Frames for Undergraduates

We present a general formalism for giving a measure space paired with a separable Hilbert space a quantum version based on a normalized positive operator-valued measure. The latter are built from families of density operators labeled by points of the measure space. We especially focus on various probabilistic aspects of these constructions. Simple or more elaborate examples illustrate the procedure: circle, two-sphere, plane and half-plane. Links with Positive-Operator Valued Measure (POVM) quantum measurement and quantum statistical inference are sketched.

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“…which means that the set { √ ν i |x i } is a Parseval frame [13][14][15][16]. Such an identity is possible if N ≥ n;…”

Section: Quantum World From Classical Probabilistic Distribution?mentioning

confidence: 99%

Positive-Operator Valued Measure (POVM) Quantization

Gazeau

Heller

2014

Axioms

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“…It is recognized that, in contrast to classical orthonormal bases (m=n), overcomplete frame based modeling is a much more efficient for imaging 34,37 and results in a better wave field reconstruction accuracy. The sparsity of approximation is characterized by either the l₀ norm ||θ||₀ defined as a number of non-zero components of the vector θ or the l₁ norm ||θ||₁ as the sum of absolute values of items of the vector θ.…”

Section: Reconstruction Formalization: Constrained Maximum Likelihoodmentioning

confidence: 99%

Advanced multi-plane phase retrieval using graphic processing unit: augmented Lagrangian technique with sparse regularization

2012

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In our work we demonstrate a computational method of phase retrieval realized for various propagation models. The effects, arising due to the wave field propagation in an optical setup, lead to significant distortions in measurements; therefore the reconstructed wave fields are noisy and corrupted by different artifacts (e.g. blurring, "waves" on boards, etc.). These disturbances are hard to be specified, but could be suppressed by filtering. The contribution of this paper concerns application of an adaptive sparse approximation of the object phase and amplitude in order to improve imaging. This work is considered as a further development and improvement of the variational phase-retrieval algorithm originated in 1 . It is shown that the sparse regularization enables a better reconstruction quality and substantial enhancement of imaging. Moreover, it is demonstrated that an essential acceleration of the algorithm can be obtained by a commodity graphic processing unit, what is crucial for processing of large images.

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“…For example, if μ is the middlethird Cantor measure, the question of whether of not L 2 (μ) admits Fourier frames has implications for the Kadison-Singer conjecture. For some of these results regarding frames, the reader is referred to [15,17,21,23,32].…”

Section: Overview Of Prior Literaturementioning

confidence: 99%

Families of Spectral Sets for Bernoulli Convolutions

Jørgensen

Kornelson

Shuman

2010

J Fourier Anal Appl

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We study the harmonic analysis of Bernoulli measures μ λ , a one-parameter family of compactly supported Borel probability measures on the real line. The parameter λ is a fixed number in the open interval (0, 1). The measures μ λ may be understood in any one of the following three equivalent ways: as infinite convolution measures of a two-point probability distribution; as the distribution of a random power series; or as an iterated function system (IFS) equilibrium measure determined by the two transformations λ(x ± 1). For a given λ, we consider the harmonic analysis in the sense of Fourier series in the Hilbert space L 2 (μ λ ). For L 2 (μ λ ) to have infinite families of orthogonal complex exponential functions e 2πis(·) , it is known that λ must be a rational number of the form m 2n , where m is odd. We show that L 2 (μ 1 2n ) has a variety of Fourier bases; i.e. orthonormal bases of exponential functions. For some other rational values of λ, we exhibit maximal Fourier families that are not orthonormal bases.

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Frames for Undergraduates

Cited by 107 publications

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Positive-Operator Valued Measure (POVM) Quantization

Positive-Operator Valued Measure (POVM) Quantization

Advanced multi-plane phase retrieval using graphic processing unit: augmented Lagrangian technique with sparse regularization

Families of Spectral Sets for Bernoulli Convolutions

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