Pointwise weighted approximation of functions with inner singularities by combinations of Bernstein operators

Modified Bernstein Polynomials and Jacobi Polynomials in q-Calculus

We introduce here a generalization of the modified Bernstein polynomials for Jacobi weights using the $q$-Bernstein basis proposed by G.M. Phillips to generalize classical Bernstein Polynomials. The function is evaluated at points which are in geometric progression in $]0,1[$. Numerous properties of the modified Bernstein Polynomials are extended to their $q$-analogues: simultaneous approximation, pointwise convergence even for unbounded functions, shape-preserving property, Voronovskaya theorem, self-adjointness. Some properties of the eigenvectors, which are $q$-extensions of Jacobi polynomi…

Singularities of Fredholm maps with one-dimensional kernels, II: Local behaviour and pointwise conditions

In analogy to what happens in finite dimensions we state the Normal Form Theorem for k-singularities, introduced in the previous paper of the series. By means of that we study the local behaviour near a singularity i.e. we deduce local results of existence and multiplicity of solutions for the equation F(x) = y where F is a 0-Fredholm map and x belongs to a suitable neighbourhood of a singular point x_(o), once x_(o) is identified as one of the three kinds of singularities defined in the first paper. To this end we also start to seek alternative strategies for the determination of the type of …

Wavelets techniques for pointwise anti-Holderian irregularity

In this paper, we introduce a notion of weak pointwise Holder regularity, starting from the de nition of the pointwise anti-Holder irregularity. Using this concept, a weak spectrum of singularities can be de ned as for the usual pointwise Holder regularity. We build a class of wavelet series satisfying the multifractal formalism and thus show the optimality of the upper bound. We also show that the weak spectrum of singularities is disconnected from the casual one (denoted here strong spectrum of singularities) by exhibiting a multifractal function made of Davenport series whose weak spectrum …

利用伯恩斯坦技术重新审视四阶Hénon方程的Fazly-Wei-Xu逐点估计

Revisiting Fazly-Wei-Xu pointwise estimates for the fourth-order Hénon equation via Bernstein's technique

本文利用伯恩斯坦技术替代原有迭代方法，重新得到四阶Hénon方程的Fazly-Wei-Xu逐点估计，放宽了解的有界性条件，证明系数最优并给出共形度量存在性结果。

伯恩斯坦型对数算子的 Kantorovich 版本

A Kantorovich version of Bernstein-type logarithmic operators

本文提出了一种伯恩斯坦型对数算子的 Kantorovich 版本，研究了其收敛性、渐近公式及逼近阶数的估计。

Pointwise Values and Fundamental Theorem in the Algebra of Asymptotic Functions

We show that the algebra of asymptotic functions $^ρ\mathcal{E}(Ω)$ (introduced in another paper by the author of this article jointly with M. Oberguggenberger) is isomorphic to a class of pointwise functions in the field of A. Robinson asymptotic numbers $^ρ\mathbb{C}$. Since the algebra $^ρ\mathcal{E}(Ω)$, contains a copy of the Schwartz distributions $\mathcal{D}^\prime(Ω)$, it follows that every Schwartz distribution is a pointwise function in the field $^ρ\mathbb{C}$. The class of asymptotic functions $^ρ\mathcal{E}(Ω)$ is an algebra of generalized functions of Colombeau's type. The field…

加权Weyl-Sonine算子的半群方法：绕过Bernstein函数实现适定性

A Semigroup Approach to Weighted Weyl-Sonine Operators: Bypassing Bernstein Functions for Well-Posedness

本文建立加权Weyl-Sonine分数阶算子的泛函分析框架，通过引入遗忘权重绕过Bernstein函数限制，统一处理扩散弛豫和振荡非局部动力学，并证明相关抽象分数阶Cauchy问题的适定性。

一类新的正线性算子保持对数函数

A new class of positive linear operators preserving logarithmic functions

本文提出了一类新的正线性算子，能够再现对数函数，并研究了其收敛性、误差估计及应用

Quasilinear and Hessian equations of Lane-Emden type

The existence problem is solved, and global pointwise estimates of solutions are obtained for quasilinear and Hessian equations of Lane-Emden type, including the following two model problems: $ -Δ_p u = u^q + μ$ and $F_k[-u] = u^q + μ$, $u > 0$, on $R^n$, or on a bounded domain $Ω$. Here $Δ_p$ is the p-Laplacian, and $F_k[u]$ is the $k$-Hessian defined as the sum of $k\times k$ principal minors of the Hessian matrix $D^2 u$; $μ$ is a nonnegative measurable function (or measure) on $Ω$. The solvability of these classes of equations in the renormalized (entropy) or viscosity sense has been an op…

A Second-Quantized Kolmogorov-Chentsov Theorem via the Operator Product Expansion

We establish a direct connection between two fundamental topics: one in probability theory and one in quantum field theory. The first topic is the problem of pointwise multiplication of random Schwartz distributions which has been the object of recent progress thanks to Hairer's theory of regularity structures and the theory of paracontrolled distributions introduced by Gubinelli, Imkeller and Perkowski. The second topic is Wilson's operator product expansion which is a general property of models of quantum field theory and a cornerstone of the bootstrap approach to conformal field theory. Our…

半离散光滑模及其在单边和双边误差估计中的应用

Semi-discrete moduli of smoothness and their applications in one- and two- sided error estimates

本文引入一种新的半离散光滑模，推广了Kolomoitsev和Lomako在2023年的定义，并利用Steklov积分的正则化与逼近性质，在非限制性假设下建立了点态线性算子的广义单边和双边误差估计，得到了比经典平均光滑模更精确的估计，同时建立了Rathore型定理并引入等价的K-泛函。