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Rigidity of sets of independent functions in symmetric spaces

T0 review · 3 major / 8 minor · reviewed 2026-07-08 · glm-5.2

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Pith's one-line read Independent functions resist low-dimensional approximation in spaces near L_2

desk verdict Solid paper extending approximation rigidity of independent functions beyond L_p; deserves a serious referee read the letter →

arxiv 2607.06530 v1 pith:KQDNKK2E submitted 2026-07-07 math.FA

Sergey V. Astashkin , Yu. V. Malykhin This is my paper · ORCID
classification math.FA MSC 41A4646E3046B09
keywords KolmogorovwidthindependentfunctionssymmetricfunctionspaceKruglovpropertyLorentzOrliczlower2-estimaterigidity
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper characterizes which symmetric function spaces X have the property (IR): that any N independent mean-zero functions of norm at least 1 cannot be approximated to within a constant error by any subspace of dimension proportional to but smaller than N. The authors prove that (IR) holds when X satisfies a lower 2-estimate, contains L_2, and its associate space X' has the Kruglov property (a condition ensuring sums of independent functions are controlled by their disjoint rearrangements). Conversely, (IR) fails when the fundamental function of X grows faster than t^{1/2} near zero, or when a copy of ℓ_q for q>2 can be found in the lattice structure of X. Applied to Lorentz spaces L_{p,q}, this yields a complete classification: (IR) holds exactly when 12. The boundary case L_{2,q} for q<2 remains open.

What carries the argument

The positive direction constructs a biorthogonal system {g_k} in X' conjugate to the independent functions {f_k} in X, then applies the ℓ_2-estimate (guaranteed by the Kruglov property and lower 2-estimate) to invoke a lemma yielding averaged width bounds of order (1-n/N)^{1/2}. The negative direction constructs explicit sparse independent functions (taking values 0 or ±M with small probability) and uses width estimates for the Euclidean ball in ℓ_∞^N to show they can be approximated by O(N^γ)-dimensional subspaces.

What would settle it

If one could exhibit a symmetric space X satisfying a lower 2-estimate and X⊃L_2 but whose associate X' lacks the Kruglov property, and show that (IR) still holds for X, then the Kruglov condition would be revealed as sufficient but not necessary, narrowing the true criterion.

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Extended reading notes

Core claim

The central mechanism is the duality between a space X and its associate space X': the (IR) property for X reduces to whether sums of independent functions in X' satisfy an ℓ_2-type norm inequality (inequality (16)), which in turn depends on X' having the Kruglov property and X satisfying a lower 2-estimate. The Kruglov property — the requirement that a Poisson sum of independent copies of any function in X' remains in X' — is the load-bearing structural condition that allows comparing norms of independent sums with disjoint sums, and from there, applying classical Kolmogorov width estimates for Euclidean balls and octahedra. The failure results use a different mechanism: when the space is '

Load-bearing premise

The positive results depend on the Kruglov property of the associate space X', which ensures that sums of independent functions in X' are norm-controlled by their disjoint rearrangements. This property is not automatic — for instance, exponential Orlicz spaces Exp L_p fail it for p>1 — and without it, the duality argument that produces the width lower bounds collapses entirely.

Editorial extensions

If this is right

  • For Lorentz spaces L_{p,q}, the (IR) property is now fully classified except at the boundary p=2, q<2, giving a clean phase diagram for rigidity of independent functions.
  • The duality mechanism — reducing (IR) for X to an ℓ_2-estimate in X' — suggests that any symmetric space whose associate satisfies a von Bahr–Esseen type inequality will inherit rigidity, potentially extending results to non-Orlicz, non-Lorentz settings.
  • The construction of a space X with both X and X' failing (IR) (Corollary 3.10) shows that rigidity is not self-dual, raising the question of whether L_2 is the unique self-dual rigid space.
  • The partial rigidity results (Proposition 3.19, Corollary 3.21) show that even when full (IR) fails, control on the support sizes or distributional properties of the independent functions can restore lower width bounds.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The open problem about L_{2,q} for q<2 suggests that the condition X⊃L_2 in Theorem 1 may be an artifact of the proof technique rather than a genuine threshold; the true dividing line might be whether the space is '2-dominated' in a weaker sense.
  • The connection between the Kruglov property and (IR) hints that rigidity of independent functions is fundamentally a statement about how a space handles probabilistic summation — spaces close enough to L_2 to control random sums will be rigid, while spaces that allow heavier-tailed behavior will not.
  • The phase transition at the L_2 boundary for Lorentz spaces mirrors the Hilbert-space optimality in classical approximation theory, suggesting that the (IR) property may be characterizable purely through the Boyd index interval containing 1/2.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 8 minor

