Spectral decomposition of   normal operator in   real Hilbert space

Oreshina, Maria

doi:10.13108/2017-9-4-85

Cited by 6 publications

(12 citation statements)

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“…By Cauchy's Integral Formula, we know that P R (λ) = (λ + a) −1 if λ ∈ Ω R and is zero otherwise [45]. Taking the limit R → ∞, we obtain Using functional calculus for unbounded normal operators we can extend P(λ) to an operator-valued function P(L) [34]. Moreover, one can show that the spectrum of , computed from the IVP (7.9).…”

Section: 1mentioning

confidence: 99%

FEAST for Differential Eigenvalue Problems

Horning¹,

Townsend²

2020

SIAM J. Numer. Anal.

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An operator analogue of the FEAST matrix eigensolver is developed to compute the discrete part of the spectrum of a differential operator in a region of interest in the complex plane. Unbounded search regions are handled with a novel rational filter for the right half-plane. If the differential operator is normal or self-adjoint, then the operator analogue preserves that structure and robustly computes eigenvalues to near machine precision accuracy. The algorithm is particularly adept at computing high-frequency modes of differential operators that possess self-adjoint structure with respect to weighted Hilbert spaces.

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Section: 1mentioning

confidence: 99%

FEAST for Differential Eigenvalue Problems

Horning¹,

Townsend²

2020

SIAM J. Numer. Anal.

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“…Figure 4 shows a typical time–frequency diagram of the group wave in the near field of the blasting. From the time–frequency diagram [21,22], the P-wave frequency range corresponding to each node is determined in the time–frequency domain, which relies on the propagation speed dominance and frequency dominance of the P-wave.…”

Section: The Principle Of High Precision Time-difference Informatimentioning

confidence: 99%

A Method of FPGA-Based Extraction of High-Precision Time-Difference Information and Implementation of Its Hardware Circuit

Yan

et al. 2019

Sensors

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The positioning technology to find shallow underground vibration sources based on a wireless sensor network is receiving great interest in the field of underground position measurements. The slow peaking and strong multi-waveform aliasing typical of the underground vibration signal result in a low extraction accuracy of the time difference and a poor source-positioning accuracy. At the same time, the transmission of large amounts of sensor data and the host computer’s slow data processing speed make locating a source a slow process. To address the above problems, this paper proposes a method for high-precision time-difference measurements in near-field blasting and a method for its hardware implementation. First, based on the broadband that is typical of blast waves, the peak frequency of the P-wave was obtained in the time–frequency domain, taking advantage of the difference in the propagation speed of the P-wave, S-wave, and the surface wave. Second, the phase difference between two sensor nodes was found by means of a spectral decomposition and a correlation measurement. Third, the phase ambiguity was eliminated using the time interval of the first break and the dynamic characteristics of the sensors. Finally, following a top-down design idea, the hardware system was designed using Field Programmable Gate Array(FPGA). Verification, using both numerical simulations and experiments, suggested that compared with generalized cross-correlation-based time-difference measurement methods, the proposed method produced a higher time-difference resolution and accuracy. Compared with the traditional host computer post-position positioning method, the proposed method was significantly quicker. It can be seen that the proposed method provides a new solution for solving high-precision and quick source-location problems, and affords a technical means for developing high-speed, real-time source-location instruments.

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“…For many years, it has been preponderantly used in the framework of complex Hilbert spaces. For some reasons, mainly related to certain consistent implications, spectral decompositions of normal operators were systematically studied only quite recently in the background of real Hilbert spaces (see [8]). A first approach to normal operators in the framework of quaternionic Hilbert spaces apparently goes back to [11], and other investigations concerning their spectral decompositions, in contexts different of that of ours, are presented in [16], and in [3] and its references.…”

Section: Introductionmentioning

confidence: 99%

“…In the present work, we start with an approach to the normal operators in real Hilbert spaces, subsequently trying to apply it to quatrnionic normal operators, regarded as a special class of real normal operators. Although studied in [8], we shall present a new, simplified approach, leading to a more natural formula for the associated spectral measures. For the case of real normal operators, and unlike in [8], we think useful to start with the bounded case, the unbounded one being subsequently approached, using some arguments from the former case.…”

Section: Introductionmentioning

confidence: 99%

“…Although studied in [8], we shall present a new, simplified approach, leading to a more natural formula for the associated spectral measures. For the case of real normal operators, and unlike in [8], we think useful to start with the bounded case, the unbounded one being subsequently approached, using some arguments from the former case. Our main results in this respect are Theorem 1 for the bounded case, and Theorem 2 for the unbounded one.…”

Section: Introductionmentioning

confidence: 99%

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