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1.
LetH be any complex inner product space with inner product <·,·>. We say thatf: ℂ→ℂ is Hermitian positive definite onH if the matrix
is Hermitian positive definite for all choice ofz
1,…,z
n inH for alln. It is strictly Hermitian positive definite if the matrix (*) is also non-singular for any choice of distinctz
1,…,z
n inH. In this article, we prove that if dimH≥3, thenf is Hermitian positive definite onH if and only if
whereb
k,l
≥0 for allk, l in ℤ, and the series converges for allz in ℂ. We also prove thatf of the form (**) is strictly Hermitian positive definite on anyH if and only if the setJ={(k,l):b
k,l
>0} is such that (0,0)∈J, and every arithmetic sequence in ℤ intersects the values {k−l: (k, l)∈J} an infinite number of times. 相似文献
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2.
In this paper, some necessary and sufficient conditions for the existence of the positive definite solutions for the matrix equation X + A*X?αA = Q with α ∈ (0, ∞) are given. Iterative methods to obtain the positive definite solutions are established and the rates of convergence of the considered methods are obtained. Copyright © 2006 John Wiley & Sons, Ltd. 相似文献
3.
Danping Yang 《Numerical Methods for Partial Differential Equations》2001,17(3):229-249
A miscible displacement of one compressible fluid by another in a porous medium is governed by a nonlinear parabolic system. A new mixed finite element method, in which the mixed element system is symmetric positive definite and the flux equation is separated from pressure equation, is introduced to solve the pressure equation of parabolic type, and a standard Galerkin method is used to treat the convection‐diffusion equation of concentration of one of the fluids. The convergence of the approximate solution with an optimal accuracy in L2‐norm is proved. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 229–249, 2001 相似文献
4.
Compactly supported positive definite radial functions 总被引:3,自引:0,他引:3
Zongmin Wu 《Advances in Computational Mathematics》1995,4(1):283-292
We provide criteria for positive definiteness of radial functions with compact support. Based on these criteria we will produce a series of positive definite and compactly supported radial functions, which will be very useful in applications. The simplest ones arecut-off polynomials, which consist of a single polynomial piece on [0, 1] and vanish on [1, ∞). More precisely, for any given dimensionn and prescribedC k smoothness, there is a function inC k (? n ), which is a positive definite radial function with compact support and is a cut-off polynomial as a function of Euclidean distance. Another example is derived from odd-degreeB-splines. 相似文献
5.
Jinman Kim 《Applicable analysis》2013,92(8):987-1000
We characterize positive definite temperature functions, i.e., positive definite solutions of the heat equation, on the Heisenberg group in terms of the initial values. We also obtain an integral representation for positive definite and U(n)-invariant temperature functions with polynomial growth, where U(n) is the group of all n× n unitary matrices. 相似文献
6.
Positive definite dot product kernels in learning theory 总被引:1,自引:0,他引:1
In the classical support vector machines, linear polynomials corresponding to the reproducing kernel K(x,y)=xy are used. In many models of learning theory, polynomial kernels K(x,y)=l=0Nal(xy)l generating polynomials of degree N, and dot product kernels K(x,y)=l=0+al(xy)l are involved. For corresponding learning algorithms, properties of these kernels need to be understood. In this paper, we consider their positive definiteness. A necessary and sufficient condition for the dot product kernel K to be positive definite is given. Generally, we present a characterization of a function f:RR such that the matrix [f(xixj)]i,j=1m is positive semi-definite for any x1,x2,...,xmRn, n2.
Supported by CERG Grant No. CityU 1144/01P and City University of Hong Kong Grant No. 7001342.AMS subject classification 42A82, 41A05 相似文献
7.
J Kuelbs 《Journal of Mathematical Analysis and Applications》1973,42(2):413-426
If Φ is a positive definite function on a real linear space E of infinite dimension and Φ enjoys certain symmetry conditions we are able to show that Φ is expressible as a certain Laplace-Stieltjes transform. Conversely, if Φ is given by such a transform we can often show that Φ is positive definite on E. In particular, our results apply to the Lp spaces, 0 < p < ∞, as well as to other Orlicz spaces. We also are able to show that the only positive definite continuous sup-norm symmetric functions on C(T), the space of bounded real continuous functions on T, are constants whenever C(T) contains a sequence of functions with sup-norm one and disjoint support. Finally, we apply these ideas to obtain a result on radial exponentially convex functions on a Hilbert space. 相似文献
8.
