Random Fourier Series with Applications to Harmonic Analysis. (AM-101)

Marcus, Michael B.; Pisier, Gilles

doi:10.1515/9781400881536

Cited by 104 publications

(141 citation statements)

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“…So we give in the appendix a quicker proof of a result that suffices for our needs (where 2 √ n − 1 is replaced by C ′ √ n, C ′ being a numerical constant) and which in several respects gives us better estimates than Lemma 1.8. Lemma 1.9 below can be viewed as a non-commutative variant of results in [24] (see also [23] where the non-commutative case is already considered) in the style of [18] (see also [7,21]). We view this as a (weak) sort of non-commutative Sauer lemma, that it might be worthwhile to strengthen.…”

Section: Quantum Expandersmentioning

confidence: 99%

Quantum expanders and geometry of operator spaces

Pisier¹

2014

J. Eur. Math. Soc.

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We show that there are well separated families of quantum expanders with asymptotically the maximal cardinality allowed by a known upper bound. This has applications to the "growth" of certain operator spaces: It implies asymptotically sharp estimates for the growth of the multiplicity of M N -spaces needed to represent (up to a constant C > 1) the M N -version of the n-dimensional operator Hilbert space OH n as a direct sum of copies of M N . We show that, when C is close to 1, this multiplicity grows as exp βnN 2 for some constant β > 0. The main idea is to relate quantum expanders with "smooth" points on the matricial analogue of the Euclidean unit sphere. This generalizes to operator spaces a classical geometric result on n-dimensional Hilbert space (corresponding to N=1). In an appendix, we give a quick proof of an inequality (related to Hastings's previous work) on random unitary matrices that is crucial for this paper.The term "Quantum Expander" is used by Hastings in [11] and by Ben-Aroya and Ta-Shma in [2] to designate a sequencen ) of N × N unitary matrices such that there is an ε > 0 satisfying the following "spectral gap" condition:where . 2 denotes the Hilbert-Schmidt norm on M N . More generally, the term is extended to the case when this is only defined for infinitely many N 's, and also to n-tuples of matrices satisfying merely= nI. We will say that an n-tuple U (N ) satisfying (0.1) is a ε-quantum expander. We refer the reader to the survey [3] for more information and references on quantum expanders.In analogy with the classical expanders (see below), one seeks to exhibit (and hopefully to construct explicitly ) sequences {U (Nm) | m ≥ 1} of n-tuples of N m × N m unitary matrices that are ε-quantum expanders with N m → ∞ while n and ε > 0 remain fixed.When G is a finite group generated by S = {t 1 , · · · , t n } the associated Cayley graph G(G, S) is said to have a spectral gap if the left regular representation λ G satisfies (0.2) λ G (t j ) |I ⊥ < n(1 − ε) * Partially supported by ANR-2011-BS01-008-01.

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Section: Quantum Expandersmentioning

confidence: 99%

Quantum expanders and geometry of operator spaces

Pisier¹

2014

J. Eur. Math. Soc.

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“…by the entries of a quantized Rademacher system R, then we obtain an operator space R p (E) which is Banach isomorphic but not completely isomorphic to R q (E) whenever 1 ≤ p = q < ∞. This equivalence of norms, which fails to be complete, follows from a version of the Khinchin-Kahane inequalities for R stated in [10]. Therefore, in contrast with (1), each choice of the exponent 1 ≤ q < ∞ in Definition 2.4 gives a different notion of Rademacher type and cotype.…”

Section: The Kwapień Theorem For Operator Spacesmentioning

confidence: 99%

Quantized orthonormal systems: A non-commutative Kwapień theorem

Garcı́a-Cuerva

Parcet

2003

Studia Math.

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Abstract. The concepts of Riesz type and cotype of a given Banach space are extended to a non-commutative setting. First, the Banach space is replaced by an operator space. The notion of quantized orthonormal system, which plays the role of an orthonormal system in the classical setting, is then defined. The Fourier type and cotype of an operator space with respect to a non-commutative compact group fit in this context. Also, the quantized analogs of Rademacher and Gaussian systems are treated. All this is used to obtain an operator space version of the classical theorem of Kwapień characterizing Hilbert spaces by means of vector-valued orthogonal series. Several approaches to this result with different consequences are given.

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“…See for example [1,2,12,16,19], as well as the references therein. With this in mind, we present an abstract probability estimate for general random fields.…”

Section: Probabilistic Estimatesmentioning

confidence: 99%

Probabilistic and numerical validation of homology computations for nodal domains

Day

Kalies

Mischaikow

et al. 2007

Electron. Res. Announc. Amer. Math. Soc.

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Abstract. Homology has long been accepted as an important computable tool for quantifying complex structures. In many applications these structures arise as nodal domains of real-valued functions and are therefore amenable only to a numerical study, based on suitable discretizations. Such an approach immediately raises the question of how accurate the resulting homology computations are. In this paper we present a probabilistic approach to quantifying the validity of homology computations for nodal domains of random Fourier series in one and two space dimensions, which furnishes explicit probabilistic apriori bounds for the suitability of certain discretization sizes. In addition, we introduce a numerical method for verifying the homology computation using interval arithmetic.

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Random Fourier Series with Applications to Harmonic Analysis. (AM-101)

Cited by 104 publications

References 0 publications

Quantum expanders and geometry of operator spaces

Quantum expanders and geometry of operator spaces

Quantized orthonormal systems: A non-commutative Kwapień theorem

Probabilistic and numerical validation of homology computations for nodal domains

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