Approximation Theory XVI: Nashville, TN, USA, May 19-22, 2019 (Springer Proceedings in Mathematics & Statistics, 336)

These proceedings are based on the international conference Approximation Theory XVI held on May 19–22, 2019 in Nashville, Tennessee. The conference was the sixteenth in a series of meetings in Approximation Theory held at various locations in the United States. Over 130 mathematicians from 20 countries attended. The book contains two longer survey papers on nonstationary subdivision and Prony’s method, along with 11 research papers on a variety of topics in approximation theory, including Balian-Low theorems, butterfly spline interpolation, cubature rules, Hankel and Toeplitz matrices, phase retrieval, positive definite kernels, quasi-interpolation operators, stochastic collocation, the gradient conjecture, time-variant systems, and trivariate finite elements. The book should be of interest to mathematicians, engineers, and computer scientists working in approximation theory, computer-aided geometric design, numerical analysis, and related approximation areas. Preface Contents Time-Variant System Approximation via Later-Time Samples 1 Introduction 2 Laplace Equation 3 Variable Coefficient Wave Equation References C1-Quartic Butterfly-Spline Interpolation on Type-1 Triangulations 1 Introduction 2 Notations and Preliminaries 3 C1 Quartic Quasi-interpolating Splines on D2 4 C1 Quartic Interpolating Splines on D1 4.1 The Modified Butterfly Interpolatory Subdivision Scheme 4.2 C1 Quartic Interpolating Splines 5 Numerical Results 6 Conclusions References Approximation with Conditionally Positive Definite Kernels on Deficient Sets 1 Introduction 2 Approximation on Deficient Sets 3 Examples 3.1 Numerical Differentiation of Laplacian on a Grid 3.2 Interpolation of Data on Ellipse References Non-stationary Subdivision Schemes: State of the Artand Perspectives 1 Introduction 2 Classical Subdivision Schemes 2.1 Binary, Linear, and Stationary Subdivision Schemes 2.2 Examples of Subdivision Schemes 2.3 Main Applications 2.3.1 Geometric Modelling and CAGD 2.3.2 Generation of Refinable Functions and Wavelets 2.4 Analysis Tools 3 Motivation for Non-stationary Subdivision Schemes 3.1 Reproduction of Conics and Quadrics and Use of Level Dependent Tension Parameters in CAGD 3.2 Non-stationary Wavelets and Non-stationary Interpolatory Subdivision Schemes 3.3 Image Segmentation: Active Contours and Active Surfaces 4 Analysis Tools for Non-stationary Subdivision Schemes 4.1 Masks of Fixed Support 4.2 Non-stationary Schemes with Extraordinary Vertices/Faces 4.3 Masks of Growing Support 5 Open Problems in Non-stationary Subdivision References Cubature Rules Based on Bivariate Spline Quasi-Interpolation for Weakly Singular Integrals 1 Introduction 2 The Problem 3 Cubature Rules Based on Tensor-Product Spline Quasi-Interpolation 4 Numerical Results 5 Conclusions References On DC Based Methods for Phase Retrieval 1 Introduction 1.1 Phase Retrieval 1.2 Our Contribution 1.3 Organization 2 On Existence and Number of Phase Retrieval Solutions 3 A DC Based Algorithm for Phase Retrieval 4 Computation of the Inner Minimization (14) 5 Sparse Phase Retrieval 6 Numerical Results 6.1 Phase Retrieval for Real and Complex Signals 6.2 Phase Retrieval of Sparse Signals Appendix References Modifications of Prony's Method for the Recovery and Sparse Approximation with Generalized Exponential Sums 1 Introduction 1.1 The Classical Prony Method 1.2 Content of This Paper 2 Operator Based View to Prony's Method 3 Recovery of Generalized Exponential Sums 3.1 Expansion into Eigenfunctions of a Linear Differential Operator 3.2 Expansion into Eigenfunctions of a Generalized Shift Operator 3.3 Application to Special Expansions 3.3.1 Classical Exponential Sums 3.3.2 Expansions into Shifted Gaussians 3.3.3 Expansions into Functions of the Form exp( αj sinx) 4 Numerical Treatment of the Generalized Prony Method 4.1 The Simple Prony Algorithm 4.2 ESPRIT for the Generalized Prony Method 4.3 Simplification in Case of Partially Known Frequency Parameters 5 Modified Prony Method for Sparse Approximation 5.1 The Nonlinear Least-Squares Problem 5.2 Gauß-Newton and Levenberg-Marquardt Iteration References On Eigenvalue Distribution of Varying Hankel and Toeplitz Matrices with Entries of Power Growth or Decay 1 Introduction and Results 2 Hankel Matrices 3 Toeplitz Matrices 4 Proof of Theorem 2.1 and Corollary 2.2 5 Proof of Theorem 3.1 and Corollary 3.2 References On the Gradient Conjecture for Quadratic Polynomials 1 Introduction 2 Notations and Historic Preliminaries 3 Main Results 4 Application to Derivations Operators 5 One More Case of the Gradient Conjecture References Balian-Low Theorems in Several Variables 1 Introduction 1.1 Extension to Several Variables 1.2 Finite Nonsymmetric BLTs 1.3 Applications of the Continuous Quantitative BLT 2 Preliminaries: The Zak Transform and Quasiperiodic Functions 3 Proof of Theorem 1.6 4 Proof of Theorem 1.7 5 Nonsymmetric Finite BLT and Applications of the Quantitative BLTs 6 Further Questions References Quasi-Interpolant Operators and the Solution of Fractional Differential Problems 1 Introduction 2 The Cardinal B-splines 3 The Fractional Derivative of the Cardinal B-splines 4 Quasi-Interpolant Operators 5 The Quasi-Interpolant Collocation Method 6 Numerical Tests 6.1 Example 1 6.2 Example 2 6.3 Example 3 7 Conclusion References Stochastic Collocation with Hierarchical Extended B-Splines on Sparse Grids 1 Introduction 2 Sparse Grids 2.1 Regular Sparse Grids 2.2 Spatial Adaptivity 3 Basis Functions 3.1 B-Splines 3.2 Not-a-Knot B-Splines 3.3 Modified Not-a-Knot B-Splines 3.4 Extended Not-a-Knot B-Splines 4 Expansion Methods 4.1 Stochastic Collocation 4.2 Polynomial Chaos Expansion 5 Numerical Results 5.1 Exponential Objective Function 5.2 Borehole Model 6 Conclusions and Outlook References Trivariate Interpolated Galerkin Finite Elements for the Poisson Equation 1 Introduction 2 The P2 Nonconforming Interpolated Galerkin Finite Element 3 The P3 Nonconforming Interpolated Galerkin Finite Element 4 The Pk, k≥4, Conforming Interpolated Galerkin Finite Element 5 Convergence Theory 6 Numerical Tests References Index