Generalized Bochner’s Theorem for radial function
Abstract A radial function Φ(x) can be expressed by its generator ϕ(·) through Φ(x)=ϕ(‖x‖). The positive de finite of the function Φ plays an important role in the radial basis interpolation. We can naturally use Bochner’s Theorem to check if Φ is positive de finite. This requires however a n-dimens...
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Gespeichert in:
| Autor*in: |
Zongmin, Wu [verfasserIn] |
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| Format: |
Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
1997 |
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| Schlagwörter: |
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| Anmerkung: |
© Springer 1997 |
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| Übergeordnetes Werk: |
Enthalten in: Approximation theory and its applications - Editorial Board of Analysis in Theory and Applications, 1984, 13(1997), 3 vom: Sept., Seite 47-57 |
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| Übergeordnetes Werk: |
volume:13 ; year:1997 ; number:3 ; month:09 ; pages:47-57 |
| Links: |
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| DOI / URN: |
10.1007/BF02837010 |
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| Katalog-ID: |
OLC2111342270 |
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| 520 | |a Abstract A radial function Φ(x) can be expressed by its generator ϕ(·) through Φ(x)=ϕ(‖x‖). The positive de finite of the function Φ plays an important role in the radial basis interpolation. We can naturally use Bochner’s Theorem to check if Φ is positive de finite. This requires however a n-dimensional Fourier transformation and it is not very easy to calculate. Furthermore in a lot of cases we will use ϕ for spaces of various dimensions too, then for every fixed n we need do the Fourier transformation once to check if the function is positive definite in the n-dimensional space. The completely monotone function, which is discussed in [4], is positive definite for arbitrary space dimensions. With this technique we can very easily characterize the positive definite of a radial function through its generator. Unfortunately there is only a very small subset of radial function which is completely monotone. Thus this criterion excluded a lot of interesting functions such as compactly supported radial function, which are very use ful in application. Can we find some conditions (as the completely monotone function) only for the 1-dimensional Fourier transform of the generator ϕ to characterize a radial function Φ, which is positive definite in n-dimensional (fixed n) space? In this paper we defined a kind of incompletely monotone function of order α, for α=0, 1/2, 1, 3/2, 2, … (we denote the function class by ICM), in this sence a normal positvie function is in $ ICM_{0} $; a positive monotone decreasing function is in $ ICM_{1} $ and a positive monotone decreasing and convex function is in $ ICM_{2} $. Based on this definition we get a generalized Bochner’s Theorem for radial function: If1-dimensional Fourier transform of the generator of a radial function can be written as$$F{}_1\varphi (t) = \tilde F(\frac{{t^2 }}{2})$$, then corresponding radial function Φ(x) is positive definite as a n-variate function iff$$\tilde F$$ is an incompletely monotone function of order α=(n-1)/2 (or simply$$\tilde F \in ICM_{\frac{{n - 1}}{2}} $$. | ||
| 650 | 4 | |a Radial Basis Function | |
| 650 | 4 | |a Monotone Function | |
| 650 | 4 | |a Radial Function | |
| 650 | 4 | |a Positive Definite Function | |
| 650 | 4 | |a Radial Basis Function Interpolation | |
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10.1007/BF02837010 doi (DE-627)OLC2111342270 (DE-He213)BF02837010-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Zongmin, Wu verfasserin aut Generalized Bochner’s Theorem for radial function 1997 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer 1997 Abstract A radial function Φ(x) can be expressed by its generator ϕ(·) through Φ(x)=ϕ(‖x‖). The positive de finite of the function Φ plays an important role in the radial basis interpolation. We can naturally use Bochner’s Theorem to check if Φ is positive de finite. This requires however a n-dimensional Fourier transformation and it is not very easy to calculate. Furthermore in a lot of cases we will use ϕ for spaces of various dimensions too, then for every fixed n we need do the Fourier transformation once to check