Test Function Space Research Articles

Gelfand Triplets, Ladder Operators and Coherent States

Inspired by a similar construction on Hermite functions, we construct two series of Gelfand triplets, each one spanned by Laguerre–Gauss functions with a fixed positive value of one parameter, considered as the fundamental one. We prove the continuity of different types of ladder operators on these triplets. Laguerre–Gauss functions with negative values of the fundamental parameter are proven to be continuous functionals on one of these triplets. Different sorts of coherent states are considered and proven to be in some spaces of test functions corresponding to Gelfand triplets.

On the convergence of Fourier representations and Schwartz distributions

While Fourier theory remains a cornerstone for analyzing and interpreting the spectral content of diverse signals, its limitations often surface in practical applications. Notably, popular signals like sinusoids, Dirac deltas, signums, and unit steps lack convergent Fourier representations within the conventional framework. This discrepancy necessitates distribution theory to build and interpret suitable representations for such signals. However, existing literature in signal processing and communication engineering often glosses over these intricacies, leaving researchers grappling with obscure concepts regarding the very existence of Fourier representations for non-conforming signals. This work bridges this critical gap by offering a comprehensive exploration of the conditions guaranteeing the existence of Fourier representations. We introduce a novel linear space – the Gauss–Schwartz (GS) function space – and its corresponding class of tempered superexponential (TSE) distributions. We demonstrate that the Fourier transform (FT) acts as an isomorphism on the GS space of test functions and, by duality, on TSE distributions. Crucially, the GS space proves to be minimal in the sense that its dual, encompassing TSE distributions, represents the largest possible linear space over which the FT can be defined through duality. This theoretical advancement signifies a twofold contribution. Firstly, it clarifies the existence and interpretation of Fourier representations for prevalent signals that evade conventional analysis. Secondly, the introduction of the GS-TSE framework expands the reach of the FT to encompass a broader spectrum of functions, enabling novel applications in diverse fields. Ultimately, this work paves the way for a more robust and accessible understanding of Fourier analysis, empowering researchers and practitioners to leverage its full potential in exploring and processing a wider range of signals.

Analysis of a class of spectral volume methods for linear scalar hyperbolic conservation laws

AbstractIn this article, we study the spectral volume (SV) methods for scalar hyperbolic conservation laws with a class of subdivision points under the Petrov–Galerkin framework. Due to the strong connection between the DG method and the SV method with the appropriate choice of the subdivision points, it is natural to analyze the SV method in the Galerkin form and derive the analogous theoretical results as in the DG method. This article considers a class of SV methods, whose subdivision points are the zeros of a specific polynomial with a parameter in it. Properties of the piecewise constant functions under this subdivision, including the orthogonality between the trial solution space and test function space, are provided. With the aid of these properties, we are able to derive the energy stability, optimal a priori error estimates of SV methods with arbitrary high order accuracy. We also study the superconvergence of the numerical solution with the correction function technique, and show the order of superconvergence would be different with different choices of the subdivision points. In the numerical experiments, by choosing different parameters in the SV method, the theoretical findings are confirmed by the numerical results.

Wick multiplication and its relationship with integration and stochastic differentiation on spaces of nonregular test functions in the Lévy white noise analysis

We deal with spaces of nonregular test functions in the Lévy white noise analysis, which are constructed using Lytvynov's generalization of a chaotic representation property. Our goal is to study properties of a natural multiplication $-$ a Wick multiplication on these spaces, and to describe the relationship of this multiplication with integration and stochastic differentiation. More exactly, we establish that the Wick product of nonregular test functions is a nonregular test function; show that when employing the Wick multiplication, it is possible to take a time-independent multiplier out of the sign of a generalized stochastic integral; establish an analog of this result for a Pettis integral (a weak integral); obtain a representation of the generalized stochastic integral via formal Pettis integral from the Wick product of the original integrand by a Lévy white noise; and prove that the operator of stochastic differentiation of first order on the spaces of nonregular test functions satisfies the Leibnitz rule with respect to the Wick multiplication.

