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1.
Dirichlet splines as fractional integrals of Bsplines
Open Access
Title:
Dirichlet splines as fractional integrals of Bsplines
Author:
Wolfgang zu Castell
Wolfgang zu Castell
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Description:
: Using Dirichlet averages we generalize the notion of a classical divided dierence of a function by introducing a parameter r in R k+1 + . The case r in N k+1 is related to divided dierences with multiple knots. We give an interpretation of these generalized dierences in terms of fractional operators applied to classical divided differences con...
: Using Dirichlet averages we generalize the notion of a classical divided dierence of a function by introducing a parameter r in R k+1 + . The case r in N k+1 is related to divided dierences with multiple knots. We give an interpretation of these generalized dierences in terms of fractional operators applied to classical divided differences considered as functions of their knots. The result is then applied to show that Dirichlet splines can be seen as fractional derivatives of Bsplines. 2000 AMS subj. class.: 41A15, 62H10, 26A33 Keywords: Dirichlet averages, Dirichlet splines, divided dierences, Bsplines, Dirichlet distribution, fractional integrals and derivatives 1.
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Year of Publication:
20090415
Source:
http://www.gsf.de/ibb/preprints/2001/pp0115.ps
http://www.gsf.de/ibb/preprints/2001/pp0115.ps
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Document Type:
text
Language:
en
Subjects:
Dirichlet averages ; Dirichlet splines ; divided dierences ; Bsplines ; Dirichlet distribution ; fractional integrals and derivatives
Dirichlet averages ; Dirichlet splines ; divided dierences ; Bsplines ; Dirichlet distribution ; fractional integrals and derivatives
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URL:
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.20.7773
http://www.gsf.de/ibb/preprints/2001/pp0115.ps
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.20.7773
http://www.gsf.de/ibb/preprints/2001/pp0115.ps
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2.
Generalized Bessel functions for pradial functions
Open Access
Title:
Generalized Bessel functions for pradial functions
Author:
Wolfgang Zu Castell
Wolfgang Zu Castell
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Abstract: Suppose that d ∈ N and p> 0. In this paper, we study the generalized Bessel functions for the surface {v ∈ Rd: vp = 1}, introduced by D.St.P. Richards. We derive a recurrence relation for these functions and utilize a series representation to relate them to the classical symmetric functions. These generalized Bessel functions are sym...
Abstract: Suppose that d ∈ N and p> 0. In this paper, we study the generalized Bessel functions for the surface {v ∈ Rd: vp = 1}, introduced by D.St.P. Richards. We derive a recurrence relation for these functions and utilize a series representation to relate them to the classical symmetric functions. These generalized Bessel functions are symmetric with respect to the action of the hyperoctahedral group Wd, which is the symmetry group of the ℓp unit sphere. By means of this symmetry under Wd, we further express these generalized Bessel functions in terms of Bessel functions for certain finite reflection groups. For the case in which p = 2, our representations lead to known relations for the classical Bessel functions of order d−2 2. For the case in which p = 1, the generalized Bessel functions have been studied by Berens and Xu in the analysis of summability problems for 1radial functions, and we show how their results may be framed within our more general context.
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Year of Publication:
20080701
Source:
http://ibb.gsf.de/preprints/2006/pp0601.pdf
http://ibb.gsf.de/preprints/2006/pp0601.pdf
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Document Type:
text
Language:
en
Subjects:
radial functions ; complete symmetric functions ; monomial symmetric functions ; FourierBessel transform ; divided differences ; characteristic functions ; multivariate symmetric distributions ; reflection symmetry ; finite reflection groups ; Gelfand
radial functions ; complete symmetric functions ; monomial symmetric functions ; FourierBessel transform ; divided differences ; characteristic functions ; multivariate symmetric distributions ; reflection symmetry ; finite reflection groups ; Gelfand
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DDC:
510 Mathematics
(computed)
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http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.67.5064
http://ibb.gsf.de/preprints/2006/pp0601.pdf
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.67.5064
http://ibb.gsf.de/preprints/2006/pp0601.pdf
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3.