if the function is positive definite in the n-dimensional space. The completely monotone function, which is discussed in [4], is positive definite for arbitrary space dimensions. With this technique we can very easily characterize the positive definite of a radial function through its generator. Unfortunately there is only a very small subset of radial function which is completely monotone. Thus this criterion excluded a lot of interesting functions such as compactly supported radial function, which are very use ful in application. Can we find some conditions (as the completely monotone function) only for the 1-dimensional Fourier transform of the generator ϕ to characterize a radial function Φ, which is positive definite in n-dimensional (fixed n) space? In this paper we defined a kind of incompletely monotone function of order α, for α=0, 1/2, 1, 3/2, 2, … (we denote the function class by ICM), in this sence a normal positvie function is in $ ICM_{0} $; a positive monotone decreasing function is in $ ICM_{1} $ and a positive monotone decreasing and convex function is in $ ICM_{2} $. Based on this definition we get a generalized Bochner’s Theorem for radial function: If1-dimensional Fourier transform of the generator of a radial function can be written as$$F{}_1\varphi (t) = \tilde F(\frac{{t^2 }}{2})$$, then corresponding radial function Φ(x) is positive definite as a n-variate function iff$$\tilde F$$ is an incompletely monotone function of order α=(n-1)/2 (or simply$$\tilde F \in ICM_{\frac{{n - 1}}{2}} $$. Radial Basis Function Monotone Function Radial Function Positive Definite Function Radial Basis Function Interpolation Enthalten in Approximation theory and its applications Editorial Board of Analysis in Theory and Applications, 1984 13(1997), 3 vom: Sept., Seite 47-57 (DE-627)130631124 (DE-600)798435-2 (DE-576)016136691 1000-9221 nnns volume:13 year:1997 number:3 month:09 pages:47-57 https://doi.org/10.1007/BF02837010 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_20 GBV_ILN_22 GBV_ILN_32 GBV_ILN_40 GBV_ILN_70 GBV_ILN_2088 GBV_ILN_4314 GBV_ILN_4316 GBV_ILN_4318 GBV_ILN_4325 AR 13 1997 3 09 47-57 |
| spelling |
10.1007/BF02837010 doi (DE-627)OLC2111342270 (DE-He213)BF02837010-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Zongmin, Wu verfasserin aut Generalized Bochner’s Theorem for radial function 1997 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer 1997 Abstract A radial function Φ(x) can be expressed by its generator ϕ(·) through Φ(x)=ϕ(‖x‖). The positive de finite of the function Φ plays an important role in the radial basis interpolation. We can naturally use Bochner’s Theorem to check if Φ is positive de finite. This requires however a n-dimensional Fourier transformation and it is not very easy to calculate. Furthermore in a lot of cases we will use ϕ for spaces of various dimensions too, then for every fixed n we need do the Fourier transformation once to check if the function is positive definite in the n-dimensional space. The completely monotone function, which is discussed in [4], is positive definite for arbitrary space dimensions. With this technique we can very easily characterize the positive definite of a radial function through its generator. Unfortunately there is only a very small subset of radial function which is completely monotone. Thus this criterion excluded a lot of interesting functions such as compactly supported radial function, which are very use ful in application. Can we find some conditions (as the completely monotone function) only for the 1-dimensional Fourier transform of the generator ϕ to characterize a radial function Φ, which is positive definite in n-dimensional (fixed n) space? In this paper we defined a kind of incompletely monotone function of order α, for α=0, 1/2, 1, 3/2, 2, … (we denote the function class by ICM), in this sence a normal positvie function is in $ ICM_{0} $; a positive monotone decreasing function is in $ ICM_{1} $ and a positive monotone decreasing and convex function is in $ ICM_{2} $. Based on this definition we get a generalized Bochner’s Theorem for radial function: If1-dimensional Fourier transform of the generator of a radial function can be written as$$F{}_1\varphi (t) = \tilde F(\frac{{t^2 }}{2})$$, then corresponding radial function Φ(x) is positive definite as a n-variate function iff$$\tilde F$$ is an incompletely monotone function of order α=(n-1)/2 (or simply$$\tilde F \in ICM_{\frac{{n - 1}}{2}} $$. Radial Basis Function Monotone Function Radial Function Positive Definite Function Radial Basis Function Interpolation Enthalten in Approximation theory and its applications Editorial Board of Analysis in Theory and Applications, 1984 13(1997), 3 vom: Sept., Seite 47-57 (DE-627)130631124 (DE-600)798435-2 (DE-576)016136691 1000-9221 nnns volume:13 year:1997 number:3 month:09 pages:47-57 https://doi.org/10.1007/BF02837010 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_20 GBV_ILN_22 GBV_ILN_32 GBV_ILN_40 GBV_ILN_70 GBV_ILN_2088 GBV_ILN_4314 GBV_ILN_4316 GBV_ILN_4318 GBV_ILN_4325 AR 13 1997 3 09 47-57 |
