Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · math/0612756

On kernel theorems for (LF)-spaces

A. G. Smirnov
View original source ↗Read source document ↗Export metadata ↓

Abstract

A convenient technique for proving kernel theorems for (LF)-spaces (countable inductive limits of Frechet spaces)is developed. The proposed approach is based on introducing a suitable modification of the functor of the completed inductive topological tensor product. Using such modified tensor products makes it possible to prove kernel theorems without assuming the completeness of the considered (LF)-spaces. The general construction is applied to proving kernel theorems for a class of spaces of entire analytic functions arising in nonlocal quantum field theory.

Explore related subjects

math.FAmath-phmath.MP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

A. G. Smirnov. 2006-12-25. On kernel theorems for (LF)-spaces. https://arxiv.org/abs/math/0612756

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

KEEP EXPLORING

Related papers

arXiv2026

On optimal endpoints for integral kernel operators

In this paper, we study the behavior of integral kernel operators acting on functions of a single real variable. In particular, we attempt to find possible candidates for their optimal endpoint spaces in the class of Banach spaces endowed with a rearrangement-invariant quasinorm. To achieve this, we characterize the existence of a concrete pointwise estimate for their nonincreasing rearrangement. We also build a theory of optimal endpoint spaces for the Calderón operator given by this pointwise estimate. We use these results to propose optimal endpoint spaces for some integral kernel operators.

math.FA↗
arXiv2026

Local-to-Global Isometries in Standard-Norm Generalized Gyrovector Spaces: Thompson and Hilbert Geometries of JB-Cones

We introduce standard-norm generalized gyrovector spaces: generalized gyrovector spaces whose norm-value line has the ordinary real operations and whose fixed scalar maps are continuous. We prove that every bijective gyrometric isometry between nonempty connected open subsets of two such spaces extends uniquely to a global bijective gyrometric isometry. The proof combines gyromidpoints, point reflections, continuation of local isometry germs, and monodromy; joint continuity of scalar multiplication is derived, rather than assumed. The class includes all nonzero real normed spaces and standardized Möbius, Einstein, and proper-velocity models, and is stable under finite and countable $\ell^p$-products. We also obtain bounded mapping-space examples and, under natural continuity assumptions, standard-norm function-space GGVs $C(K,G)$ for compact $K$. These yield non-Hilbertian and hybrid examples. For every unital JB-algebra, the positive invertible cone carries a one-parameter family of standard-norm GGV structures whose gyrometrics are conjugate to Thompson's metric by power maps. The projective positive cone has a parallel family, based on the variation norm modulo the unit, conjugate to Hilbert's projective metric. The projective structures are contractible. Together with global JB-cone isometry classifications, the abstract theorem yields local-to-global extension and classification results for Thompson and Hilbert geometries. Full-rank density matrices and normalized state spaces of finite-dimensional Euclidean Jordan algebras, including the Albert algebra, give concrete projective examples.

math.FA↗
arXiv2026

Relevant sampling of non-decaying signals in Orlicz and mixed-norm Orlicz spaces defined on a locally compact group

We study the sampling problem for non-decaying signals defined on a locally compact group. The signals are modeled as elements of suitable subspaces of weighted Orlicz and weighted mixed-norm Orlicz spaces, thereby allowing controlled growth at infinity. Our principal focus is on image spaces of idempotent integral operators. Under suitable oscillation estimates on the associated integral kernels, we establish deterministic sampling theorems that guarantee stable reconstruction from pointwise samples. We further prove an average sampling result in the same setting. In addition, we prove random sampling theorems for functions whose norm is essentially concentrated on a compact subset of the group. In particular, we show that, with high probability, sampling is possible from $\mathcal{O}\!\left(μ(K)\log μ(K)\right)$ randomly chosen sample points, where $K$ denotes the compact set on which the signal, belonging to Orlicz space, is essentially norm concentrated. In the same spirit, a random sampling result is proved for signals in mixed-norm Orlicz space. As applications of the abstract theory, we obtain sampling theorems for weighted shift-invariant spaces and Orlicz modulation spaces.

math.FA↗