The geometry of two-valued subsets of Lp-spaces
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Anthony Weston
Abstract
Let 𝓜(Ω, μ) denote the algebra of all scalar-valued measurable functions on a measure space (Ω, μ). Let B ⊂ 𝓜(Ω, μ) be a set of finitely supported measurable functions such that the essential range of each f ∈ B is a subset of {0,1}. The main result of this paper shows that for any p ∈ (0, ∞), B has strict p-negative type when viewed as a metric subspace of Lp(Ω, μ) if and only if B is an affinely independent subset of 𝓜(Ω, μ) (when 𝓜(Ω, μ) is considered as a real vector space). It follows that every two-valued (Schauder) basis of Lp(Ω, μ) has strict p-negative type. For instance, for each p ∈ (0, ∞), the system of Walsh functions in Lp[0,1] is seen to have strict p-negative type. The techniques developed in this paper also provide a systematic way to construct, for any p ∈ (2, ∞), subsets of Lp(Ω, μ) that have p-negative type but not q-negative type for any q > p. Such sets preclude the existence of certain types of isometry into Lp-spaces.
Dedicated to Nancy Joy Weston
Acknowledgement
I am indebted to the referees for their thoughtful comments on the preliminary version of this paper. The outcome is a far more general and appealing paper. I would also like to thank the CCRDS research group in the Department of Decision Sciences at the University of South Africa for their kind support and insightful input during the preparation of this paper.
References
[1] Bretagnolle, J.—Dacunha-Castelle, D.—Krivine, J. L.: Lois stables et espaces Lp, Ann. Inst. H. Poincaré Sect. B (N.S.) 2 (1966), 231–259.10.1007/BFb0061106Search in Google Scholar
[2] Cayley, A.: On a theorem in the geometry of position, Cambridge Mathematical Journal II (1841), 267–271.10.1017/CBO9780511703676.002Search in Google Scholar
[3] Dor, L. E.: Potentials and isometric embeddings in L1, Israel J. Math. 24 (1976), 260–268.10.1007/BF02834756Search in Google Scholar
[4] Doust, I.—Weston, A.: Enhanced negative type for finite metric trees, J. Funct. Anal. 254 (2008), 2336–2364.10.1016/j.jfa.2008.01.013Search in Google Scholar
[5] Enflo, P.: On a problem of Smirnov, Ark. Mat. 8 (1969), 107–109.10.1007/BF02589550Search in Google Scholar
[6] Faver, T.—Kochalski, K.—Murugan, M.—Verheggen, H.—Wesson, E.—Weston, A.: Roundness properties of ultrametric spaces, Glasgow Math. J. 56 (2014), 519–535.10.1017/S0017089513000438Search in Google Scholar
[7] Kelleher, C.—Miller, D.—Osborn T.—Weston, A.: Polygonal equalities and virtual degeneracy in Lp-spaces, J. Math. Anal. Appl. 415 (2014), 247–268.10.1016/j.jmaa.2014.01.063Search in Google Scholar
[8] Koldobsky, A.: Generalised Lévy representation of norms and isometric embeddings into Lp-spaces, Ann. Inst. H. Poincaré, 28 (1992), 335–353.Search in Google Scholar
[9] Lennard, C. J.—Tonge, A. M.—Weston, A.: Generalized roundness and negative type, Michigan Math. J. 44 (1997), 37–45.10.1307/mmj/1029005619Search in Google Scholar
[10] Li, H.—Weston, A.: Strict p-negative type of a metric space, Positivity 14 (2010), 529–545.10.1007/s11117-009-0035-2Search in Google Scholar
[11] Menger, K.: Die Metrik des Hilbert-Raumes, Akad. Wiss. Wien Abh. Math.-Natur. K1 65 (1928), 159–160.Search in Google Scholar
[12] Misiewicz, J.: Positive definite functions on
[13] Murugan, M.: Supremal p-negative type of vertex transitive graphs, J. Math. Anal. Appl. 391 (2012), 376–381.10.1016/j.jmaa.2012.02.042Search in Google Scholar
[14] Nickolas, P.—Wolf, R.: Distance geometry in quasihypermetric spaces. III, Math. Nachr. 284 (2011), 747–760.10.1002/mana.200810216Search in Google Scholar
[15] Schoenberg, I. J.: Metric spaces and positive definite functions, Trans. Amer. Math. Soc. 44 (1938), 522–536.10.1090/S0002-9947-1938-1501980-0Search in Google Scholar
[16] Weston, A.: On the generalized roundness of finite metric spaces, J. Math. Anal. Appl. 192 (1995), 323–334.10.1006/jmaa.1995.1174Search in Google Scholar
© 2017 Mathematical Institute Slovak Academy of Sciences
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Articles in the same Issue
- Some results in local Hilbert algebras
- A note on residuated po-groupoids and lattices with antitone involutions
- The first law of cubology for the Rubik’s Revenge
- Conditional associativity in orthomodular lattices
- A monotonicity property for generalized Fibonacci sequences
- Systems of conditionally independent sets
- On the k-th order lfsr sequence with public key cryptosystems
- On parametric spaces of bicentric quadrilaterals
- Character sums with an explicit evaluation
- Some remarks on formal power series and formal Laurent series
- The co-rank of the fundamental group: The direct product, the first Betti number, and the topology of foliations
- On prime and semiprime generalized hyperaction of hypermonoid
- On O’Malley ϱ-upper continuous functions
- Uniqueness theorems for finitely additive probabilities on quantum structures
- Some results on differential-difference analogues of Brück conjecture
- Oscillation of neutral second order half-linear differential equations without commutativity in deviating arguments
- On support, separation and decomposition theorems for t-Wright-concave functions
- A characterization of holomorphic bivariate functions of bounded index
- Asymptotic expansion of double Laplace-type integrals with a curve of minimal points and application to an exit time problem
- The geometry of two-valued subsets of Lp-spaces
- Hemi-slant submanifolds as warped products in a nearly Kaehler manifold
- A comparison of two tests in multivariate models
- Modelling prescription behaviour of general practitioners
- Characterization of congruence lattices of principal p-algebras