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10.1007/BF02837010 doi (DE-627)OLC2111342270 (DE-He213)BF02837010-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Zongmin, Wu verfasserin aut Generalized Bochner’s Theorem for radial function 1997 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer 1997 Abstract A radial function Φ(x) can be expressed by its generator ϕ(·) through Φ(x)=ϕ(‖x‖). The positive de finite of the function Φ plays an important role in the radial basis interpolation. We can naturally use Bochner’s Theorem to check if Φ is positive de finite. This requires however a n-dimensional Fourier transformation and it is not very easy to calculate. Furthermore in a lot of cases we will use ϕ for spaces of various dimensions too, then for every fixed n we need do the Fourier transformation once to check if the function is positive definite in the n-dimensional space. The completely monotone function, which is discussed in [4], is positive definite for arbitrary space dimensions. With this technique we can very easily characterize the positive definite of a radial function through its generator. Unfortunately there is only a very small subset of radial function which is completely monotone. Thus this criterion excluded a lot of interesting functions such as compactly supported radial function, which are very use ful in application. Can we find some conditions (as the completely monotone function) only for the 1-dimensional Fourier transform of the generator ϕ to characterize a radial function Φ, which is positive definite in n-dimensional (fixed n) space? In this paper we defined a kind of incompletely monotone function of order α, for α=0, 1/2, 1, 3/2, 2, … (we denote the function class by ICM), in this sence a normal positvie function is in $ ICM_{0} $; a positive monotone decreasing function is in $ ICM_{1} $ and a positive monotone decreasing and convex function is in $ ICM_{2} $. Based on this definition we get a generalized Bochner’s Theorem for radial function: If1-dimensional Fourier transform of the generator of a radial function can be written as$$F{}_1\varphi (t) = \tilde F(\frac{{t^2 }}{2})$$, then corresponding radial function Φ(x) is positive definite as a n-variate function iff$$\tilde F$$ is an incompletely monotone function of order α=(n-1)/2 (or simply$$\tilde F \in ICM_{\frac{{n - 1}}{2}} $$. Radial Basis Function Monotone Function Radial Function Positive Definite Function Radial Basis Function Interpolation Enthalten in Approximation theory and its applications Editorial Board of Analysis in Theory and Applications, 1984 13(1997), 3 vom: Sept., Seite 47-57 (DE-627)130631124 (DE-600)798435-2 (DE-576)016136691 1000-9221 nnns volume:13 year:1997 number:3 month:09 pages:47-57 https://doi.org/10.1007/BF02837010 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_20 GBV_ILN_22 GBV_ILN_32 GBV_ILN_40 GBV_ILN_70 GBV_ILN_2088 GBV_ILN_4314 GBV_ILN_4316 GBV_ILN_4318 GBV_ILN_4325 AR 13 1997 3 09 47-57 |
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10.1007/BF02837010 doi (DE-627)OLC2111342270 (DE-He213)BF02837010-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Zongmin, Wu verfasserin aut Generalized Bochner’s Theorem for radial function 1997 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer 1997 Abstract A radial function Φ(x) can be expressed by its generator ϕ(·) through Φ(x)=ϕ(‖x‖). The positive de finite of the function Φ plays an important role in the radial basis interpolation. We can naturally use Bochner’s Theorem to check if Φ is positive de finite. This requires however a n-dimensional Fourier transformation and it is not very easy to calculate. Furthermore in a lot of cases we will use ϕ for spaces of various dimensions too, then for every fixed n we need do the Fourier transformation once to check if the function is positive definite in the