Constructs non-asymptotic confidence regions for unknown functions in RKHS.
problem Global probabilistic confidence regions for unknown functions in RKHS.
method Reduces confidence region construction to estimating RKHS norm.
result Valid confidence regions can be constructed non-asymptotically.
We prove a Paley-Wiener Theorem for a class of symmetric spaces of the compact type, in which all root multiplicities are even. This theorem characterizes functions of small support in terms of holomorphic extendability and exponential type of their (discrete) Fourier transforms. We also provide three independent new p…
The paper creates nonparametric confidence bands for band-limited functions.
problem Estimating confidence bands for band-limited functions with finite samples and unknown noise.
method Uses Paley-Wiener reproducing kernel Hilbert spaces and gradient-perturbation methods.
result Non-asymptotic guarantees for confidence regions without assuming a parametric model.
Optimal smooth subspaces approximate large data sets efficiently.
problem Approximating large data sets with invariant subspaces.
method Smooth functions under lattice translations or crystallographic groups, with optimal selection of Paley-Wiener space.
result Optimal lattice selection enhances approximation efficiency.
The paper improves nonparametric confidence bands for band-limited functions.
problem Constructing nonparametric simultaneous confidence bands with nonasymptotic and distribition-free guarantees.
method Based on Paley-Wiener reproducing kernel Hilbert spaces, the paper relaxes assumptions, improves noise estimation, and tightens constraints.
result Enhanced confidence bands with improved efficiency and tighter constraints.
Study approximates operator learning for PDEs using Fourier multipliers.
problem Approximating operator behavior for PDE simulations.
method Approximation of operator symbols in Fourier domain using semi-norms.
result Identifies conditions for achieving predefined approximation error.
Study on stochastic covariant derivatives in curved space-time.
problem Analyzing covariant derivatives in curved space-time under stochastic processes.
method Using Itô-Wiener processes and stochastic calculus, including Besov spaces, Schrödinger operators, and white noise.
result Developed a framework for stochastic geodesics and white noise in fractoid spaces.
It is proved that if a Paley-Wiener family of eigenfunctions of the Laplace operator in R3 vanishes on a real analytically ruled two-dimensional surface S⊂R3 then S is a union of cones, each of which is contained in a translate of the zero set of a nonzero harmonic homogeneous polynomial…
In this paper, we specify what functions induce the bounded composition operators on a reproducing kernel Hilbert space (RKHS) associated with an analytic positive definite function defined on Rd. We prove that only affine transforms can do so in a pretty large class of RKHS. Our result covers not only the …
Solves wave equation on non-flat harmonic manifolds using Abel transform and Fourier analysis.
problem Wave equation on non-flat harmonic manifolds with specific curvature conditions.
method Explicit representation using inverse dual Abel transform and Fourier transform.
result Shows asymptotic Huygens principle and equidistribution of energy.
The paper improves confidence regions for band-limited functions using tighter norm bounds and majority voting.
problem Constructing reliable confidence regions for band-limited functions from noisy data.
method Improved norm bounds using Hoeffding's inequality and empirical Bernstein bound, majority voting to aggregate intervals.
result Confidence intervals retain their simultaneous coverage guarantee even when aggregated from random subsamples.
The paper introduces a sampling theory for graphons with a Poincaré inequality and proves consistency.
problem Sampling on large graphs is challenging due to their non-Euclidean nature.
method The paper introduces a signal sampling theory for graphons, proving a Poincaré inequality and showing consistency.
result Unique sampling sets for graphon signals are consistent across graph sequences.
Quantifies polynomial approximation rates for smooth functions under various distributions.
problem Approximating smooth functions with polynomials under different distributional constraints.
method Develops a quantitative analogue of Carleman's theorem using complex analysis.
result Establishes superexponential rates of approximation for certain function classes over general distributions.