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Status:
Available5.0
26 reviews
ISBN 10: 036737563X
ISBN 13: 9780367375638
Author: Willi Freeden, M Zuhair Nashed
Part I: Central Theme
1. From Lattice Point to Shannon-Type Sampling Identities
1.1 Classical Framework of Shannon Sampling
1.2 Transition From Shannon to Shannon-Type Sampling
1.3 Novel Framework of Shannon-Type Sampling
2. Obligations, Ingredients, Achievements, and Innovations
2.1 Obligations and Ingredients
2.2 Achievements and Innovative Results
2.3 Methods and Tools
3. Layout
3.1 Structural Organisation
3.2 Relationship to Other Monographs
Part II: Univariate Poisson-Type Summation Formulas and Shannon-Type Sampling
4. Euler/Poisson-Type Summation Formulas and Shannon-Type Sampling
4.1 Classical Euler Summation Formula
4.2 Variants of the Euler Summation Formula
4.3 Poisson-Type Summation Formula over Finite Intervals
4.4 Shannon Sampling Based on the Poisson Summation-Type Formula
4.5 Shannon-Type Sampling Based on Poisson Summation-Type Formulas
4.6 Fourier Transformed Values–Based Shannon-Type Sampling (Finite Intervals)
4.7 Functional Values–Based Shannon-Type Sampling (Finite Intervals)
4.8 Paley–Wiener Reproducing Kernel Hilbert Spaces
4.9 Poisson-Type Summation Formula over the Euclidean Space
4.10 Functional Values–Based Shannon-Type Sampling (Euclidean Space)
4.11 Fourier Transformed Values–Based Shannon-Type Sampling (Euclidean Space)
Part III: Preparatory Material for Multivariate Lattice Point Summation and Shannon-Type Sampling
5. Preparatory Tools of Vector Analysis
5.1 Cartesian Notation and Settings
5.2 Spherical Notation and Settings
5.3 Regular Regions and Integral Theorems
6. Preparatory Tools of the Theory of Special Functions
6.1 Homogeneous Harmonic Polynomials
6.2 Bessel Functions
6.3 Asymptotic Expansions
7. Preparatory Tools of Lattice Point Theory
7.1 Lattices in Euclidean Spaces
7.2 Figure Lattices in Euclidean Spaces
7.3 Basic Results of the Geometry of Numbers
7.4 Lattice Points Inside Spheres
8. Preparatory Tools of Fourier Analysis
8.1 Stationary Point Asymptotics
8.2 Periodic Polynomials and Fourier Expansions
8.3 Fourier Transform over Euclidean Spaces
8.4 Periodization and Classical Poisson Summation Formula
8.5 Gauss–Weierstrass Transform over Euclidean Spaces
8.6 Hankel Transform and Discontinuous Integrals
Part IV: Multivariate Euler-Type Summation Formulas over Regular Regions
9. Euler–Green Function and Euler-Type Summation Formula
9.1 Euler–Green Function
9.2 Euler-Type Summation Formulas over Regular Regions Based on Euler–Green Functions
9.3 Iterated Euler–Green Function
9.4 Euler-Type Summation Formulas over Regular Regions Based on Iterated Euler–Green Functions
Part V: Bivariate Lattice Point/Ball Summation and Shannon-Type Sampling
10. Hardy–Landau-Type Lattice Point Identities (Constant Weight)
10.1 Integral Mean Asymptotics for the Euler–Green Function
10.2 Hardy–Landau-Type Identity
10.3 Discrepancy Asymptotics
11. Hardy–Landau-Type Lattice Point Identities (General Weights)
11.1 Pointwise Fourier Inversion Formula for Regular Regions
11.2 General Geometry and Homogeneous Boundary Weight
11.3 Circles and General Weights
11.4 Smooth Convex Regions and General Weights
12. Bandlimited Shannon-Type Sampling (Preparatory Results)
12.1 From Hardy–Landau-Type Identities to Shannon-Type Sampling
