Pego theorem for Hilbert space-valued functions on compact groups

Abstract: We prove a Hilbert space-valued analogue of Pego's compactness theorem on compact groups. For square-integrable functions taking values in a Hilbert space rather than in the complex numbers, we show that a bounded family is precompact exactly when it is simultaneously well behaved in two complementary senses: its members do not change much under small translations of the group, and their Fourier coefficients decay uniformly across the family. This equivalence holds without restriction when the Hilbert space is finite-dimensional, and it specializes to the known scalar-valued theorem when the Hilbert space is just the complex numbers. We then construct an explicit example showing that the equivalence genuinely breaks down once the Hilbert space is allowed to be infinite-dimensional. To repair this, we introduce a uniform tightness condition and we show that under this extra hypothesis the equivalence is restored regardless of the dimension of the Hilbert space. Along the way we establish the Plancherel isometry, the Hausdorff-Young inequality and its inverse for this vector-valued Fourier transform.

• The paper establishes a vector-valued Plancherel isometry and Hausdorff–Young inequality for Hilbert-space-valued functions on compact groups, showing that Hilbert structure is essential.
• For finite-dimensional value spaces, bounded families are precompact exactly when they are uniformly translation-equicontinuous and uniformly Fourier-decaying, with truncation using N(S) × dim(H) scalar coefficients.
• In infinite-dimensional spaces, the equivalence fails without additional control, but uniform tightness restores it by restricting functions uniformly to finite-dimensional subspaces and enabling finite-frequency compression.

Context and motivation

Pego's 1985 compactness theorem characterizes precompact subsets of $L^2(\mathbb{R}^n)$ by two dual conditions: uniform continuity under translation in the physical variable and uniform decay of Fourier coefficients in the frequency variable. The theorem has since been transported to several transform settings — short-time Fourier and wavelet transforms, Laplace, Laguerre, and Hankel transforms — and to groups: Górka extended it to locally compact abelian groups, Kumar to general compact (not necessarily abelian) groups via the operator-valued Fourier transform on the unitary dual $\widehat{G}$, and Krukowski proved an $L^1$ analogue over locally compact abelian groups. All of these results concern scalar-valued functions. The paper under review, by Lakmon and Mensah (2608.13142), develops the vector-valued analogue: functions on a compact group $G$ taking values in a complex Hilbert space $\mathcal{H}$.

The authors emphasize that this is not a routine extension. The scalar proof of Kumar relies on the Fourier transform being an isometry, and the paper shows this fact depends essentially on the Hilbert structure of the value space.

Vector-valued Plancherel theory

The Fourier transform of $f \in L^1(G,\mathcal{H})$ is defined following Assiamoua as a family of sesquilinear maps $$\widehat{f}(\sigma): H_\sigma \times H_\sigma \to \mathcal{H}$$, and the Assiamoua spaces $\mathscr{S}_p(\widehat{G},\mathcal{H})$ collect families with finite weighted $\ell^p$ norm over the index set $(\sigma,i,j)$. The central structural result is that the Fourier transform is an isometric isomorphism from $\widehat{G}$0 onto $\widehat{G}$1. The proof identifies $\widehat{G}$2 canonically with $\widehat{G}$3, uses the Peter-Weyl basis tensored with an orthonormal basis of $\widehat{G}$4, and applies Parseval twice — once in $\widehat{G}$5 and once in $\widehat{G}$6. Surjectivity follows by constructing the inverse series from prescribed coefficients.

This Hilbert-space hypothesis is sharp. The paper gives an explicit counterexample on $\widehat{G}$7 with $\widehat{G}$8 equipped with the max-norm $\widehat{G}$9, which violates the parallelogram law. For $L^1$0, one has $L^1$1 while $L^1$2: the Fourier transform fails to be an isometry. This counterexample also answers negatively Question 2 of Mensah's earlier work on Assiamoua spaces. The obstruction is precisely the failure of the parallelogram law, so no Banach-valued version of the isometry can hold in general.

