Related papers: Functional analysis and partial differential equat…
The aim of the paper is to investigate the solutions of special inhomogeneous linear functional equations by using spectral analysis in a translation invariant closed linear subspace of additive/multiadditive functions containing the…
We derive an estimator of the spectral density of a functional time series that is the output of a multilayer perceptron neural network. The estimator is motivated by difficulties with the computation of existing spectral density estimators…
While calculations and measurements of single-particle spectral properties often offer the most direct route to study correlated electron systems, the underlying physics may remain quite elusive, if information at higher particle levels is…
Subspaces obtained by the orthogonal projection of locally supported square-integrable vector fields onto the Hardy spaces $H_+(\mathbb{S})$ and $H_-(\mathbb{S})$, respectively, play a role in various inverse potential field problems since…
The composition of the Fourier transform in $\mathbb{R}^n$ with a suitable pseudodifferential operator is called a Fourier operator. It is compact in appropriate function spaces. The paper deals with its spectral theory. This is based on…
In this article, we focus on the construction of multivariate fractal functions in Lebesgue spaces along with some properties of associated fractal operator. First, we give a detailed construction of the fractal functions belonging to…
A burgeoning line of research leverages deep neural networks to approximate the solutions to high dimensional PDEs, opening lines of theoretical inquiry focused on explaining how it is that these models appear to evade the curse of…
A plentitude of applications in scientific computing requires the approximation of mappings between Banach spaces. Recently introduced Fourier Neural Operator (FNO) and Deep Operator Network (DeepONet) can provide this functionality. For…
Rapidly growing demands for high-performance computing, powerful data processing systems, and big data necessitate the advent of novel optical devices to perform demanding computing processes effectively. Due to its unprecedented growth in…
Function encoders are a recent technique that learn neural network basis functions to form compact, adaptive representations of Hilbert spaces of functions. We show that function encoders provide a principled connection to feature learning…
This work is concerned with variational analysis of so-called spectral functions and spectral sets of matrices that only depend on eigenvalues of the matrix. Based on our previous work [H. T. B\`ui, M. N. B\`ui, and C. Clason, Convex…
We develop a variational framework to understand the properties of functions learned by fitting deep neural networks with rectified linear unit activations to data. We propose a new function space, which is reminiscent of classical bounded…
In this article, we study properties of multilinear Fourier integral operators on weighted modulation spaces. In particular, using the theory of Gabor frames, we study boundedness of multilinear Fourier integral operators on products of…
We study fractional variants of the quasi-norms introduced by Brezis, Van Schaftingen, and Yung in the study of the Sobolev space $\dot W^{1,p}$. The resulting spaces are identified as a special class of real interpolation spaces of…
A new notion of stochastic transformation is proposed and applied to the study of both weak and strong symmetries of stochastic differential equations (SDEs). The correspondence between an algebra of weak symmetries for a given SDE and an…
We explain an array of basic functional analysis puzzles on the way to general spectral flow formulae and indicate a direction of future topological research for dealing with these puzzles.
The classical development of neural networks has primarily focused on learning mappings between finite dimensional Euclidean spaces or finite sets. We propose a generalization of neural networks to learn operators, termed neural operators,…
We construct fractional Sobolev spaces on arbitrary time scales, both in one dimension and on product time scales. In 1D, we define $W^{\alpha(\cdot),p}_{\mathrm{rd}}(\mathcal I)$ through a variable-order Gagliardo-type seminorm and prove…
To describe longitudinal fine structure like microbunching within a particle beam, a classical approach is to define a bunching factor which is the Fourier transform of the particle longitudinal density distribution. Such a 1D definition of…
The recent development of spectral method has been praised for its high-order convergence in simulating complex physical problems. The combination of embedded boundary method and spectral method becomes a mainstream way to tackle…