Classical-quantum semigroups

We consider approximation methods defined by translates of a positive definite function on a compact group. A characterization of the native space generated by a positive definite function on a compact group is presented. Starting from Bochner's theorem, we construct examples of well-localized positive definite central functions on the rotation group (3). Finally, the stability of the interpolation problem and the error analysis for the given examples are studied in detail.

Let S be a subset of an amenable group G such that e ∈ S and S–1 = S. The main result of this paper states that if the Cayley graph of G with respect to S has a certain combinatorial property, then every positive definite operator-valued function on S can be extended to a positive definite function on G. Several known extension results are obtained as corollaries. New applications are also presented.

Fourier–Stieltjes Algebras of Locally Compact Groupoids

Stability results for approximation by positive definite functions on SO(3)

The main purpose of this paper is an explicit construction of generalized Gaussian process with function tb(V) = bH(V), where H(V) = n - h(V), h(V) is the number of singletons in a pair-partition V ∈ 𝒫2(2n). This gives another proof of Theorem of A. Buchholtz15 that tb is positive definite function on the set of all pair-partitions. Here there are some new combinatorial formulas presented. Connections with free additive convolutions probability measure on ℝ are also done. There are new positive definite functions on permutations presented. What is more, it is proven that the function H is norm (on the group S(∞) = ⋃S(n)).

In this paper we study strictly positive definite functions on the unit sphere of the m m -dimensional Euclidean space. Such functions can be used for solving a scattered data interpolation problem on spheres. Since positive definite functions on the sphere were already characterized by Schoenberg some fifty years ago, the issue here is to determine what kind of positive definite functions are actually strictly positive definite. The study of this problem was initiated recently by Xu and Cheney (Proc. Amer. Math. Soc. 116 (1992), 977–981), where certain sufficient conditions were derived. A new approach, which is based on a critical connection between this problem and that of multivariate polynomial interpolation on spheres, is presented here. The relevant interpolation problem is subsequently analyzed by three different complementary methods. The first is based on the de Boor-Ron general “least solution for the multivariate polynomial interpolation problem”. The second, which is suitable only for m = 2 m=2 , is based on the connection between bivariate harmonic polynomials and univariate analytic polynomials, and reduces the problem to the structure of the integer zeros of bounded univariate exponentials. Finally, the last method invokes the realization of harmonic polynomials as the polynomial kernel of the Laplacian, thereby exploiting some basic relations between homogeneous ideals and their polynomial kernels.

Strictly positive definite functions by integral transforms

A number of consistent models describing stationary positive definite functions on stem from Bochner’s celebrated theorem characterizing continuous and stationary positive definite functions on If denotes the unit sphere in the same is true of positive definite functions on which are radial with respect to the component and stationary with respect to the component. In this paper, we summarize results on these topics, mainly those that somehow characterize the positive definiteness of the function through Fourier transforms of the sections of the function itself. We present a new perspective on the existent results in the literature along with new characterizations and applications.