From the Plancherel isometry at $L^1$3 and the trivial bound $L^1$4 at $L^1$5, Calderón-type complex interpolation yields the Hausdorff-Young inequality

$L^1$6

and its inverse for the reconstruction map $L^1$7 from $L^1$8 into $L^1$9. Notably, the interpolation argument bypasses the theory of Fourier type entirely.

The Pego theorem in finite dimensions

The two key definitions are uniform $G$0-equicontinuity of $G$1 under right translations, and uniform $G$2-decay of $G$3 (a single finite set $G$4 controls all tails uniformly over the family). Two lemmas establish their equivalence for bounded sets:

• Equicontinuity implies decay: convolution with a normalized indicator kernel $G$5 approximates every $G$6 within $G$7, and the coordinate-wise Cauchy-Schwarz estimate shows $G$8 outside a finite set, forcing $G$9 uniformly.
• Decay implies equicontinuity: band-limiting by $\mathcal{H}$0 reduces the problem to finitely many matrix coefficients, each bounded by $\mathcal{H}$1; uniform continuity of the finitely many matrix coefficients $\mathcal{H}$2 then gives the three-term $\mathcal{H}$3 decomposition.

The main theorem states that for bounded $\mathcal{H}$4 with $\mathcal{H}$5, precompactness, uniform equicontinuity, and uniform Fourier decay are equivalent. The implication (iii)$\mathcal{H}$6(i) uses that the range of $\mathcal{H}$7 is linearly isometric (up to fixed rescaling) to $\mathcal{H}$8 with $\mathcal{H}$9, hence finite-dimensional; boundedness plus Bolzano-Weierstrass gives total boundedness. When $f \in L^1(G,\mathcal{H})$0 the theorem reduces exactly to Kumar's scalar result.

Failure in infinite dimensions and tightness restoration

Finite-dimensionality cannot be dropped. The paper exhibits a decisive counterexample: let $f \in L^1(G,\mathcal{H})$1 be infinite-dimensional with orthonormal sequence $f \in L^1(G,\mathcal{H})$2 and let $f \in L^1(G,\mathcal{H})$3 where $f \in L^1(G,\mathcal{H})$4 are constant functions. Then $f \in L^1(G,\mathcal{H})$5 is bounded ($f \in L^1(G,\mathcal{H})$6), trivially equicontinuous (each $f \in L^1(G,\mathcal{H})$7 is translation-invariant), and $f \in L^1(G,\mathcal{H})$8 has uniform decay supported on the trivial representation alone — yet $f \in L^1(G,\mathcal{H})$9 for $$\widehat{f}(\sigma): H_\sigma \times H_\sigma \to \mathcal{H}$$0, so $$\widehat{f}(\sigma): H_\sigma \times H_\sigma \to \mathcal{H}$$1 has no Cauchy subsequence. Both analytic conditions hold while precompactness fails, because bounded sets in infinite-dimensional spaces need not be precompact.

To repair the equivalence, the authors introduce uniform tightness: for every $$\widehat{f}(\sigma): H_\sigma \times H_\sigma \to \mathcal{H}$$2 there exists a finite-dimensional subspace $$\widehat{f}(\sigma): H_\sigma \times H_\sigma \to \mathcal{H}$$3 such that $$\widehat{f}(\sigma): H_\sigma \times H_\sigma \to \mathcal{H}$$4 for all $$\widehat{f}(\sigma): H_\sigma \times H_\sigma \to \mathcal{H}$$5, where $$\widehat{f}(\sigma): H_\sigma \times H_\sigma \to \mathcal{H}$$6 is the orthogonal projection onto $$\widehat{f}(\sigma): H_\sigma \times H_\sigma \to \mathcal{H}$$7. This is the vector-valued counterpart of Krukowski's equivanishing condition, supplying control in the value direction that neither translation-equicontinuity nor frequency-decay provides when $$\widehat{f}(\sigma): H_\sigma \times H_\sigma \to \mathcal{H}$$8. Under boundedness plus uniform tightness, the full three-way equivalence holds for arbitrary $$\widehat{f}(\sigma): H_\sigma \times H_\sigma \to \mathcal{H}$$9. The proof of (iii)$\mathscr{S}_p(\widehat{G},\mathcal{H})$0(i) exploits that $\mathscr{S}_p(\widehat{G},\mathcal{H})$1 commutes with the Bochner integral defining the Fourier transform, so $\mathscr{S}_p(\widehat{G},\mathcal{H})$2 lies in a finite-dimensional subspace spanned by $\mathscr{S}_p(\widehat{G},\mathcal{H})$3 with $\mathscr{S}_p(\widehat{G},\mathcal{H})$4 and $\mathscr{S}_p(\widehat{G},\mathcal{H})$5 ranging over a basis of $\mathscr{S}_p(\widehat{G},\mathcal{H})$6.

