The Wavelet Element Method (WEM) provides a construction of multiresolution systems and biorthogo... more The Wavelet Element Method (WEM) provides a construction of multiresolution systems and biorthogonal wavelets on fairly general domains. These are split into subdomains that are mapped to a single reference hypercube. Tensor products of scaling functions and wavelets defined on the unit interval are used on the reference domain. By introducing appropriate matching conditions across the interelement boundaries, a globally continuous biorthogonal wavelet basis on the general domain is obtained. This construction does not uniquely define the basis functions but rather leaves some freedom for fulfilling additional features. In this paper we detail the general construction principle of the WEM to the 1D, 2D and 3D cases. We address additional features such as symmetry, vanishing moments and minimal support of the wavelet functions in each particular dimension. The construction is illustrated by using biorthogonal spline wavelets on the interval.
Estimates for matrix coefficients of unitary representations of semisimple Lie groups have been s... more Estimates for matrix coefficients of unitary representations of semisimple Lie groups have been studied for a long time, starting with the seminal work by Bargmann, by Ehrenpreis and Mautner, and by Kunze and Stein. Two types of estimates have been established: on the one hand, $L^p$ estimates, which are a dual formulation of the Kunze--Stein phenomenon, and which hold for all matrix coefficients, and on the other pointwise estimates related to asymptotic expansions at infinity, which are more precise but only hold for a restricted class of matrix coefficients. In this paper we prove a new type of estimate for the irreducibile unitary representations of $\mathrm{SL}(2,\mathbb{R})$ and for the so-called metaplectic representation, which we believe has the best features of, and implies, both forms of estimate described above. As an application outside representation theory, we prove a new $L^2$ estimate of dispersive type for the free Schrodinger equation in $\mathbb{R}^n$.
A detailed analysis of the electric field integral equation (EFIE) at low frequencies is presente... more A detailed analysis of the electric field integral equation (EFIE) at low frequencies is presented. The analysis, that is based on the Sobolev space mapping properties of the EFIE, explains the conditioning growth of the majority of currently available quasi-Helmholtz decomposition methods (such as Loop-Tree/Star, Loop-Rearranged Trees, etc.) that cure the EFIE low-frequency breakdown. It is shown that these methods have a conditioning that grows polynomially with the number of unknowns. To solve this problem, in this work we present a quasi-Helmholtz decomposition method that leads to an EFIE whose conditioning grows only logaritmically with the number of unknowns. This result is obtained by properly regularizing both the solenoidal and non-solenoidal part of the EFIE. The regularization is obtained by introducing a new set of loop hierarchical basis functions. Numerical tests are provided to confirm the results obtained by the theory.
We consider trees with root at infinity endowed with flow measures, which are nondoubling measure... more We consider trees with root at infinity endowed with flow measures, which are nondoubling measures of at least exponential growth and which do not satisfy the isoperimetric inequality. In this setting, we develop a Calderón–Zygmund theory and we define BMO and Hardy spaces, proving a number of desired results extending the corresponding theory as known in more classical settings.
A. We consider an infinite homogeneous tree V endowed with the usual metric d defined Hebisch and... more A. We consider an infinite homogeneous tree V endowed with the usual metric d defined Hebisch and Steger [8] developed a new Calderón-Zygmund theory which can be applied also to nondoubling metric measure spaces and showed that such a theory can be applied to the space (V , d, µ). In particular they proved that there exists a family of appropriate sets in V , which are called Calderón-Zygmund sets, which replace the family of balls in the classical Calderón-Zygmund theory. We mention also that some properties of the space (V , d, µ) were investigated in more detail in [1]. Key words and phrases. Hardy spaces; homogeneous trees; exponential growth. The second and third authors are members of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). .
Advances in Microlocal and Time-Frequency Analysis, 2020
A. We consider an infinite homogeneous tree V endowed with the usual metric d defined Hebisch and... more A. We consider an infinite homogeneous tree V endowed with the usual metric d defined Hebisch and Steger [8] developed a new Calderón-Zygmund theory which can be applied also to nondoubling metric measure spaces and showed that such a theory can be applied to the space (V , d, µ). In particular they proved that there exists a family of appropriate sets in V , which are called Calderón-Zygmund sets, which replace the family of balls in the classical Calderón-Zygmund theory. We mention also that some properties of the space (V , d, µ) were investigated in more detail in [1]. Key words and phrases. Hardy spaces; homogeneous trees; exponential growth. The second and third authors are members of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). .