A weighted combination of reproducing kernel particle shape functions with cardinal functions of scalable polyharmonic spline radial kernel utilized in Galerkin weak form of a mathematical model related to anti-angiogenic therapy

In this manuscript, a new localized meshfree (meshless) approximation, i.e., a combination of the reproducing kernel particle (RKP) shape functions with the cardinal functions of the scalable polyharmonic spline radial kernel with polynomial augmentation (PHS+poly) is introduced. It is called the RKP+PHS+poly approximation, and its convergence rate is of order O(hm+1), where m is the total degree of polynomials. We apply this method to construct the spaces of trial and test functions in a Galerkin scheme of an extended version of the biological mathematical model in two dimensions describing the interactions between endothelial cells, fibronectin, angiogenic growth factors, and fasentin concentrations. By considering the row-sum method, the eigenvalue stability is also numerically carried out for the discrete equations corresponding to the obtained weak formulation. Accordingly, a semi-implicit form of the backward difference method of order 1 (SBDF1) has been utilized to approximate the weak form in time. We complete our numerical algorithm by solving the obtained full-discretized problem overtime via the biconjugate gradient stabilized (BiCGSTAB) solver with a proper preconditioner. Some simulation results are investigated by estimating the maximum velocity parameter of an enzymatic reaction from the experimental dose–response curve of fasentin to demonstrate the effect of using fasentin drug.

Adaptive Deep Fourier Residual method via overlapping domain decomposition

The Deep Fourier Residual (DFR) method is a specific type of variational physics-informed neural network (VPINN). It provides a robust neural network-based solution to partial differential equations (PDEs). The DFR strategy is based on minimizing the dual norm of the weak residual of a PDE, which is equivalent to minimizing the energy norm of the error. To compute the dual norm of the weak residual, the DFR method employs an orthonormal spectral basis of the test space, known for rectangles or cuboids for multiple function spaces.In this work, we extend the DFR method with ideas of traditional domain decomposition (DD). This enables two improvements: (a) to solve problems in more general polygonal domains and (b) to develop an adaptive refinement technique in the test space using a Döfler marking algorithm. In the former case, we retain the desirable equivalence between the employed loss function and the H1 error under non-restrictive assumptions, numerically demonstrating adherence to explicit bounds in the case of the L-shaped domain problem. In the latter, we show how refinement strategies lead to potentially significant improvements against a reference, classical DFR implementation with a test function space of significantly lower dimensionality, allowing us to better approximate singular solutions at a more reasonable computational cost.

Classical and Quantized Maxwell Fields Deduced from Algebraic Many-Photon Theory

The deduction starts with the (non-relativistic) one-photon Hilbert space \mathcal{H} , equipped with the one-photon Hamiltonian and basic symmetry generators, as the only input information. We recall in the functorially associated boson Fock space the multi-photon dynamics and symmetry transformations, as well as the field operator (as the scaled self-adjoint part of the creation operator) and the q(uasi)-classical states. There is no reference to a presupposed classical Maxwell theory. By abstraction, we go over to the algebraic formulation of the multi-photon theory in terms of a C*-Weyl algebra. Its test function space E\subset \mathcal{H} is constructed as a nuclear Fréchet space, in which – via infrared damping – the dynamics and symmetries are nuclear continuous and their generators bounded. Each w*-closed, singular subspace of the continuous dual E' determines non-Fock coherent states and their mixtures lead to a representation von Neumann algebra with non-trivial center. The symmetry generators restricted to the center can be transformed into the Maxwell form by means of a symplectic transformation and involve the well-known conservation quantities of electrodynamics. This identifies the central part of the represented photon field operator as composed of the two classical canonical electrodynamic field components. We have obtained, therefore, in free space a kind of fusion of the multi-photon theory and the Maxwell theory of transverse electrodynamic fields, where the latter arise as derived quantities. By means of a Bogoliubov transformation one also gets a fusion of the quantized with the classical Maxwell theory, deduced from the photon concept. A sketch of non-relativistic gauging is added in the appendix to gain longitudinal, cohomological, and scalar potentials.