Summary. This paper introduces the (IR) property for symmetric function spaces on [0,1], which captures the rigidity of finite sets of mean-zero independent functions against approximation by low-dimensional linear subspaces. The authors establish broad sufficient conditions for (IR) (Theorem 1, Theorem 4) based on lower 2-estimates, the embedding X⊃L₂, and the Kruglov property of the associate space X'. They also prove failure of (IR) under two types of conditions: when the fundamental function grows faster than t^{1/2} (Theorem 2), and when ℓ_q for q>2 is roughly lattice finitely representable in a space with the Kruglov property (Theorem 5). These results yield a near-complete classification for Lorentz L_{p,q} spaces and partial results for Orlicz spaces. The proofs use Johnson–Schechtman disjointification inequalities, Kruglov property machinery, Kolmogorov width estimates for finite-dimensional bodies, and a duality argument via biorthogonal systems constructed from conditional expectations.

Significance. The paper extends the L_p rigidity theory to the broader setting of symmetric spaces, which is a natural and substantive generalization. The positive results (Theorems 1 and 4) provide a clean, checkable framework: lower 2-estimate plus X⊃L₂ plus X'∈K suffices for (IR), with the key inequality (16) serving as the operative condition. The negative results (Theorems 2 and 5) are complementary and use explicit constructions. The Lorentz classification (Corollaries 3.9, 3.12) is clean, and the Orlicz results (Corollaries 3.7, 3.15, 3.16) cover natural scales. The open problems (3.11, 3.14, 3.18) are well-posed and indicate genuine gaps rather than omissions. The paper builds on the authors' prior work [29, 30, 31] but the extension to symmetric spaces via the Kruglov property and lattice-geometric conditions is new. The proofs are detailed and verifiable; the key steps (separability argument, conditional-expectation biorthogonal construction, application of inequality (7), width bounds) are all carried out explicitly.