Carl H FitzGerald Roger A Horn 《Journal of Mathematical Analysis and Applications》1977,61(3):633-642
Let A = (aij) be a real symmetric n × n positive definite matrix with non-negative entries. We show that Aα ≡ (aijα) is positive definite for all real α ? n ? 2. Moreover, the lower bound is sharp. We give related results for pairs of quadratic forms and discuss partial generalizations to the case in which A is a complex Hermitian matrix. 相似文献
9.
《Linear algebra and its applications》2006,412(2-3):154-160
In this paper we consider the positive definite solutions of nonlinear matrix equation X + A⋆X−δA = Q, where δ ∈ (0, 1], which appears for the first time in [S.M. El-Sayed, A.C.M. Ran, On an iteration methods for solving a class of nonlinear matrix equations, SIAM J. Matrix Anal. Appl. 23 (2001) 632–645]. The necessary and sufficient conditions for the existence of a solution are derived. An iterative algorithm for obtaining the positive definite solutions of the equation is discussed. The error estimations are found. 相似文献
10.
Positive definite and semidefinite matrices are characterized in terms of positive definiteness and semidefiniteness on arbitrary closed convex cones in Rn. These results are obtained by generalizing Moreau's polar decomposition to a conjugate decomposition. Some typical results are: The matrix A is positive definite if and only if for some closed convex cone K, A is positive definite on K and (A+AT)?1 exists and is semidefinite on the polar cone K°. The matrix A is positive semidefinite if and only if for some closed convex cone K such that either K is polyhedral or (A+AT)(K) is closed, A is positive semidefinite on both K and the conjugate cone KA={s∣xT(A+ AT)s?0?x∈K}, and (A+AT)x=0 for all x in K such that xTAx=0. 相似文献
11.
In this paper we prove Reade’s result for the positive definite C
1 kernels by using the factorisation method used by Kühn.
Received: 10 April 2007, Revised: 3 December 2007 相似文献
12.
It is known that if A is positive definite Hermitian, then A·A-1⩾I in the positive semidefinite ordering. Our principal new result is a converse to this inequality: under certain weak regularity assumptions about a function F on the positive definite matrices, A·F(A)⩾AF(A) for all positive definite A if and only if F(A) is a positive multiple of A-1. In addition to the inequality A·A-1⩾I, it is known that A·A-1T⩾I and, stronger, that λmin(A·B)⩾λmin(ABT), for A, B positive definite Hermitian. We also show that λmin(A·B)⩾λmin(AB) and note that λmin(AB) and λmin(ABT) can be quite different for A, B positive definite Hermitian. We utilize a simple technique for dealing with the Hadamard product, which relates it to the conventional product and which allows us to give especially simple proofs of the closure of the positive definites under Hadamard multiplication and of the inequalities mentioned. 相似文献
13.
This paper characterizes several classes of conditionally positive definite kernels on a domain Ω of either or . Among the classes is that composed of strictly conditionally positive definite kernels. These kernels are known to be useful in the solution of variational interpolation problems on Ω. Our study covers the case in which Ω is the sphere Sl−1 of or a similar manifold. Among other things, our results imply that the characterization of (strict) conditional positive definiteness on Ω can be obtained from a characterization of (strict) positive definiteness on Ω. The bi-zonal strictly conditionally positive definite kernels on Sl−1, l?3, are described. 相似文献
14.
Jing Cai 《Applied mathematics and computation》2010,217(1):117-4466
Nonlinear matrix equation Xs + A∗X−tA = Q, where A, Q are n × n complex matrices with Q Hermitian positive definite, has widely applied background. In this paper, we consider the Hermitian positive definite solutions of this matrix equation with two cases: s ? 1, 0 < t ? 1 and 0 < s ? 1, t ? 1. We derive necessary conditions and sufficient conditions for the existence of Hermitian positive definite solutions for the matrix equation and obtain some properties of the solutions. We also propose iterative methods for obtaining the extremal Hermitian positive definite solution of the matrix equation. Finally, we give some numerical examples to show the efficiency of the proposed iterative methods. 相似文献
15.
