Classical Fourier Analysis

“…e.g. [11,Theorem 1.2.19]) is valid as well in the vector-valued setting without any change. C(T d ; X) denotes the space of continuous functions from T d to X, equipped with the uniform norm.…”

Section: Mean Convergencementioning

confidence: 99%

Harmonic Analysis on Quantum Tori

Chen

Yin

2013

Commun. Math. Phys.

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Abstract. This paper is devoted to the study of harmonic analysis on quantum tori. We consider several summation methods on these tori, including the square Fejér means, square and circular Poisson means, and Bochner-Riesz means. We first establish the maximal inequalities for these means, then obtain the corresponding pointwise convergence theorems. In particular, we prove the noncommutative analogue of the classical Stein theorem on Bochner-Riesz means. The second part of the paper deals with Fourier multipliers on quantum tori. We prove that the completely bounded Lp Fourier multipliers on a quantum torus are exactly those on the classical torus of the same dimension. Finally, we present the Littlewood-Paley theory associated with the circular Poisson semigroup on quantum tori. We show that the Hardy spaces in this setting possess the usual properties of Hardy spaces, as one can expect. These include the quantum torus analogue of Fefferman's H 1 -BMO duality theorem and interpolation theorems. Our analysis is based on the recent developments of noncommutative martingale/ergodic inequalities and Littlewood-Paley-Stein theory.

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“…Hence, (b) is implied by this inheritance property and Proposition 3.2.12 of [21] regarding that f is a (λ ′ )-Holder function for any 0 ≤ λ ′ < λ. (C.9)…”

Section: (C7)mentioning

confidence: 83%

Optimal change point detection in Gaussian processes

Keshavarz

Scott

Nguyen

2018

Journal of Statistical Planning and Inference

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We study the problem of detecting a change in the mean of one-dimensional Gaussian process data. This problem is investigated in the setting of increasing domain (customarily employed in time series analysis) and in the setting of fixed domain (typically arising in spatial data analysis). We propose a detection method based on the generalized likelihood ratio test (GLRT), and show that our method achieves nearly asymptotically optimal rate in the minimax sense, in both settings. The salient feature of the proposed method is that it exploits in an efficient way the data dependence captured by the Gaussian process covariance structure. When the covariance is not known, we propose the plug-in GLRT method and derive conditions under which the method remains asymptotically near optimal. By contrast, the standard CUSUM method, which does not account for the covariance structure, is shown to be asymptotically optimal only in the increasing domain. Our algorithms and accompanying theory are applicable to a wide variety of covariance structures, including the Matern class, the powered exponential class, and others. The plug-in GLRT method is shown to perform well for maximum likelihood estimators with a dense covariance matrix.

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Classical Fourier Analysis

Cited by 1,090 publications

References 263 publications

Nonlinear Muckenhoupt–Wheeden type bounds on Reifenberg flat domains, with applications to quasilinear Riccati type equations

Nonlinear Muckenhoupt–Wheeden type bounds on Reifenberg flat domains, with applications to quasilinear Riccati type equations

Harmonic Analysis on Quantum Tori

Optimal change point detection in Gaussian processes

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