On the constrained mock-Chebyshev least-squares

The algebraic polynomial interpolation on uniformly distributed nodes is affected by the Runge phenomenon, also when the function to be interpolated is analytic. Among all techniques that have been proposed to defeat this phenomenon, there is the mock-Chebyshev interpolation which is an interpolation made on a subset of the given nodes whose elements mimic as well as possible the Chebyshev-Lobatto points. In this work we use the simultaneous approximation theory to combine the previous technique with a polynomial regression in order to increase the accuracy of the approximation of a given analytic function. We give indications on how to select the degree of the simultaneous regression in order to obtain polynomial approximant good in the uniform norm and provide a sufficient condition to improve, in that norm, the accuracy of the mock-Chebyshev interpolation with a simultaneous regression. Numerical results are provided.Comment: 17 pages, 9 figure

Polynomial approximation of derivatives by the constrained mock-Chebyshev least squares operator

The constrained mock-Chebyshev least squares operator is a linear approximation operator based on an equispaced grid of points. Like other polynomial or rational approximation methods, it was recently introduced in order to defeat the Runge phenomenon that occurs when using polynomial interpolation on large sets of equally spaced points. The idea is to improve the mock-Chebyshev subset interpolation, where the considered function $f$ is interpolated only on a proper subset of the uniform grid, formed by nodes that mimic the behavior of Chebyshev--Lobatto nodes. In the mock-Chebyshev subset interpolation all remaining nodes are discarded, while in the constrained mock-Chebyshev least squares interpolation they are used in a simultaneous regression, with the aim to further improving the accuracy of the approximation provided by the mock-Chebyshev subset interpolation. The goal of this paper is two-fold. We discuss some theoretical aspects of the constrained mock-Chebyshev least squares operator and present new results. In particular, we introduce explicit representations of the error and its derivatives. Moreover, for a sufficiently smooth function $f$ in $[-1,1]$, we present a method for approximating the successive derivatives of $f$ at a point $x\in [-1,1]$, based on the constrained mock-Chebyshev least squares operator and provide estimates for these approximations. Numerical tests demonstrate the effectiveness of the proposed method.Comment: 17 pages, 23 figure

Product integration rules by the constrained mock-Chebyshev least squares operator

In this paper we consider the problem of the approximation of definite integrals on finite intervals for integrand functions showing some kind of "pathological" behavior, e.g. "nearly" singular functions, highly oscillating functions, weakly singular functions, etc. In particular, we introduce and study a product rule based on equally spaced nodes and on the constrained mock-Chebyshev least squares operator. Like other polynomial or rational approximation methods, this operator was recently introduced in order to defeat the Runge phenomenon that occurs when using polynomial interpolation on large sets of equally spaced points. Unlike methods based on piecewise approximation functions, mainly used in the case of equally spaced nodes, our product rule offers a high efficiency, with performances slightly lower than those of global methods based on orthogonal polynomials in the same spaces of functions. We study the convergence of the product rule and provide error estimates in subspaces of continuous functions. We test the effectiveness of the formula by means of several examples, which confirm the theoretical estimates

A case study of the Lunger phenomenon based on multiple algorithms

In this study, we conduct a thorough and meticulous examination of the Runge phenomenon. Initially, we engage in an extensive review of relevant literature, which aids in delineating the genesis and essence of the Runge phenomenon, along with an exploration of both conventional and contemporary algorithmic solutions. Subsequently, the paper delves into a diverse array of resolution methodologies, encompassing classical numerical approaches, regularization techniques, mock-Chebyshev interpolation, the TISI (Three-Interval Interpolation Strategy), external pseudo-constraint interpolation, and interpolation strategies predicated upon Singular Value Decomposition (SVD). For each method, we not only introduce but also innovate a novel algorithm to effectively address the phenomenon. This paper executes detailed numerical computations for each method, employing visualization techniques to vividly illustrate the efficacy of various strategies in mitigating the Runge phenomenon. Our findings reveal that although traditional methods exhibit commendable performance in certain instances, novel approaches such as mock-Chebyshev interpolation and regularization-centric methods demonstrate marked superiority in specific contexts. Moreover, the paper provides a critical analysis of these methodologies, specifically highlighting the constraints and potential avenues for enhancement in SVD decomposition-based interpolation strategies. In conclusion, we propose future research trajectories and underscore the imperative of further exploration into interpolation strategies, with an emphasis on their practical application validation. This article serves not only as a comprehensive resource on the Runge phenomenon for researchers but also offers pragmatic guidance for resolving real-world interpolation challenges.Comment: 13 Figures 9 Pages. After first submission, there was a revision of the authorship order, which was the result of joint discussion

Stability inequalities for Lebesgue constants via Markov-like inequalities

We prove that L^infty-norming sets for finite-dimensional multivariatefunction spaces on compact sets are stable under small perturbations. This implies stability of interpolation operator norms (Lebesgue constants), in spaces of algebraic and trigonometric polynomials

Polynomial mapped bases: theory and applications

In this paper, we collect the basic theory and the most important applications of a novel technique that has shown to be suitable for scattered data interpolation, quadrature, bio-imaging reconstruction. The method relies on polynomial mapped bases allowing, for instance, to incorporate data or function discontinuities in a suitable mapping function. The new technique substantially mitigates the Runge's and Gibbs effects

SN Refsdal : Photometry and Time Delay Measurements of the First Einstein Cross Supernova

We present the first year of Hubble Space Telescope imaging of the unique supernova (SN) "Refsdal," a gravitationally lensed SN at z = 1.488 ± 0.001 with multiple images behind the galaxy cluster MACS J1149.6+2223. The first four observed images of SN Refsdal (images S1–S4) exhibited a slow rise (over ~150 days) to reach a broad peak brightness around 2015 April 20. Using a set of light curve templates constructed from SN 1987A-like peculiar Type II SNe, we measure time delays for the four images relative to S1 of 4 ± 4 (for S2), 2 ± 5 (S3), and 24 ± 7 days (S4). The measured magnification ratios relative to S1 are 1.15 ± 0.05 (S2), 1.01 ± 0.04 (S3), and 0.34 ± 0.02 (S4). None of the template light curves fully captures the photometric behavior of SN Refsdal, so we also derive complementary measurements for these parameters using polynomials to represent the intrinsic light curve shape. These more flexible fits deliver fully consistent time delays of 7 ± 2 (S2), 0.6 ± 3 (S3), and 27 ± 8 days (S4). The lensing magnification ratios are similarly consistent, measured as 1.17 ± 0.02 (S2), 1.00 ± 0.01 (S3), and 0.38 ± 0.02 (S4). We compare these measurements against published predictions from lens models, and find that the majority of model predictions are in very good agreement with our measurements. Finally, we discuss avenues for future improvement of time delay measurements—both for SN Refsdal and for other strongly lensed SNe yet to come