W. Bryc, W. Matysiak, R. Szwarc, J. Wesolowski
We prove a projection formula for the four-parameter family of orthogonal polynomials that are a reparameterization of the polynomials in the Askey-Wilson class. By carefully analyzing the recurrence relations we manage to avoid using the explicit expression for the orthogonality measure, which would be cumbersome due to the complexity of the reparameterization.
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We generalize Bourgain's discretized projection theorem to higher rank situations. Like Bourgain's theorem, our result yields an estimate for the Hausdorff dimension of the exceptional sets in projection theorems formulated in terms of Hausdorff dimensions. This estimate complements earlier results of Mattila and Falconer.
We consider several classical results related to the Hausdorff dimension of exceptional sets of orthogonal projections and try to find out whether they have reasonable formulations in terms of packing dimension. We also investigate the existence of category versions for Marstrand and Falconer-Howroyd-type projection results.
We construct a decomposition of the identity operator on a Riemannian manifold $M$ as a sum of smooth orthogonal projections subordinate to an open cover of $M$. This extends a decomposition of the real line by smooth orthogonal projection due to Coifman, Meyer and Auscher, Weiss, Wickerhauser, and a similar decomposition when $M$ is the sphere by the first two authors.
Box-splines orthogonal projections
本文将 Sweldens 与 Piessens 关于 B 样条正交投影误差为伯努利多项式的结论推广到箱样条，并据此用正交投影误差的渐近公式定义 Sobolev 空间半范数。
On the parabolic Hausdorff dimension of orthogonal projections
本文证明了Borel集A⊂ℝⁿ×ℝ在ℝⁿ×ℝ的通用m维线性子空间上的正交投影的抛物型Hausdorff维数的下界。
In this paper, we introduce a new differential-difference operator $T_ξ$ $(ξ\in \mathbb{R}^N)$ by using projections associated to orthogonal subsystems in root systems. Similarly to Dunkl theory, we show that these operators commute and we construct an intertwining operator between $T_ξ$ and the directional derivative $\partial_ξ$. In the case of one variable, we prove that the Kummer functions are eigenfunctions of this operator.
We construct a single smooth orthogonal projection with desired localization whose average under a group action yields the decomposition of the identity operator. For any full rank lattice $Γ\subset\mathbb R^d$, a smooth projection is localized in a neighborhood of an arbitrary precompact fundamental domain $\mathbb R^d/Γ$. We also show the existence of a highly localized smooth orthogonal projection, whose Marcinkiewicz average under the action of $SO(d)$, is a multiple of the identity on $L^2(\mathbb S^{d-1})$. As an application we construct highly localized continuous Parseval frames on the…
Orthogonal projection, dual Furstenberg problem, and discretized sum-product
该研究提出对偶 Furstenberg 问题，利用正交投影的 $L^p$ 估计取得部分结果，并借助笛卡尔积结构改进离散和积估计。
We show that, almost surely, the Hausdorff dimension $s_0$ of a random covering set is preserved under all orthogonal projections to linear subspaces with dimension $k>s_0$. The result holds for random covering sets with a generating sequence of ball-like sets, and is obtained by investigating orthogonal projections of a class of random Cantor sets.
We study approximation properties of weighted $L^2$-orthogonal projectors onto spaces of polynomials of bounded degree in the Euclidean unit ball, where the weight is of the generalized Gegenbauer form $x \mapsto (1-\|x\|^2)^α$, $α> -1$. Said properties are measured in Sobolev-type norms in which the same weighted $L^2$ norm is used to control all the involved weak derivatives. The method of proof does not rely on any particular basis of orthogonal polynomials, which allows for a short, streamlined and dimension-independent exposition.
