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Key research themes
1. How can Lagrange interpolation be extended and applied for derivative, integral, and multivariate function approximation?
This theme focuses on the methodological advancements in extending classical Lagrange interpolation beyond function value approximation to include derivatives, integrals, and multivariate functions. Such extensions address practical needs in numerical analysis where derivative and integral estimates or bivariate interpolation are essential but challenging, aiming to improve accuracy and computational efficiency.
Key finding: Proposes a novel approach that first discretely approximates derivatives and integrals from given point data, followed by Lagrange interpolation on these approximations, yielding better estimates than traditional... Read more
Key finding: Develops a practical Excel spreadsheet implementation of four-by-four bivariate Lagrange interpolation, simplifying the otherwise tedious repetitive calculations. This tool demonstrates effective approximation for bivariate... Read more
Key finding: Derives an explicit formula generalizing the classical univariate Lagrange interpolation polynomial to multivariate multinomial functions with degree n in dimension m, given n + m choose n points. The formula preserves the... Read more
Key finding: Introduces a finite difference approach to multivariate Lagrange interpolation that leads to Newton-type interpolation formulas and integral remainder representations involving simplex spline functions. This framework... Read more
Key finding: Extends Lagrange interpolation methodology to derive general finite difference formulas and error terms applicable to unequally spaced grid points. By differentiating the Lagrange interpolation formula, it obtains finite... Read more
2. What are the optimal interpolation formulas and error minimization strategies for interpolation in function spaces with smoothness constraints?
This research investigates construction and theoretical characterization of interpolation formulas that minimize approximation errors within Sobolev-type function spaces or other function classes characterized by smoothness. Understanding error functionals and obtaining explicit interpolation coefficients reveal how to produce optimally accurate approximations, especially when fitting data measured with noise or with prescribed smoothness properties.
Key finding: Develops optimal interpolation formulas in the Hilbert-Sobolev space W_2^{(m,m-1)}(0,1), where functions have m-th derivative in L^2 and (m-1)-th derivative absolutely continuous. It derives the norm of the interpolation... Read more
Key finding: Shows that for Lagrange interpolation on general convex quadrilateral finite elements of degree k ≥ 2, error estimates in Sobolev W^{1,p} spaces depend critically on the minimal interior angle when p < 3, with constants... Read more
Key finding: Extends the probabilistic Cauchy integral formula for representable holomorphic functions from finite-dimensional complex spaces to Banach spaces with separable duals. Defines Lagrange interpolation polynomials on Banach... Read more
3. How do numerical and algorithmic innovations improve convergence and error properties of iterative and interpolation methods related to Lagrange and polynomial interpolation?
This theme explores algorithmic advancements that refine interpolation formulas, reduce oscillatory behavior, optimize iterative root-finding schemes, and improve approximation quality through order manipulation, spline quasi-interpolation, and weighted smoothing. These methods tackle classical challenges such as Runge's phenomenon, numerical stability, and convergence rates by blending interpolation theory with numerical analysis and computational methods.
Key finding: Introduces a novel transform manipulating the order of interpolation formulas to achieve higher-degree polynomial approximations with fewer data points, circumventing classical limitations and reducing Runge's phenomenon... Read more
Key finding: Presents a comprehensive account of spline quasi-interpolation methods in one, two, and three dimensions, emphasizing their construction, approximation properties, and applicability to scattered and meshed data without... Read more
Key finding: Develops nonlinear bivariate C1 quadratic spline quasi-interpolants on uniform criss-cross triangulations which integrate weighted essentially non-oscillatory (WENO) techniques to suppress Gibbs oscillations near... Read more
Key finding: Proposes two new higher-order iterative root-finding methods for nonlinear equations derived by applying quadratic least squares to approximate derivatives within existing high-order iterative schemes. Numerical experiments... Read more
Key finding: Proposes an enhanced inverse-distance weighting interpolation method introducing an adjoining polynomial transition range to accelerate decline of weights beyond a cutoff, preserving smoothness and continuity. This approach... Read more