Criteria of Pólya type for radial positive definite functions

Gneiting, Tilmann

doi:10.1090/s0002-9939-01-05839-7

Cited by 45 publications

(24 citation statements)

References 25 publications

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“…For the analogous problem with the ℓ p norm • p on IR n , see e.g. Gneiting [12] and references therein. Schoenberg [27] also characterized positive definite functions of the form K(x, y) = f ( x − y ) where • is again the Euclidean norm on IR n , but X = S n−1 is the unit sphere in IR n , n ≥ 2.…”

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confidence: 99%

Strictly Hermitian positive definite functions

Pinkus

2004

J. Anal. Math.

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Let H be any complex inner product space with inner product < •, • >.We say that f* ) is Hermitian positive definite for all choice of z 1 , . . . , z n in H, all n. It is strictly Hermitian positive definite if the matrix ( * ) is also non-singular for any choice of distinct z 1 , . . . , z n in H. In this article we prove that if dim H ≥ 3, then f is Hermitian positive definite on H if and only ifwhere b k,ℓ ≥ 0, all k, ℓ in Z Z + , and the series converges for all z in | C. We also prove that f of the form ( * * ) is strictly Hermitian positive definite on any H if and only if the setis such that (0, 0) ∈ J , and every arithmetic sequence in Z Z intersects the values {k − ℓ : (k, ℓ) ∈ J } an infinite number of times.

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confidence: 99%

Strictly Hermitian positive definite functions

Pinkus

2004

J. Anal. Math.

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“…For example in dimension one the celebrated criterion of Pólya [1949] states that if F : [0, +∞[ → ‫ޒ‬ is continuous and convex with F(0) = 1 and lim r →+∞ F(r ) = 0 then f (x) = F(|x|) is positive definite. This criterion was extended to higher dimensions by numerous authors [Askey 1973;Gneiting 2001;Trigub 1989]. We mention only one extension:…”

Section: Maximum Principlementioning

confidence: 99%

Untitled

2011

APDE

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We prove several global existence theorems for spacetimes with toroidal or hyperbolic symmetry with respect to a geometrically defined time. More specifically, we prove that generically, the maximal Cauchy development of T 2-symmetric initial data with positive cosmological constant > 0, in the vacuum or with Vlasov matter, may be covered by a global areal foliation with the area of the symmetry orbits tending to zero in the contracting direction. We then prove the same result for surface symmetric spacetimes in the hyperbolic case with Vlasov matter and ≥ 0. In all cases, there is no restriction on the size of initial data. 199 6. Remarks on the strategy of the proofs 200 7. Proof of Theorem 1 204 8. Proof of Theorem 2 216 9. Proof of Theorem 3 235 10. Comments and open questions 240 Appendix A. Initial data and the constraint equations 242 Appendix B. Surface-symmetric spacetimes in areal coordinates 243 Appendix C. From symmetric initial data to symmetric spacetimes 243 Acknowledgements 244 References 244 1 See [Choquet-Bruhat 1970; 1971] for the case of Vlasov matter. 2 The conjecture was originally developed by R. Penrose [1979] and first formulated as a statement about the global geometry of the maximal Cauchy development in [Moncrief and Eardley 1981]. See also the presentation in [Christodoulou 1999]. 3 The regularity concerns here the degree of differentiability of the possible extensions and gives rise to different versions of the conjecture. For instance, the C 2 formulation of the conjecture is obtained by replacing "regular" with C 2 in the above statement of the conjecture. 4 They also contain the even more special case of polarized T 3-Gowdy spacetimes. 5 Note that the plane symmetric case is a special case of T 3-Gowdy polarized solutions. 9 Compared to the usual metric for these spacetimes, we use the square of the radius function t = r 2 as the time coordinate rather than the radius function r itself. Moreover, we have introduced the functions α and ν by analogy with the T 2 case, so as to ease the application of the method of the T 2 case to this class of spacetimes. See Appendix B for a description of the change of coordinates from the usual parametrization to this one. SYMMETRY ORBITS OF COSMOLOGICAL SPACETIMES WITH TOROIDAL OR HYPERBOLIC SYMMETRY 197 2C. The Einstein-Vlasov system. Apart from the vacuum case, where we will set T µν = 0 in (1), we will couple the Einstein equations to the Vlasov matter model, which we present in this section. Let ᏼ ⊂-ᏹ denote the set of all future-directed timelike vectors of length −1. ᏼ is classically called the mass shell. Let f denote a nonnegative function on the mass shell. The Vlasov equation for f is derived from the condition that f be preserved along geodesics. In coordinates, we therefore have p α ∂ x α f − α βγ p β p γ ∂ p α f = 0, 200 JACQUES SMULEVICI Theorem 3. Let (ᏹ, g, f) be the maximal development of k = −1 surface-symmetric initial data with Vlasov matter and ≥ 0. Suppose that the Vlasov field f does not vanish identically. Then (ᏹ, g) adm...

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“…Fundamental early work on characteristic functions by Bochner [4], Schoenberg [26], and Pólya [24] has spawned a large literature. See [2], [16], [18], [23], [27], and the references therein. The recent monograph [15] includes a useful survey of modern results, together with an extensive bibliography.…”

Section: Radial Characteristic Functionsmentioning

confidence: 99%

“…Images with multiplicative noise are considered in section 14. In applying the MNS algorithm to these images, the choice fork(ξ, η) is the same as in (18), with various values of α, β.…”

Section: Some Examplesmentioning

confidence: 99%

False Characteristic Functions and Other Pathologies in Variational Blind Deconvolution. A Method of Recovery

Carasso¹

2009

SIAM J. Appl. Math.

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Given a blurred image g(x, y), variational blind deconvolution seeks to reconstruct both the unknown blur k(x, y) and the unknown sharp image f (x, y), by minimizing an appropriate cost functional. This paper restricts its attention to a rich and significant class of infinitely divisible isotropic blurs that includes Gaussians, Lorentzians, and other heavy-tailed densities, together with their convolutions. A recently developed highly efficient nonlinear variational approach is found to produce inadmissible reconstructions, consisting of partially deblurred images f † (x, y), associated with physically impossible blurs k † (x, y). Three basic flaws in this variational procedure are identified and shown to be the cause of this phenomenon. A method is then developed that can recover useful information from k † (x, y), by constructing a physically valid rectified blur h # (x, y), based on the low frequency part of k † (x, y). A crucial step involves interpreting h # (x, y) as the p th convolution root of the true blur k(x, y), for some postulated real number p ≥ 2. Deconvolution is performed in slow motion, by solving an associated parabolic pseudo-differential equation backwards in time, with the blurred image g(x, y) as data at t = 1. Behavior of the evolution as t ↓ 0 can be monitored and used to readjust the value of p. Previously developed APEX/SECB methodologies make such ill-posed continuation feasible. This recovery method is found highly effective in several instructive examples involving synthetically blurred images.

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Criteria of Pólya type for radial positive definite functions

Cited by 45 publications

References 25 publications

Strictly Hermitian positive definite functions

Strictly Hermitian positive definite functions

Untitled

False Characteristic Functions and Other Pathologies in Variational Blind Deconvolution. A Method of Recovery

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