Applied and Computational Harmonic Analysis最新文献

In this paper, we perform the joint study of scale-invariant operators and self-similar processes of complex order. More precisely, we introduce general families of scale-invariant complex-order fractional-derivation and integration operators by constructing them in the Fourier domain. We analyze these operators in detail, with special emphasis on the decay properties of their output. We further use them to introduce a family of complex-valued stable processes that are self-similar with complex-valued Hurst exponents. These random processes are expressed via their characteristic functionals over the Schwartz space of functions. They are therefore defined as generalized random processes in the sense of Gel'fand. Beside their self-similarity and stationarity, we study the Sobolev regularity of the proposed random processes. Our work illustrates the strong connection between scale-invariant operators and self-similar processes, with the construction of adequate complex-order scale-invariant integration operators being preparatory to the construction of the random processes.

We consider the data-driven approximation of the Koopman operator for stochastic differential equations on reproducing kernel Hilbert spaces (RKHS). Our focus is on the estimation error if the data are collected from long-term ergodic simulations. We derive both an exact expression for the variance of the kernel cross-covariance operator, measured in the Hilbert-Schmidt norm, and probabilistic bounds for the finite-data estimation error. Moreover, we derive a bound on the prediction error of observables in the RKHS using a finite Mercer series expansion. Further, assuming Koopman-invariance of the RKHS, we provide bounds on the full approximation error. Numerical experiments using the Ornstein-Uhlenbeck process illustrate our results.

We describe the full structure of the frame set for the Gabor system $G(g;\alpha ,\beta ):=\left\{e^{-2\pi im\beta \cdot }g\right(\cdot -n\alpha ):m,n\in Z\}$ with the window being the Haar function $g=-{\chi }_{[-1/2,0)}+{\chi }_{[0,1/2)}$. This is the first compactly supported window function for which the frame set is represented explicitly.

The strategy of this paper is to introduce the piecewise linear transformation $M$ on the unit circle, and to provide a complete characterization of structures for its (symmetric) maximal invariant sets. This transformation is related to the famous three gap theorem of Steinhaus which may be of independent interest. Furthermore, a classical criterion on Gabor frames is improved, which allows us to establish a necessary and sufficient condition for the Gabor system $G(g;\alpha ,\beta )$ to be a frame, i.e., the symmetric invariant set of the transformation $M$ is empty.

We prove that frame set $F_g$ for imaginary shift of sinc-function$g\left(t\right)=\frac{\sin \pi b(t-iw)}{t-iw},b,w\in R\setminus \left\{0\right\}$ can be described as $F_g=\left\{\right(\alpha ,\beta ):\alpha \beta ⩽1,\beta ⩽|b\left|\right\}$.

In addition, we prove that $F_g=\left\{\right(\alpha ,\beta ):\alpha \beta ⩽1\}$ for window functions g of the form$\frac{1}{t-iw}(1-\sum _{k=1}^{\infty }{a}_k{e}^{2\pi i{b}_{k}t}),$ such that ${\sum }_{k⩾1}\left|a_k\right|e^{2\pi \left|w{b}_k\right|}<1$, $w{b}_k<0$.

This note considers the unstructured sparse recovery problems in a general form. Examples include rational approximation, spectral function estimation, Fourier inversion, Laplace inversion, and sparse deconvolution. The main challenges are the noise in the sample values and the unstructured nature of the sample locations. This note proposes the eigenmatrix, a data-driven construction with desired approximate eigenvalues and eigenvectors. The eigenmatrix offers a new way for these sparse recovery problems. Numerical results are provided to demonstrate the efficiency of the proposed method.

