Positive definite norm dependent functions on l∞

Misiewicz, Jolanta K.

doi:10.1016/0167-7152(89)90130-2

Cited by 28 publications

(16 citation statements)

References 11 publications

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“…Thus, as noted above, the generalized roundness and supremal p-negative type of any given metric space coincide. It therefore follows from the results in [2,13,8] …”

Section: Introduction: Negative Type and Generalized Roundnessmentioning

confidence: 85%

“…The cases q = ∞ and 2 < q < ∞ were settled, respectively, by Misiewicz [13] and Koldobsky [6]: if n ≥ 3 and 2 < q ≤ ∞, then ℓ (n) q is not linearly isometric to any subspace of any L p -space for which 0 < p ≤ 2. In other words, by [2, Theorem 2], ℓ (n) q does not have p-negative type for any p > 0 if n ≥ 3 and 2 < q ≤ ∞.…”

Section: Introduction: Negative Type and Generalized Roundnessmentioning

confidence: 99%

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A direct proof that ℓ∞(3) has generalized roundness zero

Doust

Sánchez

Weston

2015

Expositiones Mathematicae

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Metric spaces of generalized roundness zero have interesting non-embedding properties. For instance, we note that no metric space of generalized roundness zero is isometric to any metric subspace of any L p -space for which 0 < p ≤ 2. Lennard, Tonge and Weston gave an indirect proof that ℓ (3) ∞ has generalized roundness zero by appealing to non-trivial isometric embedding theorems of Bretagnolle, Dacunha-Castelle and Krivine, and Misiewicz. In this paper we give a direct proof that ℓ (3) ∞ has generalized roundness zero. This provides insight into the combinatorial geometry of ℓ (3) ∞ that causes the generalized roundness inequalities to fail. We complete the paper by noting a characterization of real quasi-normed spaces of generalized roundness zero.

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“…Thus, as noted above, the generalized roundness and supremal p-negative type of any given metric space coincide. It therefore follows from the results in [2,13,8] …”

Section: Introduction: Negative Type and Generalized Roundnessmentioning

confidence: 85%

Section: Introduction: Negative Type and Generalized Roundnessmentioning

confidence: 99%

A direct proof that ℓ∞(3) has generalized roundness zero

Doust

Sánchez

Weston

2015

Expositiones Mathematicae

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“…The sufficiency is shown by Fields, Ismail [10]. The necessity can be proved as follows: if x &$ (x 2 +1) &c # M (0, + ) , then it follows from (25) …”

Section: A Criterion Of Positive Definiteness In Terms Of Completely mentioning

confidence: 99%

“…Theorem C is proved by Zastavnyi [41,42,43]. It contains, as particular cases, the results for l p , 2< p (Bretagnolle, Dacunha Castelle, Krivine [4]), C [0, 1] (Einhorn [8]), l n , n 3 (Misiewiez [25]), and some later results for l n p , 2< p< , n 3 (Lisitsky [23]). Definition 1.…”

Section: General Properties Of Positive Definite Functionsmentioning

confidence: 99%

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On Positive Definiteness of Some Functions

Zastavnyi

2000

Journal of Multivariate Analysis

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On the Positive Definiteness of Certain Functions

Maack¹,

Sasvári

1997

Mathematische Nachrichten

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For a coinmutative senugoup (S, +, *) with involution and a function f : S 4 [O, m), the set S ( f ) of those p 2 0 such that f* is a positive definite function on S is a closed subsemigroup of [O, 00) containing 0. For S = (Hi, +, G* = -G) it may happen that S(f) = { kd : k E No } for some d>O,anditmayhappenthatS(f)={O}u[d,m)forsomed>O. I f a > 2 a n d i f S = ( Z , + , n * = -n ) and f ( n ) = e-lnlo or S = (NO, +,no = n ) and f ( n ) = e n U , then S(f) n (0, c) = 0 and [d, 00) C S(f) for some d 2 c > 0 . Although (with c maximal and d minimal) we have not been able to show c = d in all cases, this equality does hold if .S = z and a 2 3.4. In the last section we give sinipler proofs of previously known results concerning the positive definiteness of Ge-llzllo on normed spaces. 1991 Mathematics Subjeci Classification. Primary 42A82, 43A35. A) Given 5' and f what can be said about the semigroup S(f)? B) Given S which subsemigroups S of IR+ have the form S = S(f) with some f ?Our main concern will be with A) in some special cases. The additive semigroup No of nonnegative integers will be endowed with the identical involution n* = n while in the other cases S will be a group and we consider the group involution s+ = -s. As far as we know B) is open even for lN0, H (the additive group of integers) or IR (the additive group of real numbers).We start with the case S = J R and show that semigroups of the formoccur in B). Apart from the cases S = (0) and S = IR,+ we know of no other examples. Note that if S(f) = I R ' then f is c.alled infinitely divisible. Such functions have been thoroughly investigated and play an important role, among others, in probability theory (see e.g. [L]).

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Positive definite norm dependent functions on l∞

Cited by 28 publications

References 11 publications

A direct proof that ℓ∞(3) has generalized roundness zero

A direct proof that ℓ∞(3) has generalized roundness zero

On Positive Definiteness of Some Functions

On the Positive Definiteness of Certain Functions

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