scholarly journals The Hilbert transform in weighted spaces of integrable vector-valued functions

1987 ◽  
Vol 53 (1) ◽  
pp. 103-108 ◽  
Author(s):  
T. M. Wolniewicz
Keyword(s):  
Hilbert Transform ◽  
Weighted Spaces ◽  
The Hilbert Transform ◽  
Vector Valued Functions ◽  
Vector Valued

scholarly journals On approximation in weighted spaces of continuous vector-valued functions

1987 ◽  
Vol 29 (1) ◽  
pp. 65-68 ◽  
Author(s):  
Liaqat Ali Khan
Keyword(s):  
Continuous Functions ◽  
Weighted Spaces ◽  
Regular Space ◽  
Local Convexity ◽  
Range Space ◽  
Completely Regular ◽  
Spaces Of Continuous Functions ◽  
Vector Valued Functions ◽  
Vector Valued ◽  
Locally Convex

The fundamental work on approximation in weighted spaces of continuous functions on a completely regular space has been done mainly by Nachbin ([5], [6]). Further investigations have been made by Summers [10], Prolla ([7], [8]), and other authors (see the monograph [8] for more references). These authors considered functions with range contained in the scalar field or a locally convex topological vector space. In the present paper we prove some approximation results without local convexity of the range space.

scholarly journals A note on UMD spaces and transference in vector-valued function spaces

1996 ◽  
Vol 39 (3) ◽  
pp. 485-490 ◽  
Author(s):  
N. H. Asmar ◽  
B. P. Kelly ◽  
S. Montgomery-Smith
Keyword(s):  
Banach Space ◽  
Hilbert Transform ◽  
Function Spaces ◽  
Conjugate Functions ◽  
Riesz Theorem ◽  
Umd Spaces ◽  
The Hilbert Transform ◽  
Vector Valued Function ◽  
Vector Valued

A Banach space X is called an HT space if the Hilbert transform is bounded from Lp(X) into Lp(X), where 1 < p < ∞. We introduce the notion of an ACF Banach space, that is, a Banach space X for which we have an abstract M. Riesz Theorem for conjugate functions in Lp(X), 1 < p < ∞. Berkson, Gillespie and Muhly [5] showed that X ∈ HT ⇒ X ∈ ACF. In this note, we will show that X ∈ ACF ⇒ X ∈ UMD, thus providing a new proof of Bourgain's result X ∈ HT ⇒ X ∈ UMD.

The finite Hilbert transform in weighted spaces

1996 ◽  
Vol 126 (6) ◽  
pp. 1157-1167 ◽  
Author(s):  
Kari Astala ◽  
Lassi Päivärinta ◽  
Eero Saksman
Keyword(s):  
Hilbert Transform ◽  
Weighted Spaces ◽  
Logarithmic Kernel ◽  
Finite Hilbert Transform ◽  
Half Axis ◽  
The Hilbert Transform

The mapping properties of the finite Hilbert-transform (respectively the Hilbert transform on the half axis) are studied. Invertibility, surjectivity, injectivity and bounded ness from below of the transform are characterised in general weighted spaces. The results are applied to the restriction of the operator with logarithmic kernel.

Model of multicomponent narrow-band periodical non-stationary random signal

2020 ◽  
Vol 2020 (48) ◽  
pp. 17-24
Author(s):  
I.M. Javorskyj ◽  
◽  
R.M. Yuzefovych ◽  
P.R. Kurapov ◽  
◽  
...  
Keyword(s):  
Time Series ◽  
Real Time ◽  
Hilbert Transform ◽  
Spectral Properties ◽  
Narrow Band ◽  
Analytic Signal ◽  
Random Signal ◽  
Hilbert Transformation ◽  
The Hilbert Transform

The correlation and spectral properties of a multicomponent narrowband periodical non-stationary random signal (PNRS) and its Hilbert transformation are considered. It is shown that multicomponent narrowband PNRS differ from the monocomponent signal. This difference is caused by correlation of the quadratures for the different carrier harmonics. Such features of the analytic signal must be taken into account when we use the Hilbert transform for the analysis of real time series.

A Bayesian View on the Hilbert Transform and the Kramers-Kronig Transform of Electrochemical Impedance Data: Probabilistic Estimates and Quality Scores

2020 ◽  
Author(s):  
Jiapeng Liu ◽  
Ting Hei Wan ◽  
Francesco Ciucci
Keyword(s):  
Power Generation ◽  
Electrochemical Impedance Spectroscopy ◽  
Hilbert Transform ◽  
Electrochemical Impedance ◽  
The Self ◽  
Impedance Data ◽  
The Hilbert Transform ◽  
Validation Tests ◽  
Self Consistency ◽  
Mathematical Relations

<p>Electrochemical impedance spectroscopy (EIS) is one of the most widely used experimental tools in electrochemistry and has applications ranging from energy storage and power generation to medicine. Considering the broad applicability of the EIS technique, it is critical to validate the EIS data against the Hilbert transform (HT) or, equivalently, the Kramers–Kronig relations. These mathematical relations allow one to assess the self-consistency of obtained spectra. However, the use of validation tests is still uncommon. In the present article, we aim at bridging this gap by reformulating the HT under a Bayesian framework. In particular, we developed the Bayesian Hilbert transform (BHT) method that interprets the HT probabilistic. Leveraging the BHT, we proposed several scores that provide quick metrics for the evaluation of the EIS data quality.<br></p>

scholarly journals A Radial Basis Function Finite Difference Scheme for the Benjamin–Ono Equation

Mathematics ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 65
Author(s):  
Benjamin Akers ◽  
Tony Liu ◽  
Jonah Reeger
Keyword(s):  
Radial Basis Function ◽  
Hilbert Transform ◽  
Basis Function ◽  
Differential Operators ◽  
Convergence Rates ◽  
Traveling Wave Solutions ◽  
Algebraic Decay ◽  
Pseudo Differential Operators ◽  
Radial Basis ◽  
The Hilbert Transform

A radial basis function-finite differencing (RBF-FD) scheme was applied to the initial value problem of the Benjamin–Ono equation. The Benjamin–Ono equation has traveling wave solutions with algebraic decay and a nonlocal pseudo-differential operator, the Hilbert transform. When posed on R, the former makes Fourier collocation a poor discretization choice; the latter is challenging for any local method. We develop an RBF-FD approximation of the Hilbert transform, and discuss the challenges of implementing this and other pseudo-differential operators on unstructured grids. Numerical examples, simulation costs, convergence rates, and generalizations of this method are all discussed.

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