n-dimensional space. The completely monotone function, which is discussed in [4], is positive definite for arbitrary space dimensions. With this technique we can very easily characterize the positive definite of a radial function through its generator. Unfortunately there is only a very small subset of radial function which is completely monotone. Thus this criterion excluded a lot of interesting functions such as compactly supported radial function, which are very use ful in application. Can we find some conditions (as the completely monotone function) only for the 1-dimensional Fourier transform of the generator ϕ to characterize a radial function Φ, which is positive definite in n-dimensional (fixed n) space? In this paper we defined a kind of incompletely monotone function of order α, for α=0, 1/2, 1, 3/2, 2, … (we denote the function class by ICM), in this sence a normal positvie function is in $ ICM_{0} $; a positive monotone decreasing function is in $ ICM_{1} $ and a positive monotone decreasing and convex function is in $ ICM_{2} $. Based on this definition we get a generalized Bochner’s Theorem for radial function: If1-dimensional Fourier transform of the generator of a radial function can be written as$$F{}_1\varphi (t) = \tilde F(\frac{{t^2 }}{2})$$, then corresponding radial function Φ(x) is positive definite as a n-variate function iff$$\tilde F$$ is an incompletely monotone function of order α=(n-1)/2 (or simply$$\tilde F \in ICM_{\frac{{n - 1}}{2}} $$. Radial Basis Function Monotone Function Radial Function Positive Definite Function Radial Basis Function Interpolation Enthalten in Approximation theory and its applications Editorial Board of Analysis in Theory and Applications, 1984 13(1997), 3 vom: Sept., Seite 47-57 (DE-627)130631124 (DE-600)798435-2 (DE-576)016136691 1000-9221 nnns volume:13 year:1997 number:3 month:09 pages:47-57 https://doi.org/10.1007/BF02837010 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_20 GBV_ILN_22 GBV_ILN_32 GBV_ILN_40 GBV_ILN_70 GBV_ILN_2088 GBV_ILN_4314 GBV_ILN_4316 GBV_ILN_4318 GBV_ILN_4325 AR 13 1997 3 09 47-57 |
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10.1007/BF02837010 doi (DE-627)OLC2111342270 (DE-He213)BF02837010-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Zongmin, Wu verfasserin aut Generalized Bochner’s Theorem for radial function 1997 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer 1997 Abstract A radial function Φ(x) can be expressed by its generator ϕ(·) through Φ(x)=ϕ(‖x‖). The positive de finite of the function Φ plays an important role in the radial basis interpolation. We can naturally use Bochner’s Theorem to check if Φ is positive de finite. This requires however a n-dimensional Fourier transformation and it is not very easy to calculate. Furthermore in a lot of cases we will use ϕ for spaces of various dimensions too, then for every fixed n we need do the Fourier transformation once to check if the function is positive definite in the n-dimensional space. The completely monotone function, which is discussed in [4], is positive definite for arbitrary space dimensions. With this technique we can very easily characterize the positive definite of a radial function through its generator. Unfortunately there is only a very small subset of radial function which is completely monotone. Thus this criterion excluded a lot of interesting functions such as compactly supported radial function, which are very use ful in application. Can we find some conditions (as the completely monotone function) only for the 1-dimensional Fourier transform of the generator ϕ to characterize a radial function Φ, which is positive definite in n-dimensional (fixed n) space? In this paper we defined a kind of incompletely monotone function of order α, for α=0, 1/2, 1, 3/2, 2, … (we denote the function class by ICM), in this sence a normal positvie function is in $ ICM_{0} $; a positive monotone decreasing function is in $ ICM_{1} $ and a positive monotone decreasing and convex function is in $ ICM_{2} $. Based on this definition we get a generalized Bochner’s Theorem for radial function: If1-dimensional Fourier transform of the generator of a radial function can be written as$$F{}_1\varphi (t) = \tilde F(\frac{{t^2 }}{2})$$, then corresponding radial function Φ(x) is positive definite as a n-variate function iff$$\tilde F$$ is an incompletely monotone function of order α=(n-1)/2 (or simply$$\tilde F \in ICM_{\frac{{n - 1}}{2}} $$. Radial Basis Function Monotone Function Radial Function Positive Definite Function Radial Basis Function Interpolation Enthalten in Approximation theory and its applications Editorial Board of Analysis in Theory and Applications, 1984 13(1997), 3 vom: Sept., Seite 47-57 (DE-627)130631124 (DE-600)798435-2 (DE-576)016136691 1000-9221 nnns volume:13 year:1997 number:3 month:09 pages:47-57 https://doi.org/10.1007/BF02837010 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_20 GBV_ILN_22 GBV_ILN_32 GBV_ILN_40 GBV_ILN_70 GBV_ILN_2088 GBV_ILN_4314 GBV_ILN_4316 GBV_ILN_4318 GBV_ILN_4325 AR 13 1997 3 09 47-57 |