major comments (3)
  1. Theorem 1 (stated in the Introduction) requires X⊃L₂, but the proof in §3.1 uses this assumption to conclude X'⊂L₂, which is then needed for inequality (7) via [9, Theorem II.2.4]. However, the more general Theorem 4 in §3.1 does not list X⊃L₂ among its hypotheses (condition (i) does, but the main statement only requires inequality (16)). The relationship between Theorem 1 and Theorem 4 should be clarified: is Theorem 1 a strict corollary of Theorem 4(i), or does the proof of Theorem 1 use X⊃L₂ in an essential way beyond what Theorem 4 captures? Remark 3.2 suggests that (16) itself implies X⊃L₂, which would close the loop, but this logical relationship is not stated explicitly in the theorems. Adding a sentence clarifying the precise logical dependencies would strengthen the presentation.
  2. In the proof of Theorem 2 (§3.2), the construction fixes ε satisfying C₁(γ)ε/φ_X(ε)² ≤ 1/2 and then fixes M via (22). The parameter s* is then set by (24) as s* = ⌊Nγ²/(4C(γ)²M²)⌋. For the tail bound m(C) ≤ 2^{-s*} to yield Mφ_X(2^{-s*}) ≤ γ/2, one needs s* → ∞ as N → ∞, which requires M to be fixed (independent of N). Since M = 1/φ_X(ε) and ε is fixed, this is indeed the case. However, the argument would benefit from an explicit statement that M is a constant depending only on X and γ (not on N), as this is load-bearing for the limit s* → ∞.
  3. Proposition 3.19 (§3.3) extends the (IR)-type lower bound to functions satisfying a support condition measured by R_{1-δ}(f). The proof involves a dilation argument with factor R (line after equation (34)) that uses the bound ‖σ̃_R‖ ≤ max{1, R} on symmetric spaces on the half-axis. The application of this dilation to the Z²_{X'} norm is correct, but the step from (34) to the final estimate involves several implicit constants (C_X from the lower 2-estimate, δ^{-1}, R, C(X)). It would help the reader to state explicitly that the final constant B = δ^{-1}C(X)R_{1-δ}(f)(C_X + 2) is the one appearing in the proposition's conclusion, and to confirm that all dependencies are as claimed (B depends on X, δ, and R_{1-δ}(f)).
minor comments (8)
  1. Abstract: 'sufficienly' should read 'sufficiently'.
  2. Proof of Theorem 2 (§3.2): 'Conbining' should read 'Combining'.
  3. §2.1: 'detailes' should read 'details'.
  4. §2.2: 'duscussed' should read 'discussed'.
  5. Reference [16]: the page range contains a formatting artifact ('1048ı¿œ1052'); please correct to '1048–1052'.
  6. Reference [21]: the author list formatting ('A. Kamińska A., L. Maligranda, L.-E. Persson') appears to have a stray 'A.' after the first author's name; please verify and correct.
  7. In the proof of Theorem 1 (§3.1), the step verifying ∫f_j g_k = 0 for j≠k uses independence of f_j and E_{A_k}g'_k. It would improve readability to note explicitly that this independence follows from the independence of the σ-algebras A_j and A_k for j≠k, which is a standard but non-trivial fact.
  8. In Proposition 3.19 (§3.3), the quantity R_{1-δ}(f) is defined as a sum over k of m{|f_k| > 1-δ}, but it is used in the proof as a single number R. A brief remark that R_{1-δ}(f) is finite by the assumption R_0(f) < ∞ (in Corollary 3.20) or by the finiteness of the collection would help the reader.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for a careful reading and for three constructive comments, all of which concern clarification of logical dependencies and explicit tracking of constants. We agree with all three points and will incorporate the requested clarifications in the revised manuscript.

read point-by-point responses

  1. Referee: Theorem 1 requires X⊃L₂, but the proof in §3.1 uses this to conclude X'⊂L₂, needed for inequality (7) via [9, Theorem II.2.4]. The more general Theorem 4 does not list X⊃L₂ among its hypotheses (condition (i) does, but the main statement only requires inequality (16)). The relationship between Theorem 1 and Theorem 4 should be clarified: is Theorem 1 a strict corollary of Theorem 4(i), or does the proof of Theorem 1 use X⊃L₂ in an essential way beyond what Theorem 4 captures? Remark 3.2 suggests that (16) itself implies X⊃L₂, which would close the loop, but this logical relationship is not stated explicitly in the theorems.

    Authors: The referee is correct that the logical dependencies between Theorem 1, Theorem 4, and Remark 3.2 are not stated as explicitly as they should be. To clarify: Theorem 1 is indeed a corollary of Theorem 4(i). The proof of Theorem 1 in §3.1 proceeds by verifying the hypotheses of Theorem 4(i): the lower 2-estimate and X⊃L₂ together imply (by duality) that X' satisfies an upper 2-estimate and X'⊂L₂; combined with X'∈K, these yield inequality (16) via [9, Theorem II.2.4]. Thus X⊃L₂ is used in the proof of Theorem 1 only to establish (16), not in any additional way. Furthermore, as the referee notes, Remark 3.2 (citing [7, Theorem 38(b)]) shows that (16) itself implies both a lower 2-estimate and X⊃L₂, so condition (16) is in fact equivalent to the conjunction of these properties (given X'∈K). We will add an explicit sentence after the statement of Theorem 4 clarifying that Theorem 1 is a direct corollary of Theorem 4(i), and we will cross-reference Remark 3.2 to note that (16) implies X⊃L₂, closing the logical loop. revision: yes

