Positive polynomials in scalar and matrix variables, the spectral theorem and optimization

John William Helton; Mihai Putinar

Home/Archive/arXiv

arXiv · math/0612103

Positive polynomials in scalar and matrix variables, the spectral theorem and optimization

John William Helton·Mihai Putinar

View original source ↗Read source document ↗Export metadata ↓

Abstract

We follow a stream of the history of positive matrices and positive functionals, as applied to algebraic sums of squares decompositions, with emphasis on the interaction between classical moment problems, function theory of one or several complex variables and modern operator theory. The second part of the survey focuses on recently discovered connections between real algebraic geometry and optimization as well as polynomials in matrix variables and some control theory problems. These new applications have prompted a series of recent studies devoted to the structure of positivity and convexity in a free *-algebra, the appropriate setting for analyzing inequalities on polynomials having matrix variables. We sketch some of these developments, add to them and comment on the rapidly growing literature.

Explore related subjects

math.FA math.OC

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

John William Helton, Mihai Putinar. 2006-12-04. Positive polynomials in scalar and matrix variables, the spectral theorem and optimization. https://arxiv.org/abs/math/0612103

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Spaces with the maximal projection constant revisited

Let $n \geq 2$ be an integer such that an equiangular set of vectors $w_1, \ldots, w_d$ of the maximal possible cardinality (that is, attaining the classical Gerzon upper bound) exists in $\mathbb{K}^n$, where $\mathbb{K}=\mathbb{R}$ or $\mathbb{K}=\mathbb{C}$ (so that $d=\frac{n(n+1)}{2}$ in the real case and $d=n^2$ in the complex case). We provide a complete characterization of $n$-dimensional normed spaces whose absolute projection constant is maximal among all $n$-dimensional normed spaces over $\mathbb{K}$. The characterization states that $X$ has the maximal projection constant if and only if it is isometric to a space whose dual unit ball is contained between the absolutely convex hull of the vectors $w_1, \ldots, w_d$ and a suitably rescaled zonotope generated by the same vectors. As a consequence, we obtain that, in the considered situations, $n=2$ with $\mathbb{K}=\mathbb{R}$ is the only case in which there is, up to isometry, a unique norm on $\mathbb{K}^n$ with the maximal projection constant. In this case, the unit ball is a linear image of a regular hexagon in $\mathbb{R}^2$.

math.FA↗

arXiv2026

Subdyadic time-frequency analysis: Gabor frames, modulation spaces, and Miyachi multipliers

We present a time-frequency framework adapted to dispersive phase functions via a subdyadic geometry in phase space. On top of this geometry we construct stable frequency-adaptive Gabor-type frames with quantitative control of overlap, almost orthogonality, and off-diagonal decay. Based on these frames we introduce modulation spaces consistent with the subdyadic scale and establish window and lattice independence, identifications in the Hilbertian case, duality, and natural inclusion relations. Within this setting we study high-frequency H"ormander--Miyachi multipliers, relying on discrete block almost diagonalization and direct localization estimates for the primal and canonical dual frames, and obtain boundedness on weighted modulation spaces. Finally, we give a subdyadic Gabor-frame characterization of H"ormander's classical local wavefront set and recover the standard microlocality and ellipticity properties of order-zero pseudodifferential operators. Taken together, these results provide a unified analytical framework for time--frequency analysis, dispersive multiplier theory, and local microlocal analysis in the subdyadic geometry.

math.FA↗

arXiv2026

B-Frames, B-Riesz bases, and Their tensor products

Like g-frames, b-frames were introduced to generalize the concept of frames, allowing for broader applications in signal processing and other fields. The advantage of b-frames resides in their simpler definition, which may lead to reduced processing times. In this paper, we define dual b-frames and b-Riesz bases which were not precisely defined in previous literature and provide several characterizations of b-Riesz bases. We prove that the tensor product of two sequences, each lying in a Hilbert space, constitutes a b-frame (or a b-Riesz basis) if and only if both components of the product are b-frames (or b-Riesz bases). Finally, we establish a correspondence between b-frames and g-frames and propose a process for constructing frames induced by b-frames.

math.FA↗