construct orthogonal polynomials

This work addresses the efficient computation of linear recurrence relations satisfied by the generalized Fourier coefficients of functions that solve linear differential equations, expressed in bases of classical orthogonal polynomials. To this end, the authors propose a unified framework based on fractional representations of linear recurrence operators, wherein the desired recurrence relation is interpreted as the numerator of such a fraction. The core computational mechanism relies on a non-commutative Euclidean algorithm, which systematically integrates several existing methods into a coherent and streamlined approach. Both theoretical analysis and illustrative examples demonstrate that the proposed framework not only offers broad generality but also enhances computational efficiency and implementation simplicity compared to prior techniques.

This work addresses the challenges of discovering differential equations, approximating functions, and estimating high-dimensional integrals from noisy, non-uniformly sampled data by introducing Sparse Orthogonal Regression Technique (SORT). The method reformulates equation discovery as a spectral coefficient learning problem, directly inferring coefficients of an orthogonal basis expansion from observational data via L1 regularization—without requiring a predefined symbolic library, explicit numerical integration, or inner product evaluations. Its core innovation lies in treating basis function design as the central modeling choice, enabling consistent model order scaling and multi-task reusability. Experiments demonstrate that when the basis functions align with the underlying problem structure, SORT matches or outperforms existing approaches under sparse sampling, noisy derivatives, and representation mismatch, while low-order dominant coefficients remain stable as model complexity increases.

This work proposes a learnable framework for adaptive orthogonal bases that overcomes the rigidity of traditional fixed bases—such as Fourier or wavelet bases—in capturing data-specific structures. The target basis is treated as a point on the Lie manifold of the orthogonal group and is obtained by continuously evolving a reference basis along a path defined by an ordinary differential equation induced by a finite-rank skew-adjoint integral operator, parameterized via neural networks. Theoretically, it is shown that rank-2 generators suffice to densely approximate any orthogonal basis in the operator topology, ensuring both universality and flexibility. Experiments demonstrate successful adaptation of the Fourier basis into data-driven principal components, eigenfunctions of operators, and dynamic modes of physical systems, confirming the method’s effectiveness and broad applicability.

This work addresses the construction of orthogonal bases on path space analogous to classical orthogonal polynomials, enabling efficient representation and approximation of square-integrable functionals of stochastic paths. By orthogonalizing the signature of stochastic processes within the shuffle algebra and free Lie algebra frameworks and analyzing $L^2$ convergence, the study extends multivariate orthogonal polynomial theory—including recurrence relations and Favard’s theorem—to path space for the first time. Key contributions include demonstrating the existence of dimension-free orthogonal signatures for Brownian motion with drift, in contrast to the driftless case where such signatures do not exist; establishing the $L^p$-density of linear signature functionals over group-like elements; and constructing orthogonal signature polynomials for Brownian motion, whose efficacy in approximating path functionals under Wiener measure is validated through numerical experiments.

This paper addresses the universal approximation of continuous (including nonlinear) operators on Banach spaces. Methodologically, it introduces a novel learning framework based on orthogonal polynomial projections—marking the first integration of Leray–Schauder mapping theory into operator approximation theorems, synergizing Banach-space operator analysis with spectral approximation techniques in $L^p$ spaces. Specifically, in $L^p$ (notably $L^2$), it establishes a two-stage operator learning paradigm: “learnable projection” followed by “finite-dimensional mapping.” Theoretical contributions include: (1) a proof of universal approximation capability for the framework on arbitrary Banach spaces; (2) explicit sufficient conditions ensuring high-precision operator approximation in $L^2$; and (3) the first rigorous, unified mathematical foundation for operator neural networks.

This work proposes a novel q-orthogonal kernel function by introducing discrete q-Hermite I polynomials into support vector machine (SVM) kernel design—a first in the literature. The proposed kernel leverages the deformation parameter \( q \) to inherently mitigate numerical underflow and overflow issues without requiring explicit scaling, thereby addressing the longstanding trade-off among interpretability, numerical stability, and computational efficiency that plagues existing orthogonal polynomial kernels. The constructed kernel satisfies Mercer’s condition, ensuring theoretical soundness while maintaining numerical robustness and computational simplicity. Furthermore, it provides a foundation for quantum-inspired algorithms. Empirical evaluation across 20 benchmark datasets demonstrates that the new kernel consistently matches or outperforms both classical and state-of-the-art orthogonal polynomial kernels, confirming its practical efficacy and potential.

本文介绍了一种通过计算交换算子的联合特征向量来解决多项式根和张量分解问题的方法，并分析了其多重结构。

This work addresses the absence of readily available Gaussian quadrature rules for nonclassical weight functions by proposing a general framework that constructs such rules for arbitrary weights via the method of moments and the Stieltjes procedure. Innovatively integrating type-generic programming with adaptive high-precision arithmetic, the approach effectively controls round-off errors and, for the first time, systematically introduces tailored Gaussian quadrature methods to the statistics community. Implemented in Julia as the CustomGaussQuadrature package—accessible from R through JuliaConnectoR—the resulting quadrature rules achieve exact integration of polynomials up to degree \(2n-1\) while substantially reducing the number of function evaluations, thereby offering both high accuracy and computational efficiency.

This study addresses the acceleration of orthogonal polynomial transforms in scientific computing by proposing discrete and continuous quantum orthogonal polynomial transform algorithms encompassing the entire polynomial families of the Askey scheme. Methodologically, it reveals intrinsic connections between polynomials and Gaussian optical gates, constructs a compilation framework based on SU(2)/SU(1,1) groups, and integrates quantum optical gates, group representation theory, Chirp decomposition, and quantum Hankel transforms to achieve efficient circuit design. The core contribution lies in attaining near-logarithmic time complexity for multiple transform classes, with the Hahn transform specifically achieving a quadratic speedup. These advances establish a novel paradigm for quantum-enhanced scientific computing.

This work addresses the computational inefficiency in Bayesian semiparametric regression arising from complex design matrix structures. To mitigate this challenge, the authors propose an orthogonalization preprocessing step applied to the design日晚间 matrix prior to iterative inference, combined with a hybrid algorithm integrating Gibbs sampling and coordinate ascent variational inference. This approach reduces computational complexity to quadratic in the number of covariates, substantially accelerating both model fitting and posterior inference. Empirical evaluations across diverse experimental settings demonstrate speedups ranging from 5× to 60× compared to conventional methods, effectively alleviating the computational bottleneck induced by high-dimensional covariates.