Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2508.18529

Translation--Modulation Identities, Ergodic Log-Products and a Conditional Obstruction for Schwartz Functions

Vignon Oussa
View original source ↗Read source document ↗Export metadata ↓

Abstract

We study very smooth functions on the real line, namely Schwartz functions, that satisfy a finite identity relating their translates and a single modulation. Concretely, we assume there is a nontrivial linear combination of translates of the function that equals a fixed frequency shift of the same function. Passing to the Fourier transform turns this into a multiplicative transfer relation: the value of the Fourier transform at one point is obtained by multiplying its value at another point by a trigonometric polynomial. Iterating this relation expresses the Fourier transform along an arithmetic progression as a product of such trigonometric factors times a fixed initial value. We then recast this product in an ergodic theoretic framework by viewing it as a Birkhoff sum for a continuous observable on a compact abelian group generated by a diagonal unitary matrix. The key quantity controlling the growth or decay of the Fourier transform along the progression is a space average, namely the integral over the compact group of the logarithm of the absolute value of the associated trigonometric polynomial. The main rigorously proved statement is the following. If, for some frequency where the Fourier transform does not vanish along the entire progression, this space average is nonzero, or if the average is zero but a certain recurrence set, in the sense of Atkinson's theorem, intersects a distinguished one-parameter subgroup, then the assumed translation--modulation identity forces exponential growth of the Fourier transform along some sequence of points. This contradicts the rapid decay required of a Schwartz function.

Explore related subjects

math.FAmath.RT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Vignon Oussa. 2025-12-13. Translation--Modulation Identities, Ergodic Log-Products and a Conditional Obstruction for Schwartz Functions. https://arxiv.org/abs/2508.18529

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

KEEP EXPLORING

Related papers

arXiv2026

The Weyl-von Neumann theorem for antilinear skew-self-adjoint operators

In this article, we prove the Weyl--von Neumann theorem for bounded antilinear skew-self-adjoint operators. More specifically, we prove the following: Let $A$ be a bounded antilinear skew-self-adjoint operator on a separable Hilbert space $H$ whose kernel is either even dimensional or infinite dimensional. Let $1 0$ there exists an antilinear block skew-diagonal operator $D$ and an antilinear skew-self-adjoint Schatten $p$-class operator $K$ such that $A=K+D$ with $\|K\|_{p}<ε$. As a consequence, we prove the Weyl--von Neumann theorem for complex skew-symmetric operators: Let $τ$ be a conjugation on $H$ and let $T$ be a bounded linear operator $τ$-skew-symmetric with $\dim N(T)=\infty$ or $\dim N(T)$ is even. Let $1 0$, there exists a $τ$-skew-symmetric Schatten $p$-class operator $K$, a skew-symmetric block diagonal operator $D$ and a unitary operator $U$ such that $T=K+UDU^{tr}$ and $\|K\|_{p}<ε$, where $U^{tr}$ is the transpose of $U$ with respect to an orthonormal basis ${\{e_n:n\in \mathbb N}\}$ such that $τ(e_n)=e_n$ for each $n\in \mathbb N$. Furthermore, the above result holds even without any assumption on the dimension of $N(T)$, provided that $N(T)=N(T^*)$.

math.FA↗
arXiv2026

Schur bounded patterns, submajorisation and operator Lipschitz functions

A Schur bounded pattern is a subset $S\subset \mathbb{N}^2$ such that Schur multiplication by every bounded function on $\mathbb{N}^2$ supported on $S$ defines a bounded linear operator in the norm of $\mathcal{B}(\ell_2(\mathbb{N})).$ Schur bounded patterns were characterised by Davidson-Donsig as being unions of row-bounded and column-bounded sets. We study the analogous question for sets $S$ such that element-wise multiplication by every bounded function on $S$ is bounded in ideals of compact operators that are not closed under submajorisation, in particular the Schatten ideals $\mathcal{L}_p$ with $0<p<1$ and the weak Schatten ideal $\mathcal{L}_{1,\infty}.$ Conversely we characterise the ideals that are not closed under submajorisation by their Schur bounded patterns. This has implications for the functions which are Lipschitz in the norm of ideals that are not closed under submajorisation. In particular such functions must be differentiable and have derivative that is asymptotically constant at infinity.

math.FA↗
arXiv2026

Derivatives of Tensor Products and Applications to Spaces $C(K^n,X)$

In this paper, we develop an abstract theory of derivatives for Banach spaces based on objects that we call \emph{bidual assignments}. This framework encompasses both the Semadeni derivative and the recently introduced Semadeni--Pełczyński derivative. More generally, suitable ideals of subsets of dual spaces give rise to a broad family of derivatives within this setting. We establish direct-sum and tensor-product formulas for these derivatives, showing that they behave naturally with respect to direct sums and injective tensor products. We then obtain explicit descriptions of derivatives associated with compact trees and finite products of compact lines. In particular, we compute iterated derivatives and use them to derive isomorphic invariants for vector-valued spaces of continuous functions. As one consequence, if $K=\prod_{i=1}^n K_i$ and $L=\prod_{j=1}^m L_j$, where $n,m\geq1$ and all the factors are compact lines of uncountable character, then, for $1\leq p,q<\infty$, \[ C(K,\ell_p)\sim C(L,\ell_q) \] implies that $n=m$ and $p=q$. We also establish classification results for spaces of the form $C(K^n,X)$. In particular, for uncountable ordinals $α$ and $β$, an integer $n\geq1$, and Banach spaces $X$ satisfying suitable rigidity assumptions, we prove that \[ C([0,α]^n,X)\sim C([0,β]^n,X) \quad\text{if and only if}\quad C([0,α])\sim C([0,β]). \] This extends Kislyakov's classification of the spaces $C([0,α])$ and its vector-valued extension due to Galego to finite powers of ordinal intervals.

math.FA↗