On the dimension of orthogonal projections of self-similar measures
本文针对 $\mathbb{R}^d$ 上自相似测度的正交投影，提出了基于 IFS 正交部分生成群作用的判据，确保投影维数得以保持，改进了前人结果并证明了其最佳性。
We study approximation properties of weighted $\mathrm{L}^2$-orthogonal projectors onto spaces of polynomials of bounded degree in the Euclidean unit ball, where the weight is of the reflection-invariant form $(1-\lVert x \rVert^2)^α\prod_{i=1}^d \lvert x_i \rvert^{γ_i}$, $α, γ_1, \dots, γ_d > -1$. Said properties are measured in Dunkl-Sobolev-type norms in which the same weighted $\mathrm{L}^2$ norm is used to control all the involved differential-difference Dunkl operators, such as those appearing in the Sturm-Liouville characterization of similarly weighted $\mathrm{L}^2$-orthogonal polynom…
Applications of dimension interpolation to orthogonal projections
本文综述了维度插值方法，并重点展示傅里叶谱、中间维数和Assouad谱在正交投影维数理论中的三类应用，拓展了经典Marstrand–Mattila投影定理的研究视角。
We consider the Assouad dimension analogues of two important problems in geometric measure theory. These problems are tied together by the common theme of `passing to weak tangents'. First, we solve an analogue of Falconer's distance set problem for Assouad dimension in the plane: if a planar set has Assouad dimension strictly greater than 1, then its distance set has Assouad dimension 1. We also obtain partial results in higher dimensions. Second, we consider how Assouad dimension behaves under orthogonal projection. We extend the planar projection theorem of Fraser and Orponen to higher dime…
A Fourier analytic approach to exceptional set estimates for orthogonal projections
本研究提出基于傅里叶谱的正交投影定理与例外集估计方法，改进了平面例外集Ren--Wang界、Peres--Schlag界及Bourgain--Oberlin 0维估计等多个豪斯多夫维数相关结果。
A study guide for "On the Hausdorff dimension of Furstenberg sets and orthogonal projections in the plane" after T. Orponen and P. Shmerkin
本文为 Orponen 与 Shmerkin 关于平面 Furstenberg 集 Hausdorff 维数与正交投影的论文撰写学习指南，介绍 Furstenberg 集问题、投影例外集，并总结证明思路与核心思想。
On the Peres--Schlag orthogonal projection problem and Kakeya-type sets
研究有限域和欧几里得空间中正交投影的 Peres--Schlag 非空内部问题，通过多项式方法建立有限域上的尖锐投影结果，并在欧几里得空间中结合几何测度论与调和分析技术改进参数范围。
Algorithmic Information Bounds for Distances and Orthogonal Projections
介绍证明欧几里得空间几何对象柯尔莫哥洛夫复杂度界的新技术，运用此技术证明关于距离和正交投影的算法信息论定理，改进固定距离集豪斯多夫维数下界，推广Bourgain关于正交投影例外集的定理。
We study the restricted families of orthogonal projections in $\mathbb{R}^3$. We show that there are families of random subspaces which admit a Marstrand- Mattila type projection theorem.
Bounding the dimension of exceptional sets for orthogonal projections
本文研究正交投影中投影维数低于给定阈值的例外 Grassmann 流形维数，改进已有上界，并证明在一维及余一维情形尖锐，从而完全解决三维情形。
Cesàro $(C,δ)$ means are studied for orthogonal expansions with respect to the weight function $\prod_{i=1}^{d}|x_i|^{2\k_i}$ on the unit sphere, and for the corresponding weight functions on the unit ball and the Jacobi weight on the simplex. A sharp pointwise estimate is established for the $(C,\d)$ kernel with $\d > -1$ and for the kernel of the projection operator, which allows us to derive the exact order for the norm of the Cesàro means and the projection operator on these domains.
We prove that for an arbitrary upper semi-continuous function $ϕ\colon G(1,2) \to [0,1]$ there exists a compact set $F$ in the plane such that $\dim_{\textrm{A}} πF = ϕ(π)$ for all $π\in G(1,2)$, where $πF$ is the orthogonal projection of $F$ onto the line $π$. In particular, this shows that the Assouad dimension of orthogonal projections can take on any finite or countable number of distinct values on a set of projections with positive measure. It was previously known that two distinct values could be achieved with positive measure. Recall that for other standard notions of dimension, such as…
On dissonance and orthogonal projections of self-conformal measures
本文针对$\mathbb{R}^d$上的自共形测度$\mu$，结合自共形测度标度场景成果与López、Moreira的Marstrand型投影定理，给出了其与任意Ahlfors正则或自共形测度卷积满足维数和公式的充分条件。
Projection theorems with countably many exceptions and applications to the exact overlaps conjecture
本文针对分形测度投影中的例外参数建立了最优估计，证明在至多可数多个方向外维数保持，并应用于精确重叠猜想，得出平面测度与集合投影维数的下界。
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