An explanation of the commuting operator "miracle" in time and band limiting

Time and band limiting for matrix valued functions: an integral and a commuting differential operator

The problem of recovering a signal of finite duration from a piece of its Fourier transform was solved at Bell Labs in the $1960$'s, by exploiting a "miracle": a certain naturally appearing integral operator commutes with an explicit differential one. Here we show that this same miracle holds in a matrix valued version of the same problem.

例外多项式的时间与带宽限制

Time and band limiting for exceptional polynomials

本研究针对满足双谱性质的例外正交多项式族，借助相关Fourier代数证明了可交换微分算子的存在性，并通过Hermite、Laguerre型实例验证且给出算子的Perline形式，拓展了时间与带宽限制交换性质的研究。

Time and Band Limiting for Matrix Valued Functions, an Example

The main purpose of this paper is to extend to a situation involving matrix valued orthogonal polynomials and spherical functions, a result that traces its origin and its importance to work of Claude Shannon in laying the mathematical foundations of information theory and to a remarkable series of papers by D. Slepian, H. Landau and H. Pollak. To our knowledge, this is the first example showing in a non-commutative setup that a bispectral property implies that the corresponding global operator of "time and band limiting" admits a commuting local operator. This is a noncommutative analog of the…

具有长球形性质与双谱性的矩阵值离散-连续函数

Matrix valued discrete-continuous functions with the prolate spheroidal property and bispectrality

本文建立了离散-连续双谱函数与长球形现象的内在联系，在矩阵值函数框架下，利用非交换矩阵值 Fourier 代数，构造性地证明了自伴 Darboux 变换可生成具有交换微分算子和平移算子的积分算子，并给出了阶数与带宽的显式上界。

Reflective prolate-spheroidal operators and the adelic Grassmannian

Beginning with the work of Landau, Pollak and Slepian in the 1960s on time-band limiting, commuting pairs of integral and differential operators have played a key role in signal processing, random matrix theory and integrable systems. Previously, such pairs were constructed by ad hoc methods, which worked because a commuting operator of low order could be found by a direct calculation. We describe a general approach to these problems that proves that every point $W$ of Wilson's infinite dimensional adelic Grassmannian $\mathrm Gr^ad$ gives rise to an integral operator $T_W$, acting on $L^2(Γ)$…