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Enthalten in Approximation theory and its applications 13(1997), 3 vom: Sept., Seite 47-57 volume:13 year:1997 number:3 month:09 pages:47-57 |
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The positive de finite of the function Φ plays an important role in the radial basis interpolation. We can naturally use Bochner’s Theorem to check if Φ is positive de finite. This requires however a n-dimensional Fourier transformation and it is not very easy to calculate. Furthermore in a lot of cases we will use ϕ for spaces of various dimensions too, then for every fixed n we need do the Fourier transformation once to check if the function is positive definite in the n-dimensional space. The completely monotone function, which is discussed in [4], is positive definite for arbitrary space dimensions. With this technique we can very easily characterize the positive definite of a radial function through its generator. Unfortunately there is only a very small subset of radial function which is completely monotone. Thus this criterion excluded a lot of interesting functions such as compactly supported radial function, which are very use ful in application. Can we find some conditions (as the completely monotone function) only for the 1-dimensional Fourier transform of the generator ϕ to characterize a radial function Φ, which is positive definite in n-dimensional (fixed n) space? In this paper we defined a kind of incompletely monotone function of order α, for α=0, 1/2, 1, 3/2, 2, … (we denote the function class by ICM), in this sence a normal positvie function is in $ ICM_{0} $; a positive monotone decreasing function is in $ ICM_{1} $ and a positive monotone decreasing and convex function is in $ ICM_{2} $. 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generalized bochner’s theorem for radial function |
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Generalized Bochner’s Theorem for radial function |
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Abstract A radial function Φ(x) can be expressed by its generator ϕ(·) through Φ(x)=ϕ(‖x‖). The positive de finite of the function Φ plays an important role in the radial basis interpolation. We can naturally use Bochner’s Theorem to check if Φ is positive de finite. This requires however a n-dimensional Fourier transformation and it is not very easy to calculate. Furthermore in a lot of cases we will use ϕ for spaces of various dimensions too, then for every fixed n we need do the Fourier transformation once to check if the function is positive definite in the n-dimensional space. The completely monotone function, which is discussed in [4], is positive definite for arbitrary space dimensions. With this technique we can very easily characterize the positive definite of a radial function through its generator. Unfortunately there is only a very small subset of radial function which is completely monotone. Thus this criterion excluded a lot of interesting functions such as compactly supported radial function, which are very use ful in application. Can we find some conditions (as the completely monotone function) only for the 1-dimensional Fourier transform of the generator ϕ to characterize a radial function Φ, which is positive definite in n-dimensional (fixed n) space? In this paper we defined a kind of incompletely monotone function of order α, for α=0, 1/2, 1, 3/2, 2, … (we denote the function class by ICM), in this sence a normal positvie function is in $ ICM_{0} $; a positive monotone decreasing function is in $ ICM_{1} $ and a positive monotone decreasing and convex function is in $ ICM_{2} $. Based on this definition we get a generalized Bochner’s Theorem for radial function: If1-dimensional Fourier transform of the generator of a radial function can be written as$$F{}_1\varphi (t) = \tilde F(\frac{{t^2 }}{2})$$, then corresponding radial function Φ(x) is positive definite as a n-variate function iff$$\tilde F$$ is an incompletely monotone function of order α=(n-1)/2 (or simply$$\tilde F \in ICM_{\frac{{n - 1}}{2}} $$. © Springer 1997 |
