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Title:

Dirichlet splines as fractional integrals of B-splines

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 B-splines. 2000 AMS subj. class.: 41A15, 62H10, 26A33 Keywords: Dirichlet averages, Dirichlet splines, divided dierences, B-splines, Dirichlet distribution, fractional integrals and derivatives 1. Minimize

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The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2009-04-15

Source:

http://www.gsf.de/ibb/preprints/2001/pp01-15.ps

http://www.gsf.de/ibb/preprints/2001/pp01-15.ps Minimize

Document Type:

text

Language:

en

Subjects:

Dirichlet averages ; Dirichlet splines ; divided dierences ; B-splines ; Dirichlet distribution ; fractional integrals and derivatives

Dirichlet averages ; Dirichlet splines ; divided dierences ; B-splines ; Dirichlet distribution ; fractional integrals and derivatives Minimize

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Title:

Generalized Bessel functions for p-radial functions

Description:

Abstract: Suppose that d ∈ N and p> 0. In this paper, we study the generalized Bessel functions for the surface {v ∈ Rd: |v|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 sym...

Abstract: Suppose that d ∈ N and p> 0. In this paper, we study the generalized Bessel functions for the surface {v ∈ Rd: |v|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 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 1-radial functions, and we show how their results may be framed within our more general context. Minimize

Contributors:

The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2008-07-01

Source:

http://ibb.gsf.de/preprints/2006/pp06-01.pdf

http://ibb.gsf.de/preprints/2006/pp06-01.pdf Minimize

Document Type:

text

Language:

en

Subjects:

radial functions ; complete symmetric functions ; monomial symmetric functions ; Fourier-Bessel transform ; divided differences ; characteristic functions ; multivariate symmetric distributions ; reflection symmetry ; finite reflection groups ; Gelfand

radial functions ; complete symmetric functions ; monomial symmetric functions ; Fourier-Bessel transform ; divided differences ; characteristic functions ; multivariate symmetric distributions ; reflection symmetry ; finite reflection groups ; Gelfand Minimize

DDC:

510 Mathematics *(computed)*

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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. Minimize

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Title:

Fractional derivatives and the inverse Fourier transform of `

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Abstract: In [1] and [2] H. Berens, Y. Xu and the author proved that the inverse Fourier integral of ` 1-radial 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 ` 1-radial functions, i.e., functions which are radial w.r.t. the ` Minimize

Contributors:

The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2008-07-01

Source:

http://ibb.gsf.de/preprints/2002/pp02-09.ps

http://ibb.gsf.de/preprints/2002/pp02-09.ps Minimize

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 Minimize

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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. Minimize

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Title:

Generalized Bessel functions for p-radial functions

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 1-radial 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, Fourier-Bessel transform, divided di#erences, characteristic functions, multivariate symmetric distributions, reflection symmetry, finite reflection groups, Gelfand pairs 1 Minimize

Contributors:

The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2009-04-19

Source:

http://ibb.gsf.de/preprints/2006/pp06-01.ps

http://ibb.gsf.de/preprints/2006/pp06-01.ps Minimize

Document Type:

text

Language:

en

Subjects:

radial functions ; complete symmetric functions ; monomial symmetric functions ; Fourier-Bessel transform ; divided differences ; characteristic functions ; multivariate symmetric distributions ; reflection symmetry ; finite reflection groups ; Gelfand

radial functions ; complete symmetric functions ; monomial symmetric functions ; Fourier-Bessel transform ; divided differences ; characteristic functions ; multivariate symmetric distributions ; reflection symmetry ; finite reflection groups ; Gelfand Minimize

DDC:

510 Mathematics *(computed)*

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Title:

Strictly positive definite reflection invariant

Description:

functions

functions Minimize

Contributors:

The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2008-07-01

Source:

http://ibb.gsf.de/preprints/2004/pp04-28.pdf

http://ibb.gsf.de/preprints/2004/pp04-28.pdf Minimize

Document Type:

text

Language:

en

Subjects:

positive definite functions ; spherical functions ; Gelfand pair ; Bochner’s Theorem ; semi-direct product ; reflection group ; radial basis

positive definite functions ; spherical functions ; Gelfand pair ; Bochner’s Theorem ; semi-direct product ; reflection group ; radial basis Minimize

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Title:

Conditionally Positive Definite Kernels and Pontryagin Spaces

Author:

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. Minimize

Contributors:

The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2008-12-08

Source:

http://ibb.gsf.de/preprints/2007/pp07-16.ps

http://ibb.gsf.de/preprints/2007/pp07-16.ps Minimize

Document Type:

text

Language:

en

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Title:

Radial basis functions and corresponding

Description:

zonal series expansions on the sphere

zonal series expansions on the sphere Minimize

Contributors:

The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2008-07-01

Source:

http://ibb.gsf.de/preprints/2004/pp04-03.ps

http://ibb.gsf.de/preprints/2004/pp04-03.ps Minimize

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 Minimize

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Title:

Radial basis functions and corresponding zonal series expansions on the sphere

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 Fourier-Bessel 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 Fourier-Bessel transform of the radial function. We show that the decay of the Gegenbauer coefficients is determined by the behavior of the Fourier-Bessel transform at the origin. Minimize

Contributors:

The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2009-03-24

Source:

http://ibb.gsf.de/preprints/2004/pp04-03.pdf

http://ibb.gsf.de/preprints/2004/pp04-03.pdf Minimize

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 Minimize

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Title:

POLYNOMIAL INTERPOLATION ON THE UNIT SPHERE II

Author:

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. Minimize

Contributors:

The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2008-07-01

Source:

http://ibb.gsf.de/preprints/2004/pp04-16.pdf

http://ibb.gsf.de/preprints/2004/pp04-16.pdf Minimize

Document Type:

text

Language:

en

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Title:

Department for Eco-System and

Author:

Description:

Volume interpolation of CT images from tree trunks

Volume interpolation of CT images from tree trunks Minimize

Contributors:

The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2008-07-01

Source:

http://ibb.gsf.de/preprints/2005/pp05-23.ps

http://ibb.gsf.de/preprints/2005/pp05-23.ps Minimize

Document Type:

text

Language:

en

Subjects:

Radial basis function interpolation

Radial basis function interpolation Minimize

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