This paper proposes a mesh-free computational framework and machine learning theory for solving elliptic PDEs on unknown manifolds, identified with point clouds, based on diffusion maps (DM) and deep learning. The PDE solver is formulated as a supervised learning task to solve a least-squares regression problem that imposes an algebraic equation approximating a PDE (and boundary conditions if applicable). This algebraic equation involves a graph-Laplacian type matrix obtained via DM asymptotic expansion, which is a consistent estimator of second-order elliptic differential operators. The resulting numerical method is to solve a highly non-convex empirical risk minimization problem subjected to a solution from a hypothesis space of neural networks (NNs). In a well-posed elliptic PDE setting, when the hypothesis space consists of neural networks with either infinite width or depth, we show that the global minimizer of the empirical loss function is a consistent solution in the limit of large training data. When the hypothesis space is a two-layer neural network, we show that for a sufficiently large width, gradient descent can identify a global minimizer of the empirical loss function. Supporting numerical examples demonstrate the convergence of the solutions, ranging from simple manifolds with low and high co-dimensions, to rough surfaces with and without boundaries. We also show that the proposed NN solver can robustly generalize the PDE solution on new data points with generalization errors that are almost identical to the training errors, superseding a Nyström-based interpolation method.

Atomic norm methods have recently been proposed for spectral super-resolution with flexibility in dealing with missing data and miscellaneous noises. A notorious drawback of these convex optimization methods however is their lower resolution in the high signal-to-noise (SNR) regime as compared to conventional methods such as ESPRIT. In this paper, we devise a simple weighting scheme in existing atomic norm methods and show that in theory the resolution of the resulting convex optimization method can be made arbitrarily high in the absence of noise, achieving the so-called separation-free super-resolution. This is proved by a novel, kernel-free construction of the dual certificate whose existence guarantees exact super-resolution using the proposed method. Numerical results corroborating our analysis are provided.

The recovery of multivariate functions and estimating their integrals from finitely many samples is one of the central tasks in modern approximation theory. Marcinkiewicz–Zygmund inequalities provide answers to both the recovery and the quadrature aspect. In this paper, we put ourselves on the q-dimensional sphere $S^q$, and investigate how well continuous $L_p$-norms of polynomials f of maximum degree n on the sphere $S^q$ can be discretized by positively weighted $L_p$-sum of finitely many samples, and discuss the distortion between the continuous and discrete quantities, the number and distribution of the (deterministic or randomly chosen) sample points ${\xi }_1,\dots ,{\xi }_N$ on $S^q$, the dimension q, and the degree n of the polynomials.

For a pair of sets $T,\Omega \subset R$ the time-frequency localization operator is defined as $S_{T,\Omega }=P_T{F}^{-1}{P}_{\Omega }F{P}_T$, where $F$ is the Fourier transform and $P_T,P_{\Omega }$ are projection operators onto T and Ω, respectively. We show that in the case when both T and Ω are intervals, the eigenvalues of $S_{T,\Omega }$ satisfy ${\lambda }_n(T,\Omega )\ge 1-{\delta }^{\left|T\right|\left|\Omega \right|}$ if $n\le (1-\epsilon )\left|T\right|\left|\Omega \right|$, where $\epsilon >0$ is arbitrary and $\delta =\delta \left(\epsilon \right)<1$, provided that $\left|T\right|\left|\Omega \right|>c_{\epsilon }$. This improves the result of Bonami, Jaming and Karoui, who proved it for $\epsilon \ge 0.42$. The proof is based on the properties of the Bargmann transform.

The high efficiency of a recently proposed method for computing with Gaussian processes relies on expanding a (translationally invariant) covariance kernel into complex exponentials, with frequencies lying on a Cartesian equispaced grid. Here we provide rigorous error bounds for this approximation for two popular kernels—Matérn and squared exponential—in terms of the grid spacing and size. The kernel error bounds are uniform over a hypercube centered at the origin. Our tools include a split into aliasing and truncation errors, and bounds on sums of Gaussians or modified Bessel functions over various lattices. For the Matérn case, motivated by numerical study, we conjecture a stronger Frobenius-norm bound on the covariance matrix error for randomly-distributed data points. Lastly, we prove bounds on, and study numerically, the ill-conditioning of the linear systems arising in such regression problems.