  2. Referee: In the proof of Theorem 2 (§3.2), the construction fixes ε satisfying C₁(γ)ε/φ_X(ε)² ≤ 1/2 and then fixes M via (22). The parameter s* is set by (24) as s* = ⌊Nγ²/(4C(γ)²M²)⌋. For the tail bound m(C) ≤ 2^{-s*} to yield Mφ_X(2^{-s*}) ≤ γ/2, one needs s* → ∞ as N → ∞, which requires M to be fixed (independent of N). Since M = 1/φ_X(ε) and ε is fixed, this is indeed the case. However, the argument would benefit from an explicit statement that M is a constant depending only on X and γ (not on N), as this is load-bearing for the limit s* → ∞.

    Authors: The referee's observation is correct. The parameter ε is fixed depending only on X and γ (via the condition C₁(γ)ε/φ_X(ε)² ≤ 1/2), and M = 1/φ_X(ε) is then also fixed depending only on X and γ. Consequently s* = ⌊Nγ²/(4C(γ)²M²)⌋ → ∞ as N → ∞, which is what drives the tail estimate Mφ_X(2^{-s*}) → 0. We will add an explicit sentence after the point where ε and M are fixed, stating that both ε and M are constants depending only on X and γ (and not on N), and that this independence from N is what ensures s* → ∞ as N → ∞. revision: yes

  3. Referee: Proposition 3.19 (§3.3) extends the (IR)-type lower bound to functions satisfying a support condition measured by R_{1-δ}(f). The proof involves a dilation argument with factor R (line after equation (34)) that uses the bound ‖σ̃_R‖ ≤ max{1, R} on symmetric spaces on the half-axis. The application of this dilation to the Z²_{X'} norm is correct, but the step from (34) to the final estimate involves several implicit constants (C_X from the lower 2-estimate, δ^{-1}, R, C(X)). It would help the reader to state explicitly that the final constant B = δ^{-1}C(X)R_{1-δ}(f)(C_X + 2) is the one appearing in the proposition's conclusion, and to confirm that all dependencies are as claimed (B depends on X, δ, and R_{1-δ}(f)).

    Authors: We agree that the tracking of constants in the proof of Proposition 3.19 could be made more transparent. The final constant is indeed B = δ^{-1}C(X)R_{1-δ}(f)(C_X + 2), where C(X) is the constant from the disjointification inequality (6)/(33), C_X is the constant from the lower 2-estimate of X (equivalently, the upper 2-estimate of X'), δ is the parameter from the support condition, and R = R_{1-δ}(f) = Σ_k m{|f_k| > 1-δ}. All dependencies are as claimed: B depends on X (through C(X) and C_X), on δ, and on R_{1-δ}(f). We will add an explicit display of the constant B at the end of the proof, immediately before the concluding sentence, and confirm the dependency structure. revision: yes

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity found; the derivation is self-contained against external benchmarks with only minor self-citation.

full rationale

The paper's central theorems (Theorems 1-5) are derived from established external results: the Johnson-Schechtman inequality (6) from [19], the Kruglov property framework from Braverman [9], Kolmogorov width estimates (9)-(12) from approximation theory, and lattice duality from Lindenstrauss-Tzafriri [27]. Self-citations [29, 30, 31] provide the (IR) definition and Lemma 2.1, but Lemma 2.1 is proved in full within the paper (Section 2.5), and its proof uses only the standard orthonormal width bound (2). The key derivation steps in Theorem 1 (duality via conditional expectations, application of inequality (7), invocation of Lemma 2.1) do not reduce to the (IR) definition by construction. Theorem 2 constructs an explicit counterexample using width estimate (10) for the Euclidean ball. Theorem 5 uses disjointification via inequality (6) and width estimate (11). None of these steps are self-definitional or fitted-input-as-prediction. The self-citations are to prior work defining the problem and providing a lemma that is re-proved here, not to an unverified uniqueness theorem that would force the conclusion. The derivation chain is genuinely independent of its outputs.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new mathematical entities or postulated objects. The (IR) property is a definition, not an axiom. All background results are standard in the literature on symmetric spaces and approximation theory. The constants γ, B, δ are existence constants whose values are not fitted to data.

