L 2-average Decay of the Fourier Transform of a Characteristic Function of a Convex Set

2014 ◽  
pp. 129-135
Author(s):  
Alex Iosevich ◽  
Elijah Liflyand
Keyword(s):  
Fourier Transform ◽  
Characteristic Function ◽  
Convex Set ◽  
The Fourier Transform

The strong sweeping out property for lacunary sequences, Riemann sums, convolution powers, and related matters

1996 ◽  
Vol 16 (2) ◽  
pp. 207-253 ◽  
Author(s):  
Mustafa Akcoglu ◽  
Alexandra Bellow ◽  
Roger L. Jones ◽  
Viktor Losert ◽  
Karin Reinhold-Larsson ◽  
...  
Keyword(s):  
Fourier Transform ◽  
Characteristic Function ◽  
Ergodic Averages ◽  
Riemann Sums ◽  
Lacunary Sequences ◽  
Linearly Independent ◽  
Convolution Powers ◽  
The Fourier Transform

AbstractIn this paper we establish conditions on a sequence of operators which imply divergence. In fact, we give conditions which imply that we can find a set B of measure as close to zero as we like, but such that the operators applied to the characteristic function of this set have a lim sup equal to 1 and a lim inf equal to 0 a.e. (strong sweeping out). The results include the fact that ergodic averages along lacunary sequences, certain convolution powers, and the Riemann sums considered by Rudin are all strong sweeping out. One of the criteria for strong sweeping out involves a condition on the Fourier transform of the sequence of measures, which is often easily checked. The second criterion for strong sweeping out involves showing that a sequence of numbers satisfies a property similar to the conclusion of Kronecker's lemma on sequences linearly independent over the rationals.

Efficient variations of the Fourier transform in applications to option pricing

2014 ◽  
Vol 18 (2) ◽  
pp. 57-90 ◽  
Author(s):  
Svetlana Boyarchenko ◽  
Sergei Levendorski˘ı
Keyword(s):  
Fourier Transform ◽  
Option Pricing ◽  
The Fourier Transform

scholarly journals Strain Sensitivity Enhancement of Broadband Ultrasonic Signals in Plates Using Spectral Phase Filtering

2021 ◽  
Vol 11 (6) ◽  
pp. 2582
Author(s):  
Lucas M. Martinho ◽  
Alan C. Kubrusly ◽  
Nicolás Pérez ◽  
Jean Pierre von der Weid
Keyword(s):  
Fourier Transform ◽  
Guided Waves ◽  
Strain Level ◽  
Energy Concentration ◽  
Short Time Fourier Transform ◽  
Time Frequency ◽  
Frequency Representation ◽  
The Fourier Transform ◽  
Short Time ◽  
Sensitivity Gain

The focused signal obtained by the time-reversal or the cross-correlation techniques of ultrasonic guided waves in plates changes when the medium is subject to strain, which can be used to monitor the medium strain level. In this paper, the sensitivity to strain of cross-correlated signals is enhanced by a post-processing filtering procedure aiming to preserve only strain-sensitive spectrum components. Two different strategies were adopted, based on the phase of either the Fourier transform or the short-time Fourier transform. Both use prior knowledge of the system impulse response at some strain level. The technique was evaluated in an aluminum plate, effectively providing up to twice higher sensitivity to strain. The sensitivity increase depends on a phase threshold parameter used in the filtering process. Its performance was assessed based on the sensitivity gain, the loss of energy concentration capability, and the value of the foreknown strain. Signals synthesized with the time–frequency representation, through the short-time Fourier transform, provided a better tradeoff between sensitivity gain and loss of energy concentration.

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