Natural derivability of Axiom IV, subgroup restriction, and hidden invariant sets in Weyl's 1949 axiomatic theory of almost periodic vectors

In his 1949 paper 'Almost Periodic Invariant Vector Sets in a Metric Vector Space', Hermann Weyl founded an axiomatic theory of almost periodic vectors: a vector space carrying a Hermitian scalar product together with a second, stronger 'length', acted on isometrically by an arbitrary group, in which the translation-invariant mean of classical almost-periodicity theory is replaced by a system of axioms (I–IV) for a vector-valued mean-value integral. The additional Axiom IV, a Cauchy–Schwarz-type bound for this integral, is precisely what upgrades the weak Fourier approximation of an almost periodic vector — obtained by an Erhard Schmidt orthogonalization from the Hermitian kernel $(sf,tf)$ — to strong approximation in the length. The paper leaves open a three-part question: (a) find simpler or more natural assumptions that imply Axiom IV, or determine conditions under which it holds; (b) determine under what circumstances Axiom IV and the strong-approximation conclusion persist when the acting group is restricted to a subgroup; and (c) clarify the structure and role of the 'hidden' irreducible invariant sets — those not selected by the given almost periodic vector, which do not enter its Parseval equation.

Background

A continuous complex-valued function $f$ on the real line is called almost periodic (in the sense of Harald Bohr) if its translates form a relatively compact set in the topology of uniform convergence, or equivalently if for every $\varepsilon>0$ the set of $\varepsilon$ -almost periods — numbers $\tau$ with $\sup_x|f(x+\tau)-f(x)|\le\varepsilon$ — is relatively dense in the line. The classical theory rests on two pillars: the existence of a translation-invariant mean value, which plays the role of an integral, and the approximation theorem, which states that every almost periodic function is a uniform limit of trigonometric polynomials (finite sums of exponentials). For a compact group the corresponding statement is the Peter–Weyl theorem, i.e. the density of the matrix entries of the irreducible continuous unitary representations in the space of continuous functions. Bochner and von Neumann extended the theory to functions with values in Banach spaces. In his 1949 paper — the source of the present problem — Hermann Weyl asked what survives of these pillars when the scalar function is replaced by a vector in an abstract space on which an arbitrary group acts, so that no group integral and no topology on the group is available.

The setting is a complex vector space $\Sigma$ carrying two structures at once: a Hermitian scalar product $(f,g)$ with Hilbert norm $\|f\|=(f,f)^{1/2}$ , and a "length" $|f|$ , a second metric on vectors satisfying $\|f\|\le|f|$ , so that convergence in length is the finer of the two notions. A group $\sigma$ acts on $\Sigma$ by linear transformations that are isometric for both structures, and $\Sigma$ is closed under convergence in the length. A vector $f\in\Sigma$ is called almost periodic if for every $\varepsilon>0$ the group $\sigma$ can be covered by finitely many $f$ -circles — sets of the form $\{s\in\sigma:|sf-af|\le\varepsilon\}$ with center $a\in\sigma$ , where $sf$ denotes the action of the group element $s$ on the vector $f$ . Since no group integral is available, Weyl postulates the mean value axiomatically: Axioms I–III secure an invariant mean $\int(\cdot)\,ds$ for suitable " $f$ -continuous" functions (a numerical function $\xi$ on $\sigma$ is $f$ -continuous if $|\xi(s)-\xi(t)|$ is uniformly small whenever $|sf-tf|$ is small), and the additional Axiom IV is the Cauchy–Schwarz-type bound