free parameters (3)
  • γ = γ(X) > 0 = not specified; existence proved
    The (IR) property asserts existence of γ>0 depending only on X. The proofs show γ exists via constants B, C(X), etc., but do not compute it explicitly.
  • B = B(X) > 0 = not specified; existence proved
    The constant in the ℓ_2-estimate (16) and the width lower bound in Theorem 4. Depends on X through the Kruglov property constant and lattice constants.
  • δ = δ(X) > 0 = not specified; existence proved
    The decay exponent in Theorem 5's width upper bound d_n ≤ CN^{-δ}. Determined by q via the octahedron width estimate (11).
assumptions (5)
  • standard math Inequality (6): if X∈K, then ‖Σf_k‖_X ≤ C(X)‖Σf̄_k‖_{Z²_X} for independent mean-zero f_k
    Johnson-Schechtman [19], also in [3, 7]. Central to all positive results.
  • standard math Inequality (7): ℓ_2-estimate for sums of independent functions holds iff X satisfies upper 2-estimate and X⊂L_2
    Braverman [9, Theorem II.2.4], extended in [7, Corollary 40]. Used in proof of Theorem 1.
  • standard math Width estimates (9)-(11) for Euclidean ball and octahedron
    Classical results in approximation theory, cited from [28, 30]. Used in Theorems 2 and 5.
  • standard math 1/α_X ∈ LFR(X) for symmetric spaces
    Lindenstrauss-Tzafriri [27, Theorem 2.b.6], full proof in [5, Theorem 4]. Used in Theorem 5 corollary.
  • standard math Conditional expectation has norm 1 in maximal symmetric spaces
    Krein-Petunin-Semenov [25, Theorem II.4.9]. Used in proof of Theorem 1 to bound ‖g_k‖_{X'}.

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Pith. "Pith review of Rigidity of sets of independent functions in symmetric spaces." pith.science (2026). https://pith.science/paper/KQDNKK2E

@misc{pith2026260706530,
  author       = {Pith},
  title        = {Pith review of: Rigidity of sets of independent functions in symmetric spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KQDNKK2E}},
  note         = {Machine review of arXiv:2607.06530}
}
abstract

We say that a symmetric function space $X$ has the $(IR)$ property whenever all sets of $N$ independent mean zero functions $f_1,\ldots,f_N\in X$, $\|f_k\|_X\ge 1$, are poorly approximated by any linear combinations of arbitrary $n$ functions, if $n$ is sufficienly smaller that $N$; namely, for some $\gamma=\gamma(X)>0$ we have $d_n(\{f_1,\ldots,f_N\},X)\ge \gamma$, $n\le \gamma N$, 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 $L_{p,q}$ spaces and Orlicz spaces.

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Works this paper leans on

34 extracted references · 34 canonical work pages · cited by 1 Pith paper

  1. [1]

    Astashkin.The Rademacher system in function spaces, Birkh¨ auser, Switzerland, 2020.– 559 pp

    S.V. Astashkin.The Rademacher system in function spaces, Birkh¨ auser, Switzerland, 2020.– 559 pp. 1, 3.2

    work page 2020

  2. [2]

    Sequences of independent functions and structure of rearrangement invari- ant spaces

    S.V. Astashkin, “Sequences of independent functions and structure of rearrangement invari- ant spaces”,Russian Math. Surveys,79:3 (2024), 375–457. 1

    work page 2024

  3. [3]

    Independent functions and the geometry of Banach spaces

    S.V. Astashkin, F.A. Sukochev, “Independent functions and the geometry of Banach spaces”,Russian Math. Surveys65(396):6 (2011), 1003–1081. 1, 1, 2.4, 2.4, 3.1

    work page 2011

  4. [4]

    Independent functions in symmetric spaces and Kruglov property

    S.V. Astashkin, “Independent functions in symmetric spaces and Kruglov property”, Sbornik: Math.199:7 (2008), 945–963. 3.1

    work page 2008

  5. [5]

