Simultaneous Interpolation and Norm-Preservation

Complex Approximation and Simultaneous Interpolation on Closed Sets

Canadian Journal of Mathematics ◽

10.4153/cjm-1977-074-1 ◽

1977 ◽

Vol 29 (4) ◽

pp. 701-706 ◽

Cited By ~ 8

Author(s):

P. M. Gauthier ◽

W. Hengartner

Keyword(s):

Analytic Function ◽

Complex Plane ◽

Closed Subset ◽

Finite Complex ◽

Complex Approximation ◽

Closed Sets ◽

Limit Points ◽

Simultaneous Interpolation ◽

Complex Valued

Let ƒ be a complex-valued function denned on a closed subset F of the finite complex plane C, and let {Zn} be a sequence on F without limit points. We wish to find an analytic function g which simultaneously approximates ƒ uniformly on F and interpolates ƒ at the points {Zn}.

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Five-dimensional interpolation: Recovering from acquisition constraints

Geophysics ◽

10.1190/1.3245216 ◽

2009 ◽

Vol 74 (6) ◽

pp. V123-V132 ◽

Cited By ~ 143

Author(s):

Daniel Trad

Keyword(s):

Seismic Data ◽

Original Data ◽

Spatial Spectrum ◽

Optimal Interpolation ◽

Multidimensional Data ◽

Inversion Algorithm ◽

Five Dimensions ◽

Fourier Reconstruction ◽

Simultaneous Interpolation ◽

Seismic Amplitudes

Although 3D seismic data are being acquired in larger volumes than ever before, the spatial sampling of these volumes is not always adequate for certain seismic processes. This is especially true of marine and land wide-azimuth acquisitions, leading to the development of multidimensional data interpolation techniques. Simultaneous interpolation in all five seismic data dimensions (inline, crossline, offset, azimuth, and frequency) has great utility in predicting missing data with correct amplitude and phase variations. Although there are many techniques that can be implemented in five dimensions, this study focused on sparse Fourier reconstruction. The success of Fourier interpolation methods depends largely on two factors: (1) having efficient Fourier transform operators that permit the use of large multidimensional data windows and (2) constraining the spatial spectrum along dimensions where seismic amplitudes change slowly so that the sparseness and band limitation assumptions remain valid. Fourier reconstruction can be performed when enforcing a sparseness constraint on the 4D spatial spectrum obtained from frequency slices of five-dimensional windows. Binning spatial positions into a fine 4D grid facilitates the use of the FFT, which helps on the convergence of the inversion algorithm. This improves the results and computational efficiency. The 5D interpolation can successfully interpolate sparse data, improve AVO analysis, and reduce migration artifacts. Target geometries for optimal interpolation and regularization of land data can be classified in terms of whether they preserve the original data and whether they are designed to achieve surface or subsurface consistency.

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Simultaneous Interpolation and Approximation by a Class of Multivariate Positive Operators

Numerical Algorithms ◽

10.1023/b:numa.0000005359.72118.b6 ◽

2003 ◽

Vol 34 (2-4) ◽

pp. 147-158 ◽

Cited By ~ 22

Author(s):

Giampietro Allasia

Keyword(s):

Positive Operators ◽

Simultaneous Interpolation

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The Simultaneous Interpolation of Target Radar Cross Section in Both the Spatial and Frequency Domains by Means of Legendre Wavelets Model-Based Parameter Estimation

International Journal of Aerospace Engineering ◽

10.1155/2015/543787 ◽

2015 ◽

Vol 2015 ◽

pp. 1-12

Author(s):

Yongqiang Yang ◽

Yunpeng Ma ◽

Lifeng Wang

Keyword(s):

Parameter Estimation ◽

Cross Section ◽

Target Identification ◽

Radar Cross Section ◽

Legendre Wavelets ◽

Model Based ◽

Function Fitting ◽

Frequency Domains ◽

Spatial Domains ◽

Simultaneous Interpolation

The understanding of the target radar cross section (RCS) is significant for target identification and for radar designing and optimization. In this paper, a numerical algorithm for calculating target RCS is presented which is based on Legendre wavelet model-based parameter estimation (LW-MBPE). The Padé rational function fitting model applied for MBPE in the frequency domain is enhanced to include spatial dependence on the numerator and denominator coefficients. This allows the function to interpolate target RCS in both the frequency and spatial domains simultaneously. The combination of Legendre wavelets guarantees the convergence of the algorithm. The method is convergent by increasing the sampling frequency and spatial points. Numerical results are provided to demonstrate the validity and applicability of the new technique.

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