A biorthogonal approach to the infinite dimensional fractional Poisson measure

In this paper, we use a biorthogonal approach to the analysis of the infinite dimensional fractional Poisson measure [Formula: see text], [Formula: see text], on the dual of Schwartz test function space [Formula: see text]. The Hilbert space [Formula: see text] of complex-valued functions is described in terms of a system of generalized Appell polynomials [Formula: see text] associated to the measure [Formula: see text]. The kernels [Formula: see text], [Formula: see text], of the monomials may be expressed in terms of the Stirling operators of the first and second kind as well as the falling factorials in infinite dimensions. Associated to the system [Formula: see text], there is a generalized dual Appell system [Formula: see text] that is biorthogonal to [Formula: see text]. The test and generalized function spaces associated to the measure [Formula: see text] are completely characterized using an integral transform as entire functions.

Towards higher-order accurate mass lumping in explicit isogeometric analysis for structural dynamics

We present a mass lumping approach based on an isogeometric Petrov–Galerkin method that preserves higher-order spatial accuracy in explicit dynamics calculations irrespective of the polynomial degree of the spline approximation. To discretize the test function space, our method uses an approximate dual basis, whose functions are smooth, have local support and satisfy approximate bi-orthogonality with respect to a trial space of B-splines. The resulting mass matrix is “close” to the identity matrix. Specifically, a lumped version of this mass matrix preserves all relevant polynomials when utilized in a Galerkin projection. Consequently, the mass matrix can be lumped (via row-sum lumping) without compromising spatial accuracy in explicit dynamics calculations. We also address the imposition of Dirichlet boundary conditions and the preservation of approximate bi-orthogonality under geometric mappings. In addition, we establish a link between the exact dual and approximate dual basis functions via an iterative algorithm that improves the approximate dual basis towards exact bi-orthogonality. We demonstrate the performance of our higher-order accurate mass lumping approach via convergence studies and spectral analyses of discretized beam, plate and shell models.

Smooth multi-patch scaled boundary isogeometric analysis for Kirchhoff–Love shells

In this work, a linear Kirchhoff–Love shell formulation in the framework of scaled boundary isogeometric analysis is presented that aims to provide a simple approach to trimming for NURBS-based shell analysis. To obtain a global C^1-regular test function space for the shell discretization, an inter-patch coupling is applied with adjusted basis functions in the vicinity of the scaling center to ensure the approximation ability. Doing so, the scaled boundary geometries are related to the concept of analysis-suitable G^1 parametrizations. This yields a coupling of patch boundaries in a strong sense that is restricted to G^1-smooth surfaces. The proposed approach is advantageous to trimmed geometries due to the incorporation of the trimming curve in the boundary representation that provides an exact representation in the planar domain. Since the coupling of star-shaped blocks is feasible, a mesh generation also for complex domains is implementable. The potential of the approach is demonstrated by several problems of untrimmed and trimmed geometries of Kirchhoff–Love shell analysis evaluated against error norms and displacements. Lastly, the applicability is highlighted in the analysis of a violin structure including arbitrarily shaped patches.

A class of dimension-free metrics for the convergence of empirical measures

This paper concerns the convergence of empirical measures in high dimensions. We propose a new class of probability metrics and show that under such metrics, the convergence is free of the curse of dimensionality (CoD). Such a feature is critical for high-dimensional analysis and stands in contrast to classical metrics (e.g., the Wasserstein metric). The proposed metrics fall into the category of integral probability metrics, for which we specify criteria of test function spaces to guarantee the property of being free of CoD. Examples of the selected test function spaces include the reproducing kernel Hilbert spaces, Barron space, and flow-induced function spaces. Three applications of the proposed metrics are presented: 1. The convergence of empirical measure in the case of random variables; 2. The convergence of n-particle system to the solution to McKean–Vlasov stochastic differential equation; 3. The construction of an ɛ-Nash equilibrium for a homogeneous n-player game by its mean-field limit. As a byproduct, we prove that, given a distribution close to the target distribution measured by our metric and a certain representation of the target distribution, we can generate a distribution close to the target one in terms of the Wasserstein metric and relative entropy. Overall, we show that the proposed class of metrics is a powerful tool to analyze the convergence of empirical measures in high dimensions without CoD.