\Bigl|\int\xi(s)\,sg\,ds\Bigr|\le\Bigl(\int|\xi(s)|^{2}\,ds\Bigr)^{1/2}\,\|g\|,

required for every $f$ -continuous numerical function $\xi$ and every vector $g$ for which $s\mapsto sg$ is $f$ -continuous. For an almost periodic $f$ , Weyl then applies Erhard Schmidt's integral-equation orthogonalization to the positive semidefinite Hermitian kernel $H(s,t)=(sf,tf)$ and obtains a unitary-orthogonal sequence $g_n$ (orthonormal for the scalar product), grouped into finite blocks each spanning a $\sigma$ -invariant subspace (an invariant set of vectors), such that the Fourier coefficients $\alpha_n=(g_n,f)$ satisfy the Parseval equation $\sum_n|\alpha_n|^{2}=\|f\|^{2}$ and the partial sums converge to $f$ weakly, i.e. in the Hilbert norm, with an explicit remainder estimate in terms of the minimal number $N_\varepsilon$ of $f$ -circles needed to cover $\sigma$ . The invariant sets decompose into primitive ones, each transforming according to an irreducible unitary representation $\Omega$ of $\sigma$ (a finite-dimensional complex representation that cannot be reduced to smaller blocks); a block of $h=\deg\Omega$ vectors is an $\Omega$ -row, it is upright for $f$ if the corresponding Fourier coefficients are not all zero and hidden otherwise, and by Schur's lemma (a linear map commuting with an irreducible representation is scalar) at most $h$ linearly independent upright $\Omega$ -rows exist for fixed $\Omega$ . Under Axiom IV the Fourier series of $f$ — and of every vector of $\Sigma_f^{0}$ , the closure in length of the linear span of the translates of $f$ — converges strongly, i.e. in the length norm. Weyl verifies Axioms I–IV in the model case of vector-valued functions on a homogeneous space of $\sigma$ (a set on which $\sigma$ acts transitively) with the supremum length. In his 1950 Gibbs lecture (a lecture series of the American Mathematical Society) he recalled the construction and observed that in general "there are more than denumerably many inequivalent irreducible invariant sets of vectors; but f itself picks out those among them that matter for f". Wilhelm Maak published two companion treatments of the same axiomatic theory in 1950.

The later literature replaced rather than continued Weyl's axioms. Working in the Bochner–von Neumann framework, Kitchen developed almost periodicity for normed modules over groups, and Holdgrün revisited the "main theorem of almost periodic vectors" for $G$ -modules; the definitive modern survey by Shtern treats almost periodicity for functions with values in locally convex spaces (vector spaces whose topology is generated by convex sets and whose continuous linear functionals separate points), where the passage from weak to strong almost periodicity is governed by completeness and compactness conditions on the target space — von Neumann completeness, the Krein property (closed convex hulls of compact sets are compact), and barrelledness (a compatibility condition between the topology and the algebraic dual) — rather than by Weyl's Axiom IV. Shtern's survey cites Weyl's 1949 paper only for the observation that the scalar almost periodic functions on a group are exactly the restrictions to it of continuous functions on its Bohr compactification (the compact group through which every continuous homomorphism of the given group into a compact group factors).

Problem Statement

Let $\Sigma$ be a complex vector space equipped with a Hermitian scalar product $(\cdot,\cdot)$ , its Hilbert norm $\|f\|=(f,f)^{1/2}$ , and a second "length" $|f|$ with $\|f\|\le|f|$ ; let a group $\sigma$ act on $\Sigma$ by linear maps isometric for both structures ( $\|sf\|=\|f\|$ and $|sf|=|f|$ for all $s\in\sigma$ ), and let $\Sigma$ be closed under convergence in the length. A vector $f\in\Sigma$ is almost periodic if for every $\varepsilon>0$ finitely many $f$ -circles $\{s\in\sigma:|sf-af|\le\varepsilon\}$ , $a\in\sigma$ , cover $\sigma$ . A numerical function $\xi$ on $\sigma$ is $f$ -continuous if for every $\delta>0$ there is $\varepsilon>0$ such that $|\xi(s)-\xi(t)|\le\delta$ whenever $|sf-tf|\le\varepsilon$ . Within this framework Weyl postulates a mean-value integral $\int(\cdot)\,ds$ , invariant under the group action, for $f$ -continuous numerical functions and for vector functions $s\mapsto sg$ that are $f$ -continuous (Axioms I–III), together with the additional requirement

\Bigl|\int\xi(s)\,sg\,ds\Bigr|\le\Bigl(\int|\xi(s)|^{2}\,ds\Bigr)^{1/2}\,\|g\|

for all such $\xi$ and $g$ (Axiom IV). The theory then yields, for almost periodic $f$ , a unitary-orthogonal sequence $g_n$ grouped into finite $\sigma$ -invariant blocks with Fourier coefficients $\alpha_n=(g_n,f)$ , the Parseval equation $\sum_n|\alpha_n|^{2}=\|f\|^{2}$ , weak (Hilbert-norm) convergence of the partial sums to $f$ , and — under Axiom IV — strong convergence in the length for $f$ and for every vector of $\Sigma_f^{0}$ , the closure in length of the linear span of the translates of $f$ . The blocks decompose into primitive invariant sets ("rows") belonging to irreducible unitary representations of $\sigma$ ; a row is upright for $f$ if its Fourier coefficients are not all zero and hidden otherwise.

