A sampling theorem on homogeneous manifolds

Pesenson, Isaac Z.

doi:10.1090/s0002-9947-00-02592-7

Cited by 102 publications

(111 citation statements)

References 10 publications

(13 reference statements)

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“…Very interesting results were obtained by Pesenson, see, e.g., [16], who considered, in particular, the case of homogeneous manifolds. In [16], reconstruction works differently, however, namely by approaching the solution iteratively in a Sobolev space setting. Useful methods should also be available from the field of spectral geometry, see, e.g., [17], which studies the close relationship between the properties of a manifold and the spectrum of its Laplacian and, in particular, from the field of noncommutative geometry and the techniques based on the spectral triple, see [18].…”

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confidence: 87%

Covariant Information-Density Cutoff in Curved Space-Time

Kempf

2004

Phys. Rev. Lett.

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In information theory, the link between continuous information and discrete information is established through well-known sampling theorems. Sampling theory explains, for example, how frequency-filtered music signals are reconstructible perfectly from discrete samples. In this Letter, sampling theory is generalized to pseudo-Riemannian manifolds. This provides a new set of mathematical tools for the study of space-time at the Planck scale: theories formulated on a differentiable space-time manifold can be completely equivalent to lattice theories. There is a close connection to generalized uncertainty relations which have appeared in string theory and other studies of quantum gravity.PACS numbers: 04.60.-m, 03.67.-a, 02.90.+p It is generally assumed that the notion of distance loses operational meaning at the Planck scale, l P ≈ 10 −35 m (assuming 3+1 dimensions), due to the combined effects of general relativity and quantum theory. Namely, if one tried to resolve a spatial region with an uncertainty of less than a Planck length, then the corresponding momentum uncertainty should randomly curve and thereby significantly disturb the very region in space that was meant to be resolved. It is expected, therefore, that the existence of a smallest possible length, area or volume, at the Planck scale or above, plays a central role in the yet-to-be-found theory of quantum gravity.In the literature, no consensus has been reached as to whether this implies that space-time is discrete. On the one hand, quantization literally means discretization, and space-time discreteness is indeed naturally accommodated within the functional analytic framework of quantum theory, see, e.g., [1]. Also, most interacting quantum field theories are mathematically well-defined only on lattices. On the other hand, within the mathematical framework of general relativity, space-time is naturally described as a differentiable manifold and deep principles such as local Lorentz invariance would appear to be violated if space-time were discrete.There is the possibility that the cardinality of spacetime is between discrete and continuous, but it is strongly restricted by results of Gödel and Cohen. In [2], they proved that both are consistent with conventional (ZF) set theory: to adopt an axiom claiming the existence of sets with intermediate cardinality or to adopt an axiom claiming their non-existence. Therefore, it is not possible to explicitly construct any set of cardinality between discrete and continuous infinity from the axioms of conventional set theory. If space-time is describable as a set and if this set is of intermediate cardinality, then its description cannot be constructive and requires mathematics beyond conventional set theory.In this Letter, we consider a simpler possibility. In a concrete sense, space-time could be simultaneously discrete and continuous. Namely, in the simplest case, physical fields could be differentiable functions which possess merely a finite density of degrees of freedom. If such a field's amplitude is sam...

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confidence: 87%

Covariant Information-Density Cutoff in Curved Space-Time

Kempf

2004

Phys. Rev. Lett.

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“…On the level of abstract Hilbert spaces it was done by Pesenson in [67,70]. Moreover, he specified this abstract setup in a number of very important situations such as: Riemannian compact and noncompact manifolds of bounded geometry [69], stratified Lie groups [68], combinatorial and quantum graphs [71,72].…”

Section: Sampling In Hilbert Spaces On Riemannian Manifolds and Graphsmentioning

confidence: 99%

Seven pivotal theorems of Fourier analysis, signal analysis, numerical analysis and number theory: their interconnections

Butzer¹,

Dodson²,

Ferreira³

et al. 2014

Bull. Math. Sci.

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The present paper deals mainly with seven fundamental theorems of mathematical analysis, numerical analysis, and number theory, namely the generalized Parseval decomposition formula (GPDF), introduced 15 years ago, the well-known approximate sampling theorem (ASF), the new approximate reproducing kernel theorem, the basic Poisson summation formula, already known to Gauß, a newer version of the GPDF having a structure similar to that of the Poisson summation formula, namely, the Parseval decomposition-Poisson summation formula, the functional equation of Riemann's zeta function, as well as the Euler-Maclaurin summation formula. It will in fact be shown that these seven theorems are all equivalent to one another, in the sense that each is a corollary of the others. Since these theorems can all be deduced from each other, one of them has to be proven independently in order to verify all. It is convenient to choose the ASF, introduced in 1963. The epilogue treats possible extensions to the more general contexts of reproducing kernel theory and of abstract harmonic analysis, using locally compact abelian groups. This paper is expository in the sense that it treats a number of mathematical theorems, their interconnections, their equivalence to one another. On the other hand, the proofs of the many intricate interconnections among these theorems are new in their essential steps and conclusions.

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“…Sufficient sampling of a continuous signal ensures that the signal can be reconstructed from the samples without loss of relevant information. Theorems that dictate the necessary conditions for accurate signal reconstruction, such as the number of samples, are essential for guiding data acquisition (e.g., Shannon 1949;Pesenson 2000;McEwen & Wiaux 2011). The best known of these theorems is the Shannon-Nyquist sampling theorem of bandlimited functions (Shannon, 1949;Luke, 1999;Unser, 2000), which connects continuous 1-D signals and their discrete representations.…”

Section: Introductionmentioning

confidence: 99%

Spatial sampling of MEG and EEG revisited: From spatial-frequency spectra to model-informed sampling

Iivanainen,

Mäkinen,

Zetter

et al. 2020

Preprint

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In this paper, we analyze spatial sampling of electro-(EEG) magnetoencephalography (MEG), where the electric or magnetic field is typically sampled on a curved surface such as the scalp. Using simulated measurements, we study the spatial-frequency content in EEG as well as in on-and off-scalp MEG. The analysis suggests that on-scalp MEG would generally benefit from three times more samples than EEG or off-scalp MEG. Based on the theory of Gaussian processes and experimental design, we suggest an approach to obtain sampling locations on surfaces that are optimal with respect to prior assumptions. Additionally, the approach allows to control, e.g., the uniformity of the sampling locations in the grid. By simulating the performance of grids constructed with different priors, we show that for a low number of spatial samples, model-informed non-uniform sampling can be beneficial. For a large number of samples, uniform sampling grids yield nearly the same total information as the modelinformed grids.

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A sampling theorem on homogeneous manifolds

Cited by 102 publications

References 10 publications

Covariant Information-Density Cutoff in Curved Space-Time

Covariant Information-Density Cutoff in Curved Space-Time

Seven pivotal theorems of Fourier analysis, signal analysis, numerical analysis and number theory: their interconnections

Spatial sampling of MEG and EEG revisited: From spatial-frequency spectra to model-informed sampling

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