    On the finite representability ofl p–spaces in rearrangement invariant spaces

    S.V. Astashkin, “On the finite representability ofl p–spaces in rearrangement invariant spaces”,St. Petersburg Math. J.23:2 (2012), 257–273. 2.3

    work page 2012

  6. [6]

    On lattice properties of the Lorentz spacesL p,q

    S.V. Astashkin, “On lattice properties of the Lorentz spacesL p,q”,Math. Notes,113:1 (2023), 10–17. 3.13

    work page 2023

  7. [7]

    Disjointification inequalities in symmetric quasi- Banach spaces and their applications

    S. Astashkin, F. Sukochev and D. Zanin, “Disjointification inequalities in symmetric quasi- Banach spaces and their applications”,Pacific J. Math,214:2 (2014), 257–285. 1, 2.4, 2.4, 3.1, 3.1, 3.1, 3.2, 3.2, 3.2, 3.2, 3.14

    work page 2014

  8. [8]

    Inequalities for ther-th absolute moment of a sum of random variables, 1⩽r⩽2

    B. von Bahr and C. Esseen, “Inequalities for ther-th absolute moment of a sum of random variables, 1⩽r⩽2”,Ann. Math. Statist.,36(1965), 299–303. 3.1

    work page 1965

Show all 34 references
  1. [9]

    Braverman.Independent random variables and rearrangement invariant spaces, Cam- bridge University Press, 1994

    M.Sh. Braverman.Independent random variables and rearrangement invariant spaces, Cam- bridge University Press, 1994. — 116 p. 1, 2.4, 2.4, 3.1, 3.1, 3.8

    1994

  2. [10]

    Type and cotype in LorentzL p,q spaces

    J. Creekmore, “Type and cotype in LorentzL p,q spaces”,Indag. Math.,43(1981), 145–152. 3.1

    1981

  3. [11]

    Summing inclusion maps between symmetric sequence spaces

    A. Defant, M. Masty lo and C. Michels, “Summing inclusion maps between symmetric sequence spaces”,Trans. Am. Math. Soc.354(2002), 4473–4492. 1

    2002

  4. [12]

    Applications of summing inclusion maps to inter- polation of operators

    A. Defant, M. Masty lo and C. Michels, “Applications of summing inclusion maps to inter- polation of operators”,Quart. J. Math.54(2003), 61–72. 1

    2003

  5. [13]

    Eigenvalue estimates for operators on symmetric Banach sequence spaces

    A. Defant, M. Masty lo and C. Michels, “Eigenvalue estimates for operators on symmetric Banach sequence spaces”,Proc. Am. Math. Soc.132(2004), 513–521. 1

    2004

  6. [14]

    Special Banach lattices and their applications

    S.J. Dilworth, “Special Banach lattices and their applications”,Handbook of the Geometry of Banach Spaces, Vol. 1, ed. by W. B. Johnson and J. Lindenstrauss, Elsevier Sci., 2001, 497–532. 3.1, 3.2

    2001

  7. [15]

    Lacunary series and independent functions

    V. F. Gaposhkin, “Lacunary series and independent functions”,Russian Math. Surveys, 21:6 (1966), 1–82. 1

    1966

  8. [16]

    The widths of a Euclidean ball

    A.Yu. Garnaev, E.D. Gluskin, “The widths of a Euclidean ball”,Dokl. Akad. Nauk SSSR, 277:5 (1984), 1048¨ ı¿œ1052 (in Russian)

    1984

  9. [17]

    Onl p-complemented copies in Orlicz spaces II

    F.L. Hernandez and B. Rodriguez-Salinas, “Onl p-complemented copies in Orlicz spaces II”,Israel J. Math.,68:1 (1989), 27–55. 3.17

    1989

  10. [18]

    Gelfand Numbers of Identity Operators Between Symmetric Sequence Spaces

    A. Hinrichs, C. Michels, “Gelfand Numbers of Identity Operators Between Symmetric Sequence Spaces”,Positivity10(2006), 111–133. 1, 3.4

    2006

  11. [19]