Introduction. 1.1 Linear Time-Invariant Systems. 1.2 Nonlinear Systems. 1.3 Equilibrium Points. 1.4 First-Order Autonomous Nonlinear Systems. 1.5 Second-Order Systems: Phase-Plane Analysis. 1.6 Phase-Plane Analysis of Linear Time-Invariant Systems. 1.7 Phase-Plane Analysis of Nonlinear Systems. 1.8 Higher-Order Systems. 1.9 Examples of Nonlinear Systems. 1.10 Exercises. Mathematical Preliminaries. 2.1 Sets. 2.2 Metric Spaces. 2.3 Vector Spaces. 2.4 Matrices. 2.5 Basic Topology. 2.6 Sequences. 2.7 Functions. 2.8 Differentiability. 2.9 Lipschitz Continuity. 2.10 Contraction Mapping. 2.11 Solution of Differential Equations. 2.12 Exercises. Lyapunov Stability I: Autonomous Systems. 3.1 Definitions. 3.2 Positive Definite Functions. 3.3 Stability Theorems. 3.4 Examples. 3.5 Asymptotic Stability in the Large. 3.6 Positive Definite Functions Revisited. 3.7 Construction of Lyapunov Functions. 3.8 The Invariance Principle. 3.9 Region of Attraction. 3.10 Analysis of Linear Time-Invariant Systems. 3.11 Instability. 3.12 Exercises. Lyapunov Stability II: Nonautonomous Systems. 4.1 Definitions. 4.2 Positive Definite Functions. 4.3 Stability Theorems. 4.4 Proof of the Stability Theorems. 4.5 Analysis of Linear Time-Varying Systems. 4.6 Perturbation Analysis. 4.7 Converse Theorems. 4.8 Discrete-Time Systems. 4.9 Discretization. 4.10 Stability of Discrete-Time Systems. 4.11 Exercises. Feedback Systems. 5.1 Basic Feedback Stabilization. 5.2 Integrator Backstepping. 5.3 Backstepping: More General Cases. 5.4 Examples. 5.5 Exercises. Input-Output Stability. 6.1 Function Spaces. 6.2 Input-Output Stability. 6.3 Linear Time-Invariant Systems. 6.4 Lp Gains for LTI Systems. 6.5 Closed Loop Input-Output Stability. 6.6 The Small Gain Theorem. 6.7 Loop Transformations. 6.8 The Circle Criterion. 6.9 Exercises. Input-to-State Stability. 7.1 Motivation. 7.2 Definitions. 7.3 Input-to-State Stability (ISS) Theorems. 7.4 Input-to-State Stability Revisited. 7.5 Cascade Connected Systems. 7.6 Exercises. Passivity. 8.1 Power and Energy: Passive Systems. 8.2 Definitions. 8.3 Interconnections of Passivity Systems. 8.4 Stability of Feedback Interconnections. 8.5 Passivity of Linear Time-Invariant Systems. 8.6 Strictly Positive Real Rational Functions. Exercises. Dissipativity. 9.1 Dissipative Systems. 9.2 Differentiable Storage Functions. 9.3 QSR Dissipativity. 9.4 Examples. 9.5 Available Storage. 9.6 Algebraic Condition for Dissipativity. 9.7 Stability of Dissipative Systems. 9.8 Feedback Interconnections. 9.9 Nonlinear L2 Gain. 9.10 Some Remarks about Control Design. 9.11 Nonlinear L2-Gain Control. 9.12 Exercises. Feedback Linearization. 10.1 Mathematical Tools. 10.2 Input-State Linearization. 10.3 Examples. 10.4 Conditions for Input-State Linearization. 10.5 Input-Output Linearization. 10.6 The Zero Dynamics. 10.7 Conditions for Input-Output Linearization. 10.8 Exercises. Nonlinear Observers. 11.1 Observers for Linear Time-Invariant Systems. 11.2 Nonlinear Observability. 11.3 Observers with Linear Error Dynamics. 11.4 Lipschitz Systems. 11.5 Nonlinear Separation Principle. Proofs. Bibliography. List of Figures. Index.

Geodesically isotropic positive definite functions on compact two-point homogeneous spaces of dimension d have series representation as members of weighted Lebesgue spaces L1w([−1,1]), where the weight w(x)=wα,β(x)=(1−x)α(1+x)β is the one related to the Jacobi orthogonal polynomials P(α,β)(x) in [−1,1], and the exponents α and β are related to the dimension d. We derive some recurrence relations among the coefficients of the series representations under different exponents, and we apply them to prove inheritance of positive definiteness between dimensions. Additionally, we give bounds on the curvature at the origin of such positive definite functions with compact support, extending the existing solutions from d-dimensional spheres to compact two-point homogeneous spaces.

Extensions of positive definite functions on free groups

On differentiability and analyticity of positive definite functions

Free products of completely positive maps and spectral sets

Conditionally Positive Definite Functions and Their Application to Multivariate Interpolations

We show that extension problems for generalized Toeplitz kernels may be completely reduced to extension problems for positive definite functions, where the solution is well known.These considerations in particular imply that generalized Toeplitz kernels may be represented as Fourier transforms of positive operator-valued measures.The notion of generalized Toeplitz kernels (g.T.k.) was introduced in [5], where also a generalized Bochner theorem was proved in the discrete case.Since then many papers concerning this topic appeared, for example, [2,3].The extension problem for the discrete case was discussed in [1] and for the continuous case, for example, in [4].The proofs for the extendibility and for the generalized Bochner theorem use mainly lifting theorems for families of operators.In this paper we want to show that these questions can be reduced directly to appropriate questions for positive definite operator-valued functions.Fix Hilbert spaces HX,H2, and let 0 < a < co, 1(a) := (-2a, 2a), Ix(a) := (-2a, 0), /2(a) := (0, 2a).An operator-valued generalized Toeplitz kernel K on 1(a) x 1(a) is an operator-valued function with K(s, t) e L(Ha, HB) if s e Ia(a), t e Ip(a) and if there are four weakly continuous functions KBo: Ia(a)-IB(a)-> L(Ha,Hp), a, 8 = 1,2, such that K(s, t) = KBa(s-1) for 5 e Ia(a), t e IB(a), and Kx2(-t) = K2x(t)* for ?e (0,4a); i.e., Kxx: (-2a, 2a) -L(HX), Kx2: (-4a, 0) -L(H2, //,), K2X: (0, 4a) -L(//,, H2), K22: (-2a, 2a) -L(H2).Afor all pairs of functions tpa: I(a(a) -> Ha, a = 1, 2, with finite support.Recall that an operator-valued function F: (-2a, 2a) -> L(H) with some Hilbert space H is called positive definite if(1) Y (F(s-t)<p(s),4>(t))H>0s,t€{-a,a)