – volume: 28 start-page: 549 year: 2011 ident: 595_CR26 publication-title: Comput. Aided Geom. Design doi: 10.1016/j.cagd.2011.09.003 contributor: fullname: P Simeonov – volume: 151 start-page: 1 year: 2003 ident: 595_CR23 publication-title: J. Comput. Appl. Math. doi: 10.1016/S0377-0427(02)00733-1 contributor: fullname: H Oruc – ident: 595_CR18 – volume: 47 start-page: 29 year: 2016 ident: 595_CR8 publication-title: Comput. Aided Geom. Design doi: 10.1016/j.cagd.2016.02.019 contributor: fullname: R Goldman – volume: 32 start-page: 773 year: 1998 ident: 595_CR13 publication-title: Math. Mod. Numer. Anal. doi: 10.1051/m2an/1998320607731 contributor: fullname: P-J Laurent – volume-title: Classical and quantum orthogonal polynomials in one variable, vol. 98 year: 2009 ident: 595_CR11 contributor: fullname: MEH Ismail – volume: 86 start-page: 81 year: 1996 ident: 595_CR9 publication-title: J. Approx. Theory. doi: 10.1006/jath.1996.0056 contributor: fullname: G Gonsor – volume: 98 start-page: 316 year: 1999 ident: 595_CR20 publication-title: J. Approx. Theory. doi: 10.1006/jath.1998.3306 contributor: fullname: M-L Mazure – ident: 595_CR5 – volume: 15 start-page: 33 year: 1999 ident: 595_CR19 publication-title: Approx doi: 10.1007/s003659900096 contributor: fullname: M-L Mazure – volume: 15 start-page: 373 year: 1997 ident: 595_CR12 publication-title: Numer. Algorithms doi: 10.1023/A:1019114424971 contributor: fullname: P-J Laurent – volume: 288 start-page: 284 year: 2015 ident: 595_CR7 publication-title: J. Comput. Appl. Math. doi: 10.1016/j.cam.2015.04.027 contributor: fullname: R Goldman – volume: 10 start-page: 219 year: 1999 ident: 595_CR16 publication-title: Adv. Comput. Math. doi: 10.1023/A:1018995019439 contributor: fullname: M-L Mazure – volume: 30 start-page: 199 year: 2013 ident: 595_CR1 publication-title: Comput. Aided Geom. Design doi: 10.1016/j.cagd.2012.10.002 contributor: fullname: R Ait-Haddou – volume: 282 start-page: 17 year: 2015 ident: 595_CR3 publication-title: J. Comput. Appl. Math. doi: 10.1016/j.cam.2014.12.034 contributor: fullname: G Budakçi – volume-title: An integrated introduction to computer graphics and geometric modeling year: 2009 ident: 595_CR6 doi: 10.1201/9781439803356 contributor: fullname: R Goldman – volume: 53 start-page: 193 year: 2013 ident: 595_CR25 publication-title: BIT. doi: 10.1007/s10543-012-0395-z contributor: fullname: P Simeonov – volume: 109 start-page: 48 year: 2001 ident: 595_CR10 publication-title: J. Approx. Theory. doi: 10.1006/jath.2000.3529 contributor: fullname: TNT Goodman – volume: 247 start-page: 172 year: 2013 ident: 595_CR2 publication-title: J. Comput. Appl. Math. doi: 10.1016/j.cam.2013.01.009 contributor: fullname: R Ait-Haddou – volume: 22 start-page: 285 year: 1999 ident: 595_CR17 publication-title: Numer. Algorithms doi: 10.1023/A:1019131529398 contributor: fullname: M-L Mazure – ident: 595_CR15 – volume: 10 start-page: 181 year: 1993 ident: 595_CR24 publication-title: Comput. Aided Geom. Design doi: 10.1016/0167-8396(93)90036-3 contributor: fullname: H Pottmann |
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| Snippet | We construct a
q
-analog of the blossom for analytic functions, the analytic
q
-blossom. This
q
-analog also extends the notion of
q
-blossoming from... We construct a q-analog of the blossom for analytic functions, the analytic q-blossom. This q-analog also extends the notion of q-blossoming from polynomials... |
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| Title | q-Blossoming for analytic functions |
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