Related papers: Sampling for approximating $R$-limited functions
A deep approximation is an approximating function defined by composing more than one layer of simple functions. We study deep approximations of functions of one variable using layers consisting of low-degree polynomials or simple conformal…
The local limit theorem describes the behavior of the convolution powers of a probability distribution supported on Z. In this work, we explore the role played by positivity in this classical result and study the convolution powers of the…
Tomography is a central tool in medical applications, allowing doctors to investigate patients' interior features. The Radon transform (in two dimensions) is commonly used to model the measurement process in parallel-beam CT. Suitable…
In this article, we develop a new method to approximate numerically the fractional Laplacian of functions defined on $\mathbb R$, as well as some more general singular integrals. After mapping $\mathbb R$ into a finite interval, we…
In this paper, we investigate the convergence properties of Fourier partial sums associated with general orthonormal systems, focusing on functions that belong to specific differentiable function classes. While classical Fourier analysis…
Approximations based on random Fourier features have recently emerged as an efficient and formally consistent methodology to design large-scale kernel machines. By expressing the kernel as a Fourier expansion, features are generated based…
For set-valued functions (SVFs, multifunctions), mapping a compact interval $[a,b]$ into the space of compact non-empty subsets of ${\mathbb R}^d$, we study approximation based on the metric approach that includes metric linear…
We present here a new method for approximating functions defined on superreflexive Banach spaces by differentiable functions with $\alpha$-H\"older derivatives (for some $0<\alpha\leq 1$). The smooth approximation is given by means of an…
Uncertainty quantification for neural operators remains an open problem in the infinite-dimensional setting due to the lack of finite-sample coverage guarantees over functional outputs. While conformal prediction offers finite-sample…
The Fourier Basis Density Model (FBM) was recently introduced as a flexible probability model for band-limited distributions, i.e. ones which are smooth in the sense of having a characteristic function with limited support around the…
We introduce an adaptation of integral approximation operators to set-valued functions (SVFs, multifunctions), mapping a compact interval $[a,b]$ into the space of compact non-empty subsets of ${\mathbb R}^d$. All operators are adapted by…
We consider a class of linear integral operators with impulse responses varying regularly in time or space. These operators appear in a large number of applications ranging from signal/image processing to biology. Evaluating their action on…
The interaction between discrete and continuous mathematics lies at the heart of many fundamental problems in applied mathematics and computational sciences. In this paper we discuss the problem of discretizing vector-valued functions…
This paper addresses the deconvolution of an image that has been obtained by superimposing many copies of an underlying unknown image of interest. The superposition is assumed to not be exact due to noise, and is described using an error…
Fourier series multiscale method, a concise and efficient analytical approach for multiscale computation, will be developed out of this series of papers. The second paper is concerned with simultaneous approximation to functions and their…
When simulating sky images, one often takes a galaxy image $F(x)$ defined by a set of pixelized samples and an interpolation kernel, and then wants to produce a new sampled image representing this galaxy as it would appear with a different…
For $ 1\le k <n$, we prove that for functions $F,G$ on $ {\Bbb R}^{n}$, any $k$-dimensional affine subspace $H \subset {\Bbb R}^{n}$, and $p,q,r \ge 2$ with $\frac{1}{p}+\frac{1}{q}+\frac{1}{r}=1$, one has the estimate $$…
In this article, we present an $O(N \log N)$ rapidly convergent algorithm for the numerical approximation of the convolution integral with radially symmetric weakly singular kernels and compactly supported densities. To achieve the reduced…
In this paper we prove a uniform Fourier restriction estimate over the class of simple curves where the last coordinate function can be extended to a holomorphic function of bounded frequency in a sufficiently large disc. The proof is based…
Interpolation and approximation of functionals with conditionally positive definite kernels is considered on sets of centers that are not determining for polynomials. It is shown that polynomial consistency is sufficient in order to define…