scholarly journals Non-spectral fractal measures with Fourier frames

2017 ◽  
Vol 4 (3) ◽  
pp. 305-327 ◽  
Author(s):  
Chun-Kit Lai ◽  
Yang Wang
Keyword(s):  
Fourier Frames ◽  
Fractal Measures

scholarly journals Weighted Fourier frames on fractal measures

2016 ◽  
Vol 444 (2) ◽  
pp. 1603-1625 ◽  
Author(s):  
Dorin Ervin Dutkay ◽  
Rajitha Ranasinghe
Keyword(s):  
Fourier Frames ◽  
Fractal Measures

scholarly journals Continuous and discrete Fourier frames for fractal measures

2013 ◽  
Vol 366 (3) ◽  
pp. 1213-1235 ◽  
Author(s):  
Dorin Ervin Dutkay ◽  
Deguang Han ◽  
Eric Weber
Keyword(s):  
Fourier Frames ◽  
Fractal Measures

scholarly journals EEG Fractal Analysis Reflects Brain Impairment after Stroke

Entropy ◽  
2021 ◽  
Vol 23 (5) ◽  
pp. 592
Author(s):  
Maria Rubega ◽  
Emanuela Formaggio ◽  
Franco Molteni ◽  
Eleonora Guanziroli ◽  
Roberto Di Marco ◽  
...  
Keyword(s):  
Early Stage ◽  
Brain Activity ◽  
Phase 1 ◽  
Behavioral Changes ◽  
Stroke Survivors ◽  
Eeg Signals ◽  
Novel Treatments ◽  
Fractal Measures ◽  
Underlying Mechanisms ◽  
Random Fluctuations

Stroke is the commonest cause of disability. Novel treatments require an improved understanding of the underlying mechanisms of recovery. Fractal approaches have demonstrated that a single metric can describe the complexity of seemingly random fluctuations of physiological signals. We hypothesize that fractal algorithms applied to electroencephalographic (EEG) signals may track brain impairment after stroke. Sixteen stroke survivors were studied in the hyperacute (<48 h) and in the acute phase (∼1 week after stroke), and 35 stroke survivors during the early subacute phase (from 8 days to 32 days and after ∼2 months after stroke): We compared resting-state EEG fractal changes using fractal measures (i.e., Higuchi Index, Tortuosity) with 11 healthy controls. Both Higuchi index and Tortuosity values were significantly lower after a stroke throughout the acute and early subacute stage compared to healthy subjects, reflecting a brain activity which is significantly less complex. These indices may be promising metrics to track behavioral changes in the very early stage after stroke. Our findings might contribute to the neurorehabilitation quest in identifying reliable biomarkers for a better tailoring of rehabilitation pathways.

A sharper stability bound of Fourier frames

1999 ◽  
Vol 5 (1) ◽  
pp. 67-71
Author(s):  
Weifeng Su ◽  
Xingwei Zhou
Keyword(s):  
Stability Bound ◽  
Fourier Frames

Fractal measures and diffusion as results of learning in neural networks

1993 ◽  
Vol 174 (4) ◽  
pp. 293-297 ◽  
Author(s):  
G. Radons ◽  
H.G. Schuster ◽  
D. Werner
Keyword(s):  
Neural Networks ◽  
Fractal Measures ◽  
And Diffusion

Fractal Measures

2002 ◽  
pp. 1037-1054 ◽  
Author(s):  
J. Falconer Kenneth
Keyword(s):  
Fractal Measures

scholarly journals Fourier Frames for Surface-Carried Measures

2020 ◽  
Author(s):  
Alex Iosevich ◽  
Chun-Kit Lai ◽  
Bochen Liu ◽  
Emmett Wyman
Keyword(s):  
Compact Manifold ◽  
Measure Zero ◽  
Orthogonal Basis ◽  
Surface Measure ◽  
Locally Compact ◽  
Locally Compact Abelian Groups ◽  
Fourier Frames ◽  
Positive Gaussian Curvature ◽  
Compact Abelian Groups ◽  
Fourier Frame

Abstract In this paper, we show that the surface measure on the boundary of a convex body of everywhere positive Gaussian curvature does not admit a Fourier frame. This answers a question proposed by Lev and provides the 1st example of a uniformly distributed measure supported on a set of Lebesgue measure zero that does not admit a Fourier frame. In contrast, we show that the surface measure on the boundary of a polytope always admits a Fourier frame. We also explore orthogonal bases and frames adopted to sets under consideration. More precisely, given a compact manifold $M$ without a boundary and $D \subset M$, we ask whether $L^2(D)$ possesses an orthogonal basis of eigenfunctions. The non-abelian nature of this problem, in general, puts it outside the realm of the previously explored questions about the existence of bases of characters for subsets of locally compact abelian groups. This paper is dedicated to Alexander Olevskii on the occasion of his birthday. Olevskii’s mathematical depth and personal kindness serve as a major source of inspiration for us and many others in the field of mathematics.

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