Determine:

(a) Natural derivability of Axiom IV: identify simpler or more natural structural conditions on the quadruple $(\Sigma,(\cdot,\cdot),|\cdot|,\sigma)$ , within this original two-norm axiomatics, that imply Axiom IV or under which it holds, so that strong approximation follows from intrinsic structure rather than from an assumed axiom.

(b) Subgroup restriction: determine under what conditions on a subgroup $\sigma^{0}\subseteq\sigma$ almost periodicity, the mean-value integral, Axiom IV, and strong length-norm Fourier approximation persist when the action is restricted from $\sigma$ to $\sigma^{0}$ (all notions being read relative to $\sigma^{0}$ ); whether Axiom IV, and hence strong length-norm Fourier approximation, carries over from $\sigma$ to $\sigma^{0}$ for vectors almost periodic relative to $\sigma^{0}$ is part of what must be determined, not a presupposition.

(c) Hidden invariant sets: characterize the structure and role of the hidden irreducible invariant sets — those with zero $f$ -component, which do not enter the Parseval equation of $f$ — including their relation to the spectrum of $f$ , the circumstances in which only countably many inequivalent irreducible invariant sets are upright for $f$ while uncountably many inequivalent ones exist in total (as Weyl observed), and the constraints, if any, the hidden sets impose on the geometry of $\Sigma_f^{0}$ and the role, if any, they can play in the broader theory.

Parts (a), (b), and (c) are logically independent sub-questions, each answerable on its own terms; together they constitute the recorded three-part open problem.

References

Hermann Weyl, 'Almost Periodic Invariant Vector Sets in a Metric Vector Space', American Journal of Mathematics 71 (1949), 178–205, DOI 10.2307/2372104, https://doi.org/10.2307/2372104 (the primary source; paywalled on JSTOR)
Hermann Weyl, 'Ramifications, old and new, of the eigenvalue problem', Bulletin of the American Mathematical Society 56 (1950), 115–139, DOI 10.1090/S0002-9904-1950-09369-0, https://doi.org/10.1090/s0002-9904-1950-09369-0 (Gibbs lecture; full text freely available on the AMS website)
Wilhelm Maak, 'Fastperiodische invariante Vektormoduln in einem metrischen Vektorraum', Mathematische Annalen 122 (1950), 157–166, DOI 10.1007/BF01342893, https://doi.org/10.1007/BF01342893 (German companion treatment of the same axiomatic theory)
Wilhelm Maak, 'Almost Periodic Invariant Vector Sets in a Metric Vector Space', Proceedings of the National Academy of Sciences of the U.S.A. 36 (1950), 208–210, DOI 10.1073/pnas.36.3.208, https://doi.org/10.1073/pnas.36.3.208 (English companion note of Maak sharing its title with Weyl's paper — a distinct work; freely hosted as page scans at PMC, PMC1063163)
Joseph W. Kitchen, Jr., 'Normed modules and almost periodicity', Monatshefte für Mathematik 70 (1966), 233–243, DOI 10.1007/BF01305304, https://doi.org/10.1007/BF01305304 (adjacent line: almost periodicity for normed modules in the Bochner–von Neumann framework)
Horst S. Holdgrün, 'Zum Hauptsatz fastperiodischer Vektoren', Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 41 (1974), 28–36, DOI 10.1007/BF02993498, https://doi.org/10.1007/BF02993498 (revisits the main theorem of almost periodic vectors for G-modules)
Alexander I. Shtern, 'Almost periodic functions and representations in locally convex spaces', Russian Mathematical Surveys 60:3 (2005), 489–557, DOI 10.1070/RM2005v060n03ABEH000849, https://doi.org/10.1070/RM2005v060n03ABEH000849 (the definitive modern survey of vector-valued almost periodicity; full English text freely available via Math-Net)
S. Bochner and J. von Neumann, 'Almost Periodic Functions in Groups, II', Transactions of the American Mathematical Society 37 (1935), 21–50, DOI 10.1090/S0002-9947-1935-1501777-9, https://doi.org/10.1090/s0002-9947-1935-1501777-9 (foundation of the Banach-space-valued theory that later superseded Weyl's axiomatics)