Generalized Bessel functions for pradial functions
Open Access
Title:
Generalized Bessel functions for pradial functions
Author:
Wolfgang Zu Castell
Wolfgang Zu Castell
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Description:
Suppose that d N and p > 0. In this paper, we study the generalized Bessel functions for the surface p = 1}, introduced by D.St.P. Richards. We derive a recurrence relation for these functions and utilize a series representation to relate them to the classical symmetric functions. These generalized Bessel functions are symmetric with respect to ...
Suppose that d N and p > 0. In this paper, we study the generalized Bessel functions for the surface p = 1}, introduced by D.St.P. Richards. We derive a recurrence relation for these functions and utilize a series representation to relate them to the classical symmetric functions. These generalized Bessel functions are symmetric with respect to the action of the hyperoctahedral group W d , which is the symmetry group of the # p unit sphere. By means of this symmetry under W d , we further express these generalized Bessel functions in terms of Bessel functions for certain finite reflection groups. For the case in which p = 2, our representations lead to known relations for the classical Bessel functions of order . For the case in which p = 1, the generalized Bessel functions have been studied by Berens and Xu in the analysis of summability problems for 1radial functions, and we show how their results may be framed within our more general context. 2000 AMS subj. class.: 33C10, 42B10, 44A15, 62H10 Keywords: radial functions, complete symmetric functions, monomial symmetric functions, FourierBessel transform, divided di#erences, characteristic functions, multivariate symmetric distributions, reflection symmetry, finite reflection groups, Gelfand pairs 1
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The Pennsylvania State University CiteSeerX Archives
Year of Publication:
20090419
Source:
http://ibb.gsf.de/preprints/2006/pp0601.ps
http://ibb.gsf.de/preprints/2006/pp0601.ps
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Document Type:
text
Language:
en
Subjects:
radial functions ; complete symmetric functions ; monomial symmetric functions ; FourierBessel transform ; divided differences ; characteristic functions ; multivariate symmetric distributions ; reflection symmetry ; finite reflection groups ; Gelfand
radial functions ; complete symmetric functions ; monomial symmetric functions ; FourierBessel transform ; divided differences ; characteristic functions ; multivariate symmetric distributions ; reflection symmetry ; finite reflection groups ; Gelfand
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DDC:
510 Mathematics
(computed)
Rights:
Metadata may be used without restrictions as long as the oai identifier remains attached to it.
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URL:
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.60.4701
http://ibb.gsf.de/preprints/2006/pp0601.ps
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.60.4701
http://ibb.gsf.de/preprints/2006/pp0601.ps
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4.
Fractional derivatives and the inverse Fourier transform of `
Open Access
Title:
Fractional derivatives and the inverse Fourier transform of `
Author:
Wolfgang Zu Castell
Wolfgang Zu Castell
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Description:
Abstract: In [1] and [2] H. Berens, Y. Xu and the author proved that the inverse Fourier integral of ` 1radial functions, i.e., functions which are radial w.r.t. the `
Abstract: In [1] and [2] H. Berens, Y. Xu and the author proved that the inverse Fourier integral of ` 1radial functions, i.e., functions which are radial w.r.t. the `
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Contributors:
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Year of Publication:
20080701
Source:
http://ibb.gsf.de/preprints/2002/pp0209.ps
http://ibb.gsf.de/preprints/2002/pp0209.ps
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Document Type:
text
Language:
en
Subjects:
2000 AMS subj. class ; 33C60 ; 42B10 ; 44A15 Keywords ; radial functions ; fractional derivative ; Dirichlet splines ; Meijer's Gfunction
2000 AMS subj. class ; 33C60 ; 42B10 ; 44A15 Keywords ; radial functions ; fractional derivative ; Dirichlet splines ; Meijer's Gfunction
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URL:
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.69.1457
http://ibb.gsf.de/preprints/2002/pp0209.ps
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.69.1457
http://ibb.gsf.de/preprints/2002/pp0209.ps
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5.