Degrees of freedom and algorithmic aspects

Both main theorems yield constructive corollaries quantifying compression. In the finite-dimensional case, every $\mathscr{S}_p(\widehat{G},\mathcal{H})$7 is determined up to $\mathscr{S}_p(\widehat{G},\mathcal{H})$8-error $\mathscr{S}_p(\widehat{G},\mathcal{H})$9 by $\ell^p$0 scalar Fourier coefficients, where $\ell^p$1; under uniform tightness, $\ell^p$2 scalars suffice with $\ell^p$3. The per-frequency cost thus multiplies by the value-space dimension, extending the classical statement that equicontinuous, Fourier-decaying families admit uniform $\ell^p$4-term truncations.

The paper is explicit about what these results do not provide. The choice of $\ell^p$5 and $\ell^p$6 from data is not solved by the theorems: there is no quantitative relationship between $\ell^p$7 and $\ell^p$8 without assuming an explicit decay rate such as $\ell^p$9, which the paper does not assume or establish. A proposed data-driven scheme — thresholding empirical coefficient magnitudes for frequency selection, and Karhunen-Loève/PCA truncation of the empirical covariance operator for value-space selection — inherits only qualitative guarantees from the existence arguments. Extending the sample-based estimates to a guarantee over all of $(\sigma,i,j)$0 would require a concentration or covering-number argument (e.g., Rademacher-complexity bounds), which the authors explicitly leave open.

Limitations and open questions

Three limitations bear directly on the results. First, the entire framework requires $(\sigma,i,j)$1 to be a genuine Hilbert space; the parallelogram-law obstruction shown on $(\sigma,i,j)$2 with the max-norm rules out Banach-valued extensions of the Plancherel isometry, and hence of the whole compactness theory, in any straightforward form. Second, the equivalence fails outright in infinite dimensions without tightness, and tightness is a strong hypothesis: it demands a single finite-dimensional subspace working uniformly across the family, which may be unavailable for genuinely high-dimensional data. Third, the quantitative side is undeveloped — no rate connecting $(\sigma,i,j)$3 to $(\sigma,i,j)$4 or $(\sigma,i,j)$5 is established, and the gap between in-sample selection of $(\sigma,i,j)$6 and $(\sigma,i,j)$7 and out-of-sample guarantees over $(\sigma,i,j)$8 remains open, as do learning-theoretic routes (covering numbers, Rademacher complexity) to closing it.

Conclusion

The paper extends Pego's compactness theorem to Hilbert space-valued functions on compact groups, establishing along the way the vector-valued Plancherel isometry and Hausdorff-Young inequalities. The equivalence between precompactness, translation-equicontinuity, and uniform Fourier decay holds unconditionally in finite dimensions, fails demonstrably in infinite dimensions, and is restored in full generality by a uniform tightness condition analogous to equivanishing. The constructive proofs yield explicit degrees-of-freedom counts for compressing multi-channel group-indexed signals, while leaving the quantitative and statistical aspects of selecting the truncation parameters as clearly stated open problems.