Function integration, reconstruction and approximation using rank-$1$ lattices
Copyright policy )Function integration, reconstruction and approximation using rank-$1$ lattices
We consider rank-1 lattices for integration and reconstruction of functions with series expansion supported on a finite index set. We explore the connection between the periodic Fourier space and the non-periodic cosine space and Chebyshev space, via tent transform and then cosine transform, to transfer known results from the periodic setting into new insights for the non-periodic settings. Fast discrete cosine transform can be applied for the reconstruction phase. To reduce the size of the auxiliary index set in the associated component-by-component (CBC) construction for the lattice generating vectors, we work with a bi-orthonormal set of basis functions, leading to three methods for function reconstruction in the non-periodic settings. We provide new theory and efficient algorithmic strategies for the CBC construction. We also interpret our results in the context of general function approximation and discrete least-squares approximation.
- KU Leuven Belgium
- École Polytechnique Fédérale de Lausanne EPFL Switzerland
- University of Paris France
- French National Centre for Scientific Research France
- University of Copenhagen Faculty of Science Denmark
Trigonometric approximation, Rank-1 lattice points, Mathematics, Applied, Multidimensional problems, exact integration and approximation on finite index sets, component-by-component construction, Numerical & Computational Mathematics, quasi-Monte Carlo methods, SPACES, [MATH] Mathematics [math], 41A10 (Approximation by polynomials), Cosine space, Numerical quadrature and cubature formulas, cosine space, Approximation by polynomials, 41A10, 42A10, 41A63, 42B05, 65D30, 65D32, 65D15, L-INFINITY APPROXIMATION, TRIGONOMETRIC POLYNOMIALS, 41A63 (Multidimensional problems), 0102 Applied Mathematics, FOS: Mathematics, 4901 Applied mathematics, 65D30 (Numerical integration), Mathematics - Numerical Analysis, 0802 Computation Theory and Mathematics, Quasi-Monte Carlo methods, Componentby-component construction, Science & Technology, 0103 Numerical and Computational Mathematics, ALGORITHMS, rank-\(1\) lattice points, Fourier space, 65D15 (Algorithms for functional approximation), Numerical Analysis (math.NA), 42A10 (Trigonometric approximation), COLLOCATION, 65D32 (Quadrature and cubature formulas), Exact integration and approximation on finite index sets, Algorithms for approximation of functions, Physical Sciences, 4903 Numerical and computational mathematics, Numerical integration, RULES, 42B05 (Fourier series and coefficients), Chebyshev space, Mathematics
Trigonometric approximation, Rank-1 lattice points, Mathematics, Applied, Multidimensional problems, exact integration and approximation on finite index sets, component-by-component construction, Numerical & Computational Mathematics, quasi-Monte Carlo methods, SPACES, [MATH] Mathematics [math], 41A10 (Approximation by polynomials), Cosine space, Numerical quadrature and cubature formulas, cosine space, Approximation by polynomials, 41A10, 42A10, 41A63, 42B05, 65D30, 65D32, 65D15, L-INFINITY APPROXIMATION, TRIGONOMETRIC POLYNOMIALS, 41A63 (Multidimensional problems), 0102 Applied Mathematics, FOS: Mathematics, 4901 Applied mathematics, 65D30 (Numerical integration), Mathematics - Numerical Analysis, 0802 Computation Theory and Mathematics, Quasi-Monte Carlo methods, Componentby-component construction, Science & Technology, 0103 Numerical and Computational Mathematics, ALGORITHMS, rank-\(1\) lattice points, Fourier space, 65D15 (Algorithms for functional approximation), Numerical Analysis (math.NA), 42A10 (Trigonometric approximation), COLLOCATION, 65D32 (Quadrature and cubature formulas), Exact integration and approximation on finite index sets, Algorithms for approximation of functions, Physical Sciences, 4903 Numerical and computational mathematics, Numerical integration, RULES, 42B05 (Fourier series and coefficients), Chebyshev space, Mathematics
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