We write S∞ to denote the unit sphere of `, the set of all z in ` for which 〈z, z〉 = 1 (this set is often called the complex Hilbert sphere). A continuous complex-valued function f defined on the closed unit disc D := {ζ ∈ C : |ζ| ≤ 1} is said to be positive definite of order n on S∞ if, for any set of n distinct points z1, . . . , zn on S∞, the n×n matrix A with ij-entry given by f(〈zi, zj〉) i...

We provide a stability and performance analysis for nonlinear model predictive control (NMPC) schemes subject to input constraints. Given an exponential stabilizability detectability condition w.r.t. the employed state cost, we sufficiently long prediction horizon ensure asymptotic desired bound infinite-horizon optimal controller. Compared existing results, provided is applicable positive semi...

This paper is mainly a survey of published results. We recall the definition of positive definite and (conditionally) negative definite functions on abelian semigroups with involution, and we consider three main examples: Rk, [0,∞[k, N0–the first with the inverse involution and the two others with the identical involution. Schoenberg’s theorem explains the possibility of constructing rotation i...

A long-standing open problem asks if there can exist 7 mutually unbiased bases (MUBs) in $$\mathbb {C}^6$$ , or, more generally, $$d + 1$$ MUBs {C}^d$$ for any d that is not a prime power. Recent work of Proc Am Math Soc 146(3), 1143–1150 (2018) proposed an application the method positive definite functions (a relative Delsarte’s coding theory and Lovász’s semidefinite programming relaxation in...