Spectral methods for solving elliptic PDEs on unknown manifolds

In this paper, we propose a mesh-free numerical method for solving elliptic PDEs on unknown manifolds, identified with randomly sampled point cloud data. The PDE solver is formulated as a spectral method where the test function space is the span of the leading eigenfunctions of the Laplacian operator, which are approximated from the point cloud data. While the framework is flexible for any test functional space, we will consider the eigensolutions of a weighted Laplacian obtained from a symmetric Radial Basis Function (RBF) method induced by a weak approximation of a weighted Laplacian on an appropriate Hilbert space. In this paper, we consider a test function space that encodes the geometry of the data yet does not require us to identify and use the sampling density of the point cloud. To attain a more accurate approximation of the expansion coefficients, we adopt a second-order tangent space estimation method to improve the RBF interpolation accuracy in estimating the tangential derivatives. This spectral framework allows us to efficiently solve the PDE many times subjected to different parameters, which reduces the computational cost in the related inverse problem applications. In a well-posed elliptic PDE setting with randomly sampled point cloud data, we provide a theoretical analysis to demonstrate the convergence of the proposed solver as the sample size increases. We also report some numerical studies that show the convergence of the spectral solver on simple manifolds and unknown, rough surfaces. Our numerical results suggest that the proposed method is more accurate than a graph Laplacian-based solver on smooth manifolds. On rough manifolds, these two approaches are comparable. Due to the flexibility of the framework, we empirically found improved accuracies in both smoothed and unsmoothed Stanford bunny domains by blending the graph Laplacian eigensolutions and RBF interpolator.

Convolution on distribution spaces characterized by regularization

AbstractLocally convex convolutor spaces are studied which consist of those distributions that define a continuous convolution operator mapping from the space of test functions into a given locally convex lattice of measures. The convolutor spaces are endowed with the topology of uniform convergence on bounded sets. Their locally convex structure is characterized via regularization and function‐valued seminorms under mild structural assumptions on the space of measures. Many recent generalizations of classical distribution spaces turn out to be special cases of the general convolutor spaces introduced here. Recent topological characterizations of convolutor spaces via regularization are extended and improved. A valuable property of the convolutor spaces in applications is that convolution of distributions inherits continuity properties from those of bilinear convolution mappings between the locally convex lattices of measures.

Solving two-dimensional H(curl)-elliptic interface systems with optimal convergence on unfitted meshes

AbstractFinite element methods developed for unfitted meshes have been widely applied to various interface problems. However, many of them resort to non-conforming spaces for approximation, which is a critical obstacle for the extension to $\textbf{H}(\text{curl})$ equations. This essential issue stems from the underlying Sobolev space $\textbf{H}^s(\text{curl};\,\Omega)$ , and even the widely used penalty methodology may not yield the optimal convergence rate. One promising approach to circumvent this issue is to use a conforming test function space, which motivates us to develop a Petrov–Galerkin immersed finite element (PG-IFE) method for $\textbf{H}(\text{curl})$ -elliptic interface problems. We establish the Nédélec-type IFE spaces and develop some important properties including their edge degrees of freedom, an exact sequence relating to the $H^1$ IFE space and optimal approximation capabilities. We analyse the inf-sup condition under certain assumptions and show the optimal convergence rate, which is also validated by numerical experiments.

A new approach for hardy spaces with variable exponents on spaces of homogeneous type

In the paper, we establish and study Hardy spaces with variable exponents on spaces of homogeneous type (X, d, ?) in the sense of Coifman and Weiss, where d may have no any regularity property and ? fulfills the doubling property only. First we introduce the Hardy spaces with variable exponents Hp(?)(X) by using the wavelet Littlewood-Paley square functions and give their equivalent characterizations. Then we establish the atomic characterization theory for Hp(?)(X) via the new Calder?n-type reproducing identity and the Littlewood-Paley-Stein theory. Finally, wegive a unified method for defining these variable Hardy spaces Hp(?)(X) in terms of the same spaces of test functions and distributions. More precisely, we introduce the variable Carleson measure spaces CMOp(?) L2 (X) and characterize the variable Hardy spaces via the distributions of CMOp(?) L2 (X).

