Reconstruction of Bandlimited Functions from Space-Time Samples

Sampling the flow of a bandlimited function

We analyze the problem of reconstruction of a bandlimited function $f$ from the space-time samples of its states $f_t=ϕ_t\ast f$ resulting from the convolution with a kernel $ϕ_t$. It is well-known that, in natural phenomena, uniform space-time samples of $f$ are not sufficient to reconstruct $f$ in a stable way. To enable stable reconstruction, a space-time sampling with periodic nonuniformly spaced samples must be used as was shown by Lu and Vetterli. We show that the stability of reconstruction, as measured by a condition number, controls the maximal gap between the spacial samples. We prov…

On planar sampling with Gaussian kernel in spaces of bandlimited functions

Let $I=(a,b)\times(c,d)\subset {\mathbb R}_{+}^2$ be an index set and let $\{G_α(x) \}_{α\in I}$ be a collection of Gaussian functions, i.e. $G_α(x) = \exp(-α_1 x_1^2 - α_2 x_2^2)$, where $α= (α_1, α_2) \in I, \, x = (x_1, x_2) \in {\mathbb R}^2$. We present a complete description of the uniformly discrete sets $Λ\subset {\mathbb R}^2$ such that every bandlimited signal $f$ admits a stable reconstruction from the samples $\{f \ast G_α (λ)\}_{λ\in Λ}$.

Sampling of operators

Sampling and reconstruction of functions is a central tool in science. A key result is given by the sampling theorem for bandlimited functions attributed to Whittaker, Shannon, Nyquist, and Kotelnikov. We develop an analogous sampling theory for operators which we call bandlimited if their Kohn-Nirenberg symbols are bandlimited. We prove sampling theorems for such operators and show that they are extensions of the classical sampling theorem.

Novel sampling formulas associated with quaternionic prolate spheroidal wave functions

The Whittaker-Shannon-Kotel'nikov (WSK) sampling theorem provides a reconstruction formula for the bandlimited signals. In this paper, a novel kind of the WSK sampling theorem is established by using the theory of quaternion reproducing kernel Hilbert spaces. This generalization is employed to obtain the novel sampling formulas for the bandlimited quaternion-valued signals. A special case of our result is to show that the 2D generalized prolate spheroidal wave signals obtained by Slepian can be used to achieve a sampling series of cube-bandlimited signals. The solutions of energy concentration…

随机周期采样模式用于平移不变空间

Random periodic sampling patterns for shift-invariant spaces

本文提出随机周期采样模式用于平移不变空间信号重建，证明密度超过频率分布数量对数因子时高概率成功，提供简单替代方案并具稳定性裕度，通过Zak域测度集中方法实现均匀重建保证。

通过最小二乘法和深度学习重构频率局部化的函数

Reconstruction of frequency-localized functions from pointwise samples via least squares and deep learning

本文通过最小二乘法和深度学习方法，研究了从点wise数据中重构频率局部化函数的理论与实践问题，提出了新的恢复定理和近似保证，并通过实验比较了两种方法的性能。