Let Ω ⊂ ℝ
d
be a compact convex set of positive measure. In a recent paper, we established a definiteness theory for cubature formulae
of order two on Ω. Here we study extremal properties of those positive definite formulae that can be generated by a centroidal
Voronoi tessellation of Ω. In this connection we come across a class of operators of the form Ln[f](x): = ?i=1n fi(x)(f(yi) + á?f(yi), x-yi?)L_n[f](\boldsymbol{x}):= \sum_{i=1}^n \phi_i(\boldsymbol{x})(f(\boldsymbol{y}_i) + \langle\nabla f(\boldsymbol{y}_i), \boldsymbol{x}-\boldsymbol{y}_i\rangle), where y1,..., yn\boldsymbol{y}_1,\dots, \boldsymbol{y}_n are distinct points in Ω and {ϕ
1, ..., ϕ
n
} is a partition of unity on Ω. We present best possible pointwise error estimates and describe operators L
n
with a smallest constant in an L
p
error estimate for 1 ≤ p < ∞ . For a generalization, we introduce a new type of Voronoi tessellation in terms of a twice continuously differentiable
and strictly convex function f. It allows us to describe a best operator L
n
for approximating f by L
n
[f] with respect to the L
p
norm. 相似文献
16.
Let A and B be Hermitian matrices, and let c(A, B) = inf{|xH(A + iB)x|:6 = 1}. The eigenvalue problem Ax = λBx is called definite if c(A, B)>0. It is shown that a definite problem has a complete system of eigenvectors and that its eigenvalues are real. Under pertubations of A and B, the eigenvalues behave like the eigenvalues of a Hermitian matrix in the sense that there is a 1-1 pairing of the eigenvalues with the perturbed eigenvalues and a uniform bound for their differences (in this case in the chordal metric). Pertubation bounds are also developed for eigenvectors and eigenspaces. 相似文献
17.
Let Ω⊂Rn be an open, connected subset of Rn, and let F:Ω−Ω→C, where Ω−Ω={x−y:x,y∈Ω}, be a continuous positive definite function. We give necessary and sufficient conditions for F to have an extension to a continuous positive definite function defined on the entire Euclidean space Rn. The conditions are formulated in terms of existence of a unitary representations of Rn whose generators extend a certain system of unbounded Hermitian operators defined on a Hilbert space associated to F. Different positive definite extensions correspond to different unitary representations. 相似文献
18.
Jan Stochel 《Rendiconti del Circolo Matematico di Palermo》1991,40(2):153-176
Smoothness of positive definite functions on multiplicative semigroups of the form ]−1, 1[
d
,d≥1, is characterized in terms of their representing measures. Some extension questions concerning functions of this type are
discussed. 相似文献
19.
Jerzy K. Baksalary 《Linear and Multilinear Algebra》1984,16(1):133-139
The general nonegative definite solution to the matrix equation AXA* = B is established in a form which can be viewed as advantageous over that derived by Khatri and Mitra (1976). The problem of determining an existence criterion and a representation of a positive definite to this equation is considered. 相似文献
20.
Karl-Hermann Neeb 《Monatshefte für Mathematik》1998,126(2):125-160
In this paper we generalize the concept of an infinite positive measure on a -algebra to a vector valued setting, where we consider measures with values in the compactification of a convex coneC which can be described as the set of monoid homomorphisms of the dual coneC
* into [0, ]. Applying these concepts to measures on the dual of a vector space leads to generalizations of Bochner's Theorem to operator valued positive definite functions on locally compact abelian groups and likewise to generalizations of Nussbaum's Theorem on positive definite functions on cones. In the latter case we use the Laplace transform to realize the corresponding Hilbert spaces by holomorphic functions on tube domains. 相似文献