Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2203.09302

Change of Basis from Bernstein to Zernike

D. A. Wolfram
View original source ↗Read source document ↗Export metadata ↓

Abstract

We increase the scope of previous work on change of basis between finite bases of polynomials by defining ascending and descending bases and introducing three techniques for defining them from known ones. The minimum degrees of polynomials in an ascending basis can increase such as with bases of Bernstein and Zernike Radial polynomials. They have applications in computer-aided design and optics. We give coefficient functions for mappings from the monomials to descending bases of Bernstein polynomials, and ascending ones of Zernike Radial polynomials and prove their correctness. Allowing for parity, we define eight general change of basis matrices and the related equations for composing their coefficient functions. A main example is the change of basis from shifted Legendre polynomials to Bernstein polynomials considered by R Farouki [7]. The analysis enables us to find a more general hypergeometric function for the coefficient function, and recurrences for finding coefficient functions by matrix row and column. We show the column coefficient functions are equivalent to the Lagrange interpolation polynomials of their elements. We also provide a solution to an open problem [7] by showing there is no general closed form expression for the coefficient function from Gosper's Algorithm. Truncation, alternation and superposition increase the scope further. Alternation enables, for example, a change of basis between Bernstein and Zernike Radial polynomials. A summary shows that the groupoid of change of basis matrices is a small category, and triangular and alternating change of basis matrices are morphisms in full subcategories of it. Truncation is a covariant functor.

Explore related subjects

math.CA

Keep this discovery

BibTeXRIS

D. A. Wolfram. 2022-03-17. Change of Basis from Bernstein to Zernike. https://arxiv.org/abs/2203.09302

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On the log-concavity of the composite Bessel function $x^{\alpha}J_{\nu }\left( \beta x^{\gamma}\right) $

For a twice differentiable function $f:\left( a,b\right) \rightarrow \mathbb{R}$ define $v\left( f\right) =f^{\prime}f^{\prime}-f^{\prime\prime }f.$ It is well known that the positivity of $v\left( f\right) $ implies that the function $\left\vert f\right\vert $ is strictly log-concave on each subinterval which does not contain zeros of $f.$ In this paper we provide criteria for the positivity of $v\left( F\right) $ for the composite Bessel function $F\left( x\right) =J_{\alpha,\beta,\gamma,\nu}\left( x\right) :=x^{\alpha}J_{\nu}\left( \beta x^{\gamma}\right) $ for positive numbers $\beta$ and $\gamma$ and real numbers $\alpha$ and $\nu.$

math.CA↗
arXiv2026

Riesz capacity ratios with negative exponents

We investigate sharp inequalities for ratios of Riesz capacities with negative exponents by combining computational experiments with rigorous analysis. For finite subsets of the line, we prove positivity of equilibrium masses when $-1<p<0$, enabling numerical tests of conjectured extremal ratios. In the plane, comparisons of the disk with regular polygon vertex sets reveal a cascade of transitions among the tested competitors and suggest a precise conjecture for the equilibrium measure of odd polygons, for which we give a partial proof. Numerical intersections of equality curves show that the regions where these sets outperform the disk are not simply nested. Similar numerical intersections occur in three dimensions between the regular-simplex equality curve and those of explicit five-point and six-point configurations. Motivated by the dimensional dependence of these comparisons, we prove that for each fixed $p<-2<q<0$, the regular simplex has a larger capacity ratio than the ball in all sufficiently large dimensions. Accompanying Python and Mathematica code supports reproduction and further testing of the conjectures.

math.CA↗
arXiv2026

Shorter proof of dimension-free $L^p$ estimates for maximal Riesz transforms

We provide a shorter and more direct proof of $L^p$ estimates for maximal Riesz transforms (of an arbitrary order) in terms of the corresponding Riesz transforms, with a constant independent of the dimension of the Euclidean space $\mathbb R^d$. This result was originally proved by Mateu, Orobitg, P\'erez and Verdera with a constant depending on the dimension, and improved to a dimension-free inequality by Kucharski, Wr\'obel and Zienkiewicz.

math.CA↗