RECORD
Sobolev orthogonal polynomials on the conic surface
Classical Analysis and ODEs · Submitted 16 Sep 2022
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Our main results provide an explicit construction of an orthogonal basis and a formula for the orthogonal projection operators; the latter is used to exploit the interaction of differential operators and the projection operator, which allows us to study the convergence of the Fourier orthogonal series.
The compiled record
The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
Open the compiled record
The abstract, sentence by sentence
Orthogonal polynomials with respect to the weight function $w_{\beta,\gamma}(t) = t^\beta (1-t)^\gamma$, $\gamma > -1$, on the conic surface $\{(x,t): \|x\| = t, \, x \in \mathbb{R}^d, \, t \le 1\}$ are studied recently, and are shown to be eigenfunctions of a second order differential operator $\mathcal{D}_\gamma$ when $\beta =-1$. We extend the setting to the Sobolev inner product, defined as the integration of the $s$-th normal derivative $\mathfrak{D} = \frac{\mathrm{d}}{\mathrm{d} t} - t^{-1} \langle x, \nabla_x\rangle$ of the cone with respect to $w_{\beta+s,0}$ over the conic surface, plus a sum of integrals over the rim of the cone. Our main results provide an explicit construction of an orthogonal basis and a formula for the orthogonal projection operators; the latter is used to exploit the interaction of differential operators and the projection operator, which allows us to study the convergence of the Fourier orthogonal series. The study can be regarded as an extension of the orthogonal structure to the weight function $w_{\beta, -s}$ for a positive integer $s$. It shows, in particular, that the Sobolev orthogonal polynomials are eigenfunctions of $\mathcal{D}_{\gamma}$ when $\gamma = -1$.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Orthogonal polynomials with respect to the weight function $w_{\beta,\gamma}(t) = t^\beta (1-t)^\gamma$, $\gamma > -1$, on the conic surface $\{(x,t): \|x\| = t, \, x \in \mathbb{R}^d, \, t \le 1\}$ are studied recently, and are shown to be eigenfunctions of a second order differential operator $\mathcal{D}_\gamma$ when $\beta =-1$. | background 1.00 | 0.11 | 0.08 | asserted 1.00 |
| s1 | We extend the setting to the Sobolev inner product, defined as the integration of the $s$-th normal derivative $\mathfrak{D} = \frac{\mathrm{d}}{\mathrm{d} t} - t^{-1} \langle x, \nabla_x\rangle$ of the cone with respect to $w_{\beta+s,0}$ over the conic surface, plus a sum of integrals over the rim of the cone. | contribution 0.70 | 0.92 | 0.07 | asserted 0.95 |
| s2 | Our main results provide an explicit construction of an orthogonal basis and a formula for the orthogonal projection operators; the latter is used to exploit the interaction of differential operators and the projection operator, which allows us to study the convergence of the Fourier orthogonal series. | contribution 0.78 | 0.96 | 0.81 | asserted 1.00 |
| s3 | The study can be regarded as an extension of the orthogonal structure to the weight function $w_{\beta, -s}$ for a positive integer $s$. | interpretation 0.96 | 0.63 | 0.09 | asserted 0.74 |
| s4 | It shows, in particular, that the Sobolev orthogonal polynomials are eigenfunctions of $\mathcal{D}_{\gamma}$ when $\gamma = -1$. | result 0.93 | 0.87 | 0.22 | asserted 1.00 |
What was examined, and what was not
- arXiv listing abstract v2
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
The paper
Original abstract
Orthogonal polynomials with respect to the weight function $w_{\beta,\gamma}(t) = t^\beta (1-t)^\gamma$, $\gamma > -1$, on the conic surface $\{(x,t): \|x\| = t, \, x \in \mathbb{R}^d, \, t \le 1\}$ are studied recently, and are shown to be eigenfunctions of a second order differential operator $\mathcal{D}_\gamma$ when $\beta =-1$. We extend the setting to the Sobolev inner product, defined as the integration of the $s$-th normal derivative $\mathfrak{D} = \frac{\mathrm{d}}{\mathrm{d} t} - t^{-1} \langle x, \nabla_x\rangle$ of the cone with respect to $w_{\beta+s,0}$ over the conic surface, plus a sum of integrals over the rim of the cone. Our main results provide an explicit construction of an orthogonal basis and a formula for the orthogonal projection operators; the latter is used to exploit the interaction of differential operators and the projection operator, which allows us to study the convergence of the Fourier orthogonal series. The study can be regarded as an extension of the orthogonal structure to the weight function $w_{\beta, -s}$ for a positive integer $s$. It shows, in particular, that the Sobolev orthogonal polynomials are eigenfunctions of $\mathcal{D}_{\gamma}$ when $\gamma = -1$.
Human review
Discussion
Know something about this paper? Leave a comment, or highlight any passage above to comment on it.