Approximation based on elliptic splines

2014 ◽  
Vol 12 (05) ◽  
pp. 1461014 ◽  
Author(s):  
Zhuyuan Yang ◽  
Zongwen Yang
Keyword(s):  
Sobolev Spaces ◽  
Besov Spaces ◽  
Fourier Transforms ◽  
Projection Operators ◽  
Order Of Approximation ◽  
B Splines ◽  
Interpolation Operators ◽  
Polynomial Reproducing

In this paper, we study the elliptic splines which include the well-known polyharmonic B-splines. We analyze their Fourier transforms, decay behaviors and polynomial reproducing properties. We also study the order of approximation in Sobolev spaces and consider their characterizations of Besov spaces by the scale projection operators, quasi-interpolation operators and wavelet operators.

A lithospheric velocity anomaly beneath the Shagan river Test Site. Part 2. Imaging and inversion with amplitude transmission tomography

1992 ◽  
Vol 82 (2) ◽  
pp. 999-1017
Author(s):  
K. L. McLaughlin ◽  
J. R. Murphy ◽  
B. W. Barker
Keyword(s):  
Fourier Transforms ◽  
Test Site ◽  
Small Scale ◽  
Projection Operators ◽  
Order Partial Differential Equation ◽  
Linear Inversion ◽  
Discrete Fourier Transforms ◽  
Velocity Anomaly ◽  
Inversion Procedure ◽  
River Test

Abstract A linear inversion procedure is introduced that images weak velocity anomalies using amplitudes of transmitted seismic waves. Using projection operators from geometrical ray theory, an image of an anomaly is constructed from amplitudes recorded at arrays of receivers using arrays of sources. The image is related to the velocity anomaly by a second-order partial-differential equation that is inverted using 2-D discrete Fourier transforms. As an example of the inversion procedure, magnitude residuals for European stations recording Shagan River explosions are used to image the deep lithospheric anomaly beneath the Shagan River test site described in Part 1. This formal inversion analysis confirms the existence of a small-scale lateral heterogeneity located 50 km west-northwest of the test site at a probable depth between 80 and 100 km and indicates that it is consistent with a deterministic 1.5% peak-to-peak (or 0.5% rms) velocity anomaly with a scale length of about 3 km. 3-D dynamic raytracing is then used to verify that the inferred laterally varying structure produces amplitude fluctuations consistent with observations.

scholarly journals Traces of Besov, Triebel-Lizorkin and Sobolev Spaces on Metric Spaces

2017 ◽  
Vol 5 (1) ◽  
pp. 98-115 ◽  
Author(s):  
Eero Saksman ◽  
Tomás Soto
Keyword(s):  
Sobolev Space ◽  
Sobolev Spaces ◽  
Metric Space ◽  
Besov Spaces ◽  
Metric Spaces ◽  
Function Spaces ◽  
First Order ◽  
Trace Theorems ◽  
Euclidean Setting ◽  
Regular Subset

Abstract We establish trace theorems for function spaces defined on general Ahlfors regular metric spaces Z. The results cover the Triebel-Lizorkin spaces and the Besov spaces for smoothness indices s < 1, as well as the first order Hajłasz-Sobolev space M1,p(Z). They generalize the classical results from the Euclidean setting, since the traces of these function spaces onto any closed Ahlfors regular subset F ⊂ Z are Besov spaces defined intrinsically on F. Our method employs the definitions of the function spaces via hyperbolic fillings of the underlying metric space.

Interpolation operators in Orlicz–Sobolev spaces

2007 ◽  
Vol 107 (1) ◽  
pp. 107-129 ◽  
Author(s):  
L. Diening ◽  
M. Růžička
Keyword(s):  
Sobolev Spaces ◽  
Interpolation Operators
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