Continuity properties of the shearlet transform and the shearlet synthesis operator on the Lizorkin type spaces

AbstractWe develop a distributional framework for the shearlet transform and the shearlet synthesis operator , where is the Lizorkin test function space and is the space of highly localized test functions on the standard shearlet group . These spaces and their duals are called Lizorkin type spaces of test functions and distributions. We analyze the continuity properties of these transforms when the admissible vector ψ belongs to . Then, we define the shearlet transform and the shearlet synthesis operator of Lizorkin type distributions as transpose mappings of the shearlet synthesis operator and the shearlet transform, respectively. They yield continuous mappings from to and from to . Furthermore, we show the consistency of our definition with the shearlet transform defined by direct evaluation of a distribution on the shearlets. The same can be done for the shearlet synthesis operator. Finally, we give a reconstruction formula for Lizorkin type distributions, from which follows that the action of such generalized functions can be written as an absolutely convergent integral over the standard shearlet group.

Sequence Space Representations for Translation-Modulation Invariant Function and Distribution Spaces

We provide sequence space representations for the test function space $\mathcal{D}_{E}$ and the distribution space $\mathcal{D}^{\prime}_{E}$ associated to a Banach space $E$ belonging to a broad class of translation-modulation invariant Banach spaces of distributions. The spaces $\mathcal{D}_{E}$ and $\mathcal{D}^{\prime}_{E}$ generalize the classical Schwartz spaces $\mathcal{D}_{L^p}$ and $\mathcal{D}^{\prime}_{L^p}$, respectively. Our proof is based on Gabor frame characterizations of $\mathcal{D}_{E}$ and $\mathcal{D}^{\prime}_{E}$, which are also established here and are of independent interest. We recover in a unified way some known sequence space representations as well as obtain several new ones.

Three dimensional high order finite volume element schemes for elliptic equations

AbstractIn this work, we construct and analyze three dimensional high order finite volume element schemes for elliptic equations. In these schemes, the trial function space is chosen as the standard rth order Lagrange finite element space, where is an arbitrary positive integer; the test function space is chosen as the piecewise constant space with respect to the dual mesh of which the control volumes are constructed using Gauss points in each element of the primal mesh. We investigate the inf–sup property of these schemes and based on it, we prove that , and , where is the exact solution, is the rth order finite volume element solution and is the piecewise rth order Lagrange interpolation of . Several numerical examples are presented to verify the theoretical findings.

Application of C-Bézier and H-Bézier basis functions to numerical solution of convection-diffusion equations

Convection-diffusion equation is widely used to describe many engineering and physical problems. The finite element method is one of the most common tools for computing numerical solution. In 2003, Wang et al. proposed C-Bézier and H-Bézier basis functions which are not only a generalization of classical Bernstein basis functions but also have a free shape parameter bringing a lot of flexibility to geometrical modeling. In this paper, we adopt C-Bézier and H-Bézier basis functions to construct test and trial function spaces of finite element method to get numerical solution of convection-diffusion equations. Compared with Lagrange basis functions, numerical accuracy is improved by 1-3 order-of magnitudes which implies a much better approximation in simulating convection-diffusion problems. Several examples are presented to verify the feasibility and effectiveness of our method.

On Wick calculus and its relationship with stochastic integration on spaces of regular test functions in the Lévy white noise analysis

We deal with spaces of regular test functions in the Lévy white noise analysis, which are constructed using Lytvynov's generalization of a chaotic representation property. Our aim is to study properties of Wick multiplication and of Wick versions of holomorphic functions, and to describe a relationship between Wick multiplication and integration, on these spaces. More exactly, we establish that a Wick product of regular test functions is a regular test function; under some conditions a Wick version of a holomorphic function with an argument from the space of regular test functions is a regular test function; show that when employing the Wick multiplication, it is possible to take a time-independent multiplier out of the sign of an extended stochastic integral with respect to a Lévy process; establish an analog of this result for a Pettis integral (a weak integral); obtain a representation of the extended stochastic integral via formal Pettis integral from the Wick product of the original integrand by a Lévy white noise. As an example of an application of our results, we consider an integral stochastic equation with Wick multiplication.