POLYNOMIAL INTERPOLATION ON THE UNIT SPHERE II
Open Access
Title:
POLYNOMIAL INTERPOLATION ON THE UNIT SPHERE II
Author:
Wolfgang Zu Castell
;
Noemí Laín
;
Fernández
;
Yuan Xu
Wolfgang Zu Castell
;
Noemí Laín
;
Fernández
;
Yuan Xu
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Description:
Abstract. The problem of interpolation at (n+1) 2 points on the unit sphere S 2 by spherical polynomials of degree at most n is proved to have a unique solution for several sets of points. The points are located on a number of circles on the sphere with even number of points on each circle. The proof is based on a method of factorization of poly...
Abstract. The problem of interpolation at (n+1) 2 points on the unit sphere S 2 by spherical polynomials of degree at most n is proved to have a unique solution for several sets of points. The points are located on a number of circles on the sphere with even number of points on each circle. The proof is based on a method of factorization of polynomials. 1.
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Year of Publication:
20080701
Source:
http://ibb.gsf.de/preprints/2004/pp0416.pdf
http://ibb.gsf.de/preprints/2004/pp0416.pdf
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Document Type:
text
Language:
en
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Metadata may be used without restrictions as long as the oai identifier remains attached to it.
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URL:
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.69.8421
http://ibb.gsf.de/preprints/2004/pp0416.pdf
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.69.8421
http://ibb.gsf.de/preprints/2004/pp0416.pdf
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6.
Radial basis functions and corresponding
Open Access
Title:
Radial basis functions and corresponding
Author:
Wolfgang Zu Castell
;
Frank Filbir
Wolfgang Zu Castell
;
Frank Filbir
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Description:
zonal series expansions on the sphere
zonal series expansions on the sphere
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Year of Publication:
20080701
Source:
http://ibb.gsf.de/preprints/2004/pp0403.ps
http://ibb.gsf.de/preprints/2004/pp0403.ps
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Document Type:
text
Language:
en
Subjects:
2000 AMS subj. class ; 42C10 ; 42A82 ; 33C10 ; 33C45 Keywords ; radial basis functions ; zonal basis functions ; positive definite functions ; conditionally positive definite functions ; Bessel functions ; Gegenbauer polynomials ; kriging
2000 AMS subj. class ; 42C10 ; 42A82 ; 33C10 ; 33C45 Keywords ; radial basis functions ; zonal basis functions ; positive definite functions ; conditionally positive definite functions ; Bessel functions ; Gegenbauer polynomials ; kriging
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URL:
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.67.6092
http://ibb.gsf.de/preprints/2004/pp0403.ps
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.67.6092
http://ibb.gsf.de/preprints/2004/pp0403.ps
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7.
POLYNOMIAL INTERPOLATION ON THE UNIT SPHERE II
Open Access
Title:
POLYNOMIAL INTERPOLATION ON THE UNIT SPHERE II
Author:
Wolfgang Zu Castell
;
Noemí Laín
;
Fernández
;
Yuan Xu
Wolfgang Zu Castell
;
Noemí Laín
;
Fernández
;
Yuan Xu
Minimize authors
Description:
Abstract. The problem of interpolation at (n+1) 2 points on the unit sphere S 2 by spherical polynomials of degree at most n is proved to have a unique solution for several sets of points. The points are located on a number of circles on the sphere with even number of points on each circle. The proof is based on a method of factorization of poly...
Abstract. The problem of interpolation at (n+1) 2 points on the unit sphere S 2 by spherical polynomials of degree at most n is proved to have a unique solution for several sets of points. The points are located on a number of circles on the sphere with even number of points on each circle. The proof is based on a method of factorization of polynomials. 1.
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Year of Publication:
20080701
Source:
http://ibb.gsf.de/preprints/2004/pp0416.ps
http://ibb.gsf.de/preprints/2004/pp0416.ps
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Language:
en
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Metadata may be used without restrictions as long as the oai identifier remains attached to it.