    Sums of independent random variables in rearrange- ment invariant function spaces

    W. B. Johnson and G. Schechtman, “Sums of independent random variables in rearrange- ment invariant function spaces”,Ann. Probab.17(1989), 789–808. 1, 2.4, 2.4

    1989

  12. [20]

    Symmetric structures in Banach spaces

    W. B. Johnson, B. Maurey, G. Schechtman, and L. Tzafriri, “Symmetric structures in Banach spaces”,Mem. Amer. Math. Soc.1:217 (1979). 1, 2.4, 3.17

    1979

  13. [21]

    Type, cotype and convexity properties of Orlicz spaces

    A. Kami´ nska A., L. Maligranda, L.-E. Persson, “Type, cotype and convexity properties of Orlicz spaces”,Publ. del Depart. de Anal. Matem., Fac. de Matem., Univ. Compl. de Madrid, ¨ ı¿œ 42, 1996–97. P. 113–126. 3.1

    1996

  14. [22]

    Diameters of some finite-dimensional sets and classes of smooth functions

    B.S. Kashin, “Diameters of some finite-dimensional sets and classes of smooth functions”, Math. USSR-Izv.,11:2 (1977), 317–333

    1977

  15. [23]

    B. S. Kashin and A. A. Saakyan,Orthogonal series, Amer. Math. Soc., Providence RI,

  16. [24]

    ASTASHKIN AND YU

    2.1 24 SERGEY V. ASTASHKIN AND YU. V. MALYKHIN

  17. [25]

    M. A. Krasnosel’skii and Ya. B. Rutickii,Convex functions and Orlicz spaces, Noordhooff, Groningen, 1961. 3.1, 3.2

    1961

  18. [26]

    S. G. Krein, Yu. I. Petunin, and E. M. Semenov,Interpolation of linear operators, Amer. Math. Soc., Providence, RI, 1982. 2.2, 2.2, 2.4, 3.1, 3.2, 3.3

    1982

  19. [27]

    V. M. Kruglov, ”A note on infinitely divisible distributions” ,Theory Probab. Appl. 15:2(1970), 319–324. 3.1

    1970

  20. [28]

    Lindenstrauss, L

    J. Lindenstrauss, L. Tzafriri.Classical Banach spaces 2, Function spaces. Berlin: Springer- Verlag, 1979. — 243 pp. 2.2, 2.3, 2.4, 3.1, 3.1, 3.3

    1979

  21. [29]

    Lorentz, M

    G.G. Lorentz, M. Golitschek, Y. Makovoz,Constructive approximation: Advanced Prob- lems, Springer, 1996. 2.5

    1996

  22. [30]

    Widths and rigidity

    Yu.V. Malykhin, “Widths and rigidity”,Sb. Math.,215:4 (2024), 543–571. 1, 1, 3.1, 3.2, 3.3

    2024

  23. [31]

    Widths and rigidity of unconditional sets and random vectors

    Yu.V. Malykhin, K.S. Ryutin, “Widths and rigidity of unconditional sets and random vectors”,Izv. Math.,89:2 (2025), 261–273. 1, 1, 2.5, 3.1

    2025

  24. [32]

    Malykhin,Bounds for Kolmogorov widths and related quantites, Doctoral disserta- tion, 2025

    Yu.V. Malykhin,Bounds for Kolmogorov widths and related quantites, Doctoral disserta- tion, 2025. 1, 2.5

    2025

  25. [33]

    Matrix and tensor rigidity andL p-approximation

    Y. Malykhin, “Matrix and tensor rigidity andL p-approximation”,J. Complexity,72 (2022), 101651. 1

    2022

  26. [34]

    Krivine’s theorem and the indices of a Banach lattice

    A.R. Schep, “Krivine’s theorem and the indices of a Banach lattice”,Acta Appl. Math.27 (1992), 111–121. 2.3 Astashkin: Department of Mathematics, Samara National Research Univer- sity, Moskovskoye shosse 34, 443086, Samara, Russia; Lomonosov Moscow State University, Moscow, Ru...

    1992

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