12.2 Over- and Undersampling
13. Lattice Ball Euler Summation Formulas and Shannon-Type Sampling
13.1 Lattice Ball Euler–Green Function
13.2 Lattice Ball Euler Summation Formula
13.3 Lattice Ball Mean Shannon-Type Sampling
Part VI: Multivariate Poisson-Type Summation Formulas over Regular Regions
14. Gauss–Weierstrass Mean Euler-Type Summation Formulas and Shannon-Type Sampling
14.1 Gauss–Weierstrass Transform over Regular Regions
14.2 Gauss–Weierstrass Mean Euler–Green Function
14.3 Gauss–Weierstrass Mean Euler Summation over Regular Regions
14.4 Bandlimited Gauss–Weierstrass Shannon-Type Sampling
15. From Gauss–Weierstrass to Ordinary Lattice Point Poisson–Type Summation
15.1 Theta Function and Functional Equation
15.2 Poisson-Type Summation over Regular Regions (Gauss–Weierstrass Approach)
15.3 Poisson-Type Summation over Regular Regions (Ordinary Approach)
Part VII: Multivariate Shannon-Type Sampling Formulas over Regular Regions
16. Shannon-Type Sampling Based on Poisson-Type Summation Formulas
16.1 Fourier-Transformed Values–Based Shannon-Type Sampling (Gauss–Weierstrass Approach)
16.2 Parseval-Type Identity (Gaussian/Ordinary Approach)
16.3 Fourier-Transformed Values–Based Shannon-Type Sampling (Ordinary Approach)
16.4 Functional Values–Based Shannon-Type Sampling (Gaussian Approach)
17. Paley–Wiener Space Framework and Spline Approximation
17.1 Paley–Wiener Reproducing Kernel Structure
17.2 Spline Interpolation in Paley–Wiener Spaces
17.3 Paley–Wiener Spline Interpolatory Sampling
17.4 Paley–Wiener Spline Interpolatory Cubature
17.5 Multivariate Antenna Problem
Part VIII: Multivariate Poisson-Type Summation Formulas over Euclidean Spaces
18. Poisson-Type Summation Formulas over Euclidean Spaces
18.1 Integral Means for Iterated Euler–Green Functions
18.2 Euler-Type Summation Formula over Increasing Balls Involving Euler–Green Functions
18.3 Spherically-Reflected Convergence Criteria (Basic Differentiability Order)
18.4 Poisson-Type Summation Formula (Heuristic Approach)
18.5 Euler-Type Summation Formula over Increasing Balls Involving Iterated Euler–Green Functions
18.6 Spherically-Reflected Convergence Criteria (Higher Differentiability Orders)
18.7 Poisson-Type Summation Formula (Rigorous Approach)
18.8 Hardy–Landau-Type Identities (Spherical Harmonic Weights)
Part IX: Multivariate Shannon-Type Sampling Formulas over Euclidean Spaces
19. Shannon-Type Sampling Based on Poisson-Type Summation Formulas over Euclidean Spaces
19.1 Functional Values–Based Shannon-Type Sampling
19.2 Paley–Wiener Reproducing Kernel Structure
19.3 Fourier Transformed Values–Based Shannon-Type Sampling
19.4 Shannon-Type Sampling Involving Dilated Fundamental Cells
19.5 Bivariate Locally-Supported Sampling Functions
19.6 From Gaussian to Ordinary Non-Bandlimited Shannon-Type Sampling
Part X: Conclusion
20. Trends, Progress, and Perspectives
20.1 Trendsetting Extensions of Shannon Sampling
20.2 Methodological Progress in Sampling
20.3 Bridging Role of Sampling in Recovery Problems
20.4 SampTA Conference Series
lattice points examples
lattice point geometry
lattice signatures and bimodal gaussians
lattice points chemistry
example of lattice points
Tags: Willi Freeden, M Zuhair Nashed, Lattice Point, Shannon Type
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