Metadata may be used without restrictions as long as the oai identifier remains attached to it.
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URL:
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.72.6932
http://ibb.gsf.de/preprints/2004/pp0416.ps
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8.
Radial basis functions and corresponding zonal series expansions on the sphere
Open Access
Title:
Radial basis functions and corresponding zonal series expansions on the sphere
Author:
Wolfgang Zu Castell
;
Frank Filbir
Wolfgang Zu Castell
;
Frank Filbir
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Description:
Abstract: Since radial positive definite functions on R d remain positive definite when restricted to the sphere, it is natural to ask for properties of the zonal series expansion of such functions which relate to properties of the FourierBessel transform of the radial function. We show that the decay of the Gegenbauer coefficients is determine...
Abstract: Since radial positive definite functions on R d remain positive definite when restricted to the sphere, it is natural to ask for properties of the zonal series expansion of such functions which relate to properties of the FourierBessel transform of the radial function. We show that the decay of the Gegenbauer coefficients is determined by the behavior of the FourierBessel transform at the origin.
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Year of Publication:
20090324
Source:
http://ibb.gsf.de/preprints/2004/pp0403.pdf
http://ibb.gsf.de/preprints/2004/pp0403.pdf
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Document Type:
text
Language:
en
Subjects:
radial basis functions ; zonal basis functions ; positive definite functions ; conditionally positive definite functions ; Bessel functions ; Gegenbauer polynomials
radial basis functions ; zonal basis functions ; positive definite functions ; conditionally positive definite functions ; Bessel functions ; Gegenbauer polynomials
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http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.73.1012
http://ibb.gsf.de/preprints/2004/pp0403.pdf
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.73.1012
http://ibb.gsf.de/preprints/2004/pp0403.pdf
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9.
Strictly positive definite reflection invariant
Open Access
Title:
Strictly positive definite reflection invariant
Author:
Wolfgang Zu Castell
;
Frank Filbir
Wolfgang Zu Castell
;
Frank Filbir
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Description:
functions
functions
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Year of Publication:
20080701
Source:
http://ibb.gsf.de/preprints/2004/pp0428.pdf
http://ibb.gsf.de/preprints/2004/pp0428.pdf
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Document Type:
text
Language:
en
Subjects:
positive definite functions ; spherical functions ; Gelfand pair ; Bochner’s Theorem ; semidirect product ; reflection group ; radial basis
positive definite functions ; spherical functions ; Gelfand pair ; Bochner’s Theorem ; semidirect product ; reflection group ; radial basis
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http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.72.3080
http://ibb.gsf.de/preprints/2004/pp0428.pdf
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10.
Conditionally Positive Definite Kernels and Pontryagin Spaces
Open Access
Title:
Conditionally Positive Definite Kernels and Pontryagin Spaces
Author:
Georg Berschneider
;
Wolfgang Zu Castell
Georg Berschneider
;
Wolfgang Zu Castell
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Description:
Abstract. Conditionally positive definite kernels provide a powerful tool for scattered data approximation. Many nice properties of such methods follow from an underlying reproducing kernel structure. While the connection between positive definite kernels and reproducing kernel Hilbert spaces is well understood, the analog relation between condi...
Abstract. Conditionally positive definite kernels provide a powerful tool for scattered data approximation. Many nice properties of such methods follow from an underlying reproducing kernel structure. While the connection between positive definite kernels and reproducing kernel Hilbert spaces is well understood, the analog relation between conditionally positive definite kernels and reproducing kernel Pontryagin spaces is less known. We want to provide a theoretical framework which allows to study approximation with conditionally positive definite kernels in associated Pontryagin spaces.
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Year of Publication:
20081208
Source:
http://ibb.gsf.de/preprints/2007/pp0716.ps
http://ibb.gsf.de/preprints/2007/pp0716.ps
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http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.68.9430
http://ibb.gsf.de/preprints/2007/pp0716.ps
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(3) dirichlet splines
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(1) Computer science, knowledge & systems [00*]
(1) Engineering [62*]
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