Rigidity of sets of independent functions in symmetric spaces

Abstract: We say that a symmetric function space $X$ has the $(IR)$ property whenever all sets of $N$ independent mean zero functions $f_1,\ldots,f_N\in X$, $|f_k|<em>X\ge 1$, are poorly approximated by any linear combinations of arbitrary $n$ functions, if $n$ is sufficienly smaller that $N$; namely, for some $γ=γ(X)&gt;0$ we have $d_n({f_1,\ldots,f_N},X)\ge γ$, $n\le γN$, where $d_n(K,X)$ is the Kolmogorov $n$-width of the set $K\subset X$. The spaces $X=L_p$ satisfy this property if and only if $1\le p\le2$ or $p=\infty$. The goal of this paper is to move from $L_p$ scale to a larger class of symmetric spaces. We obtain rather broad conditions, under which such a space $X$ has the $(IR)$ property and prove precise statements for particular scales of Lorentz $L</em>{p,q}$ spaces and Orlicz spaces.

• The paper establishes that symmetric spaces with a lower 2-estimate and the Kruglov property exhibit rigidity, providing sharp lower bounds on their Kolmogorov widths.
• It extends classical results on Lᵖ-spaces to Lorentz and Orlicz spaces, clarifying conditions under which rigidity holds or fails.
• The study addresses borderline cases and open problems, offering insights into approximation theory and the geometric structure of function systems.

Rigidity of Sets of Independent Functions in Symmetric Function Spaces

Introduction and Problem Setting

The paper provides a comprehensive analysis of approximation properties of systems of independent, mean-zero functions in symmetric Banach (and more generally, quasi-Banach) function spaces, with particular focus on their Kolmogorov widths. The key concept investigated is the so-called (IR) property: a space $X$ possesses (IR) if, for every sufficiently large set of independent mean-zero functions of unit norm in $X$, any linear subspace of dimension significantly less than the set size fails to approximate the set closely. This is quantified by a nontrivial lower bound on the Kolmogorov $n$-width $d_n$ for $n \leq \gamma N$.

While dichotomies for $L^p$-spaces—where rigidity holds if and only if $1 < p \leq 2$ or $p = \infty$—are classical, the extension of these properties to broader symmetric spaces such as Lorentz and Orlicz spaces was open. This work identifies general geometric and lattice-theoretic conditions under which rigidity is retained or lost, and gives sharp estimates in diverse families of symmetric spaces.

Main Definitions and Technical Background

A detailed framework is set up around symmetric function spaces (s.s.), Kolmogorov $n$-widths, and associated lattice and convexity properties. Fundamental tools include:

• Kolmogorov Width: $d_n(K, X)$ quantifies the minimal worst-case error of approximating all elements of $X$0 by $X$1-dimensional subspaces.
• Symmetric Spaces: Banach (or quasi-Banach) spaces invariant under equimeasurable rearrangements.
• Lower and Upper $X$2-Estimates, $X$3-Concavity/Convexity: Key in extending the $X$4 structure to more general symmetric spaces.
• Kruglov Property: A central property (usually of the Köthe dual $X$5 or $X$6 itself), encoding the behaviour of sums of independent copies under the norm of $X$7.
• Boyd Indices, Finite Representability: Quantities encoding how $X$8 embeds into $X$9, crucial for precise dichotomies especially for Lorentz and Orlicz spaces.

Classes of Spaces

The analysis is carried over classical $n$0, Lorentz $n$1, and Orlicz spaces $n$2, where structural parameters (e.g., $n$3, $n$4, Orlicz function growth) crucially affect rigidity.

Results: Criteria for Rigidity and Non-Rigidity

Characterization in $n$5-Spaces

It is shown that $n$6 has the (IR) property if and only if $n$7 or $n$8, with sharp lower bounds for the Kolmogorov widths in these cases. In $n$9 and $d_n$0 for $d_n$1, rigidity fails, as previously established in [29,30].

Extension to General Symmetric Spaces: Positive Results

Theorem 1: If $d_n$2 is a symmetric space with a lower 2-estimate, contains $d_n$3, and $d_n$4 has the Kruglov property, then $d_n$5 has (IR).

• Lorentz Spaces: $d_n$6 admits (IR) if $d_n$7, $d_n$8.
• Orlicz Spaces: For $d_n$9 with $n \leq \gamma N$0 subquadratic growth and suitable conjugacy conditions (including $n \leq \gamma N$1 for $n \leq \gamma N$2), (IR) holds.
• Strong numerical lower bounds for $n \leq \gamma N$3 are derived: $n \leq \gamma N$4.

Negative Results and Sharp Transition

Theorem 2: If $n \leq \gamma N$5 and the fundamental function $n \leq \gamma N$6 grows faster than $n \leq \gamma N$7 as $n \leq \gamma N$8, rigidity fails. Similarly, if for some $n \leq \gamma N$9 the lattice $L^p$0 is finitely representable in $L^p$1 and $L^p$2 has the Kruglov property, (IR) fails.

• Lorentz Spaces: $L^p$3 fails to have (IR) if $L^p$4.
• Orlicz Spaces: For sufficiently superquadratic growth, rigidity is lost—rigidity essentially occurs only between spaces close to $L^p$5.
• Upper bounds: For appropriate spaces, one constructs systems of normalized independent functions with Kolmogorov widths decaying as $L^p$6 for some $L^p$7.

Borderline and Open Cases

The question of (IR) for $L^p$8 with $L^p$9 remains open, as the space satisfies a lower 2-estimate but does not contain $1 < p \leq 2$0, and the required dual Kruglov property is not available.

Extension to Quasi-Banach Spaces

Analogous dichotomies are established in the broader setting of quasi-Banach symmetric spaces, employing suitable modifications of convexity and norm comparison tools.

Additional Rigidity Mechanisms

Under additional regularity or normalization (e.g., identical distributions, support conditions), rigidity may be secured even in certain borderline contexts.

Implications and Theoretical Significance

These results clarify the geometric structure underlying rigidity-type phenomena for systems of independent functions in symmetric spaces. They give:

• Sharp Thresholds: The precise role of 2-concavity, the Kruglov property, and Boyd indices in ensuring or precluding the existence of "rigid" systems, mapping the transition from $1 < p \leq 2$1-type behaviour to "softer" structures.
• Width Estimates: Explicit lower bounds for Kolmogorov widths are important in Approximation Theory, contributing to the understanding of the complexity of classes of independent functions.
• Broader Functional-Analytic Impact: The results interplay with local theory of Banach lattices, finite representability, and duality, deepening the connections between functional analysis, probability, and approximation.

Future Directions

Open questions remain concerning the rigidity of certain endpoint and intermediate scale spaces (e.g., $1 < p \leq 2$2 for $1 < p \leq 2$3), and finding necessary and sufficient Orlicz function criteria for rigidity. There is potential to extend these results to non-commutative settings, to finer invariants in random matrix theory, and to the geometry of high-dimensional probability measures.

Conclusion

This paper establishes a comprehensive landscape for rigidity phenomena for systems of independent functions in symmetric function spaces. By elucidating sharp lattice and convexity conditions, it unifies and extends known results from classical $1 < p \leq 2$4-spaces to broader families including Lorentz and Orlicz spaces, providing both lower and upper estimates for the complexity of approximation. The work advances the understanding of probabilistic and geometric properties of high-dimensional function systems in rearrangement-invariant frameworks.

**** (2607.06530)