| abstractGer |
Abstract A radial function Φ(x) can be expressed by its generator ϕ(·) through Φ(x)=ϕ(‖x‖). The positive de finite of the function Φ plays an important role in the radial basis interpolation. We can naturally use Bochner’s Theorem to check if Φ is positive de finite. This requires however a n-dimensional Fourier transformation and it is not very easy to calculate. Furthermore in a lot of cases we will use ϕ for spaces of various dimensions too, then for every fixed n we need do the Fourier transformation once to check if the function is positive definite in the n-dimensional space. The completely monotone function, which is discussed in [4], is positive definite for arbitrary space dimensions. With this technique we can very easily characterize the positive definite of a radial function through its generator. Unfortunately there is only a very small subset of radial function which is completely monotone. Thus this criterion excluded a lot of interesting functions such as compactly supported radial function, which are very use ful in application. Can we find some conditions (as the completely monotone function) only for the 1-dimensional Fourier transform of the generator ϕ to characterize a radial function Φ, which is positive definite in n-dimensional (fixed n) space? In this paper we defined a kind of incompletely monotone function of order α, for α=0, 1/2, 1, 3/2, 2, … (we denote the function class by ICM), in this sence a normal positvie function is in $ ICM_{0} $; a positive monotone decreasing function is in $ ICM_{1} $ and a positive monotone decreasing and convex function is in $ ICM_{2} $. Based on this definition we get a generalized Bochner’s Theorem for radial function: If1-dimensional Fourier transform of the generator of a radial function can be written as$$F{}_1\varphi (t) = \tilde F(\frac{{t^2 }}{2})$$, then corresponding radial function Φ(x) is positive definite as a n-variate function iff$$\tilde F$$ is an incompletely monotone function of order α=(n-1)/2 (or simply$$\tilde F \in ICM_{\frac{{n - 1}}{2}} $$. © Springer 1997 |
| abstract_unstemmed |
Abstract A radial function Φ(x) can be expressed by its generator ϕ(·) through Φ(x)=ϕ(‖x‖). The positive de finite of the function Φ plays an important role in the radial basis interpolation. We can naturally use Bochner’s Theorem to check if Φ is positive de finite. This requires however a n-dimensional Fourier transformation and it is not very easy to calculate. Furthermore in a lot of cases we will use ϕ for spaces of various dimensions too, then for every fixed n we need do the Fourier transformation once to check if the function is positive definite in the n-dimensional space. The completely monotone function, which is discussed in [4], is positive definite for arbitrary space dimensions. With this technique we can very easily characterize the positive definite of a radial function through its generator. Unfortunately there is only a very small subset of radial function which is completely monotone. Thus this criterion excluded a lot of interesting functions such as compactly supported radial function, which are very use ful in application. Can we find some conditions (as the completely monotone function) only for the 1-dimensional Fourier transform of the generator ϕ to characterize a radial function Φ, which is positive definite in n-dimensional (fixed n) space? In this paper we defined a kind of incompletely monotone function of order α, for α=0, 1/2, 1, 3/2, 2, … (we denote the function class by ICM), in this sence a normal positvie function is in $ ICM_{0} $; a positive monotone decreasing function is in $ ICM_{1} $ and a positive monotone decreasing and convex function is in $ ICM_{2} $. Based on this definition we get a generalized Bochner’s Theorem for radial function: If1-dimensional Fourier transform of the generator of a radial function can be written as$$F{}_1\varphi (t) = \tilde F(\frac{{t^2 }}{2})$$, then corresponding radial function Φ(x) is positive definite as a n-variate function iff$$\tilde F$$ is an incompletely monotone function of order α=(n-1)/2 (or simply$$\tilde F \in ICM_{\frac{{n - 1}}{2}} $$. © Springer 1997 |
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