The Wavelet Element Method (WEM) combines biorthogonal wavelet systems with the philosophy of Spe... more The Wavelet Element Method (WEM) combines biorthogonal wavelet systems with the philosophy of Spectral Element Methods in order to obtain a biorthogonal wavelet system on fairly general bounded domains in some Rn. The domain of interest is split into subdomains which are mapped to a simple reference domain, heren-dimensional cubes. Thus, one has to construct appropriate biorthogonal wavelets on the
We consider the (extended) metaplectic representation of the semidirect product $\mathcal{G}={\ma... more We consider the (extended) metaplectic representation of the semidirect product $\mathcal{G}={\mathbb H}^d\rtimes Sp(d,{\mathbb R})$ between the Heisenberg group and the symplectic group. Subgroups $H=\Sigma \rtimes D$, with $\Sigma$ being a $d\times d$ symmetric matrix and $D$ a closed subgroup of $GL(d,{\mathbb R})$, are our main concern. We shall give a general setting for the reproducibility of such groups which include and assemble the ones for the single examples treated in [5]. As a byproduct, the extended metaplectic representation restricted to some classes of such subgroups is either the Schrödinger representation of ${\mathbb R}^{2d}$ or the wavelet representation of ${\mathbb R}^d\rtimes D$, with $D$ closed subgroup of $GL(d,{\mathbb R})$. Finally, we shall provide new examples of reproducing groups of the type $H=\Sigma\rtimes D$, in dimension $d=2$.
An anisotropic functional setting for convection-diffusion problems
Abstract A new functional framework for consistently stabilized discrete approximations to convec... more Abstract A new functional framework for consistently stabilized discrete approximations to convection-diffusion problems was recently proposed. The key ideas are the evaluation of the residual in an inner product of the type H –1/2 and the realization of this inner product via explicitely computable multilevel decompositions of function spaces. Here we improve such approach, by taking into account the anisotropic nature of the convection-diffusion operator. We derive uniform (in the diffusion parameter) anisotropic estimates for both the exact and the discrete solutions and we study the convergence of the approximation. To this end we develop a functional framework involving anisotropic Sobolev spaces which depend on the velocity field.
The Wavelet Element Method (WEM) is a construction of multiresolution systems and biorthogonal wa... more The Wavelet Element Method (WEM) is a construction of multiresolution systems and biorthogonal wavelets on fairly general domains. Domain decomposition and mapping to a reference domain allow the use of tensor products of scaling functions and wavelets on the unit interval. Appropriate matching conditions across the interelement boundaries yield a globally continuous biorthogonal wavelet basis. In this paper, we detail the general construction for two{dimensional domains and show how to use the WEM for the numerical solution of elliptic PDE's in an L{shaped domain.
The goal of this paper is to provide wavelet characterizations for anisotropic Besov spaces. Depe... more The goal of this paper is to provide wavelet characterizations for anisotropic Besov spaces. Depending on the anisotropy, appropriate biorthogonal tensor product bases are introduced and Jackson and Bernstein estimates are proved for two-parameter families of finite-dimensional spaces. These estimates lead to characterizations for anisotropic Besov spaces by anisotropy-dependent linear approximation spaces and lead further on to interpolation and embedding results. Finally, wavelet characterizations for anisotropic Besov spaces with respect to L pspaces with 0 < p < ∞ are derived.
The Wavelet Element Method (WEM) provides a construction of multiresolution systems and biorthogo... more The Wavelet Element Method (WEM) provides a construction of multiresolution systems and biorthogonal wavelets on fairly general domains. These are split into subdomains that are mapped to a single reference hypercube. Tensor products of scaling functions and wavelets defined on the unit interval are used on the reference domain. By introducing appropriate matching conditions across the interelement boundaries, a globally continuous biorthogonal wavelet basis on the general domain is obtained. This construction does not uniquely define the basis functions but rather leaves some freedom for fulfilling additional features. In this paper we detail the general construction principle of the WEM to the 1D, 2D and 3D cases. We address additional features such as symmetry, vanishing moments and minimal support of the wavelet functions in each particular dimension. The construction is illustrated by using biorthogonal spline wavelets on the interval.
Estimates for matrix coefficients of unitary representations of semisimple Lie groups have been s... more Estimates for matrix coefficients of unitary representations of semisimple Lie groups have been studied for a long time, starting with the seminal work by Bargmann, by Ehrenpreis and Mautner, and by Kunze and Stein. Two types of estimates have been established: on the one hand, $L^p$ estimates, which are a dual formulation of the Kunze--Stein phenomenon, and which hold for all matrix coefficients, and on the other pointwise estimates related to asymptotic expansions at infinity, which are more precise but only hold for a restricted class of matrix coefficients. In this paper we prove a new type of estimate for the irreducibile unitary representations of $\mathrm{SL}(2,\mathbb{R})$ and for the so-called metaplectic representation, which we believe has the best features of, and implies, both forms of estimate described above. As an application outside representation theory, we prove a new $L^2$ estimate of dispersive type for the free Schrodinger equation in $\mathbb{R}^n$.
A detailed analysis of the electric field integral equation (EFIE) at low frequencies is presente... more A detailed analysis of the electric field integral equation (EFIE) at low frequencies is presented. The analysis, that is based on the Sobolev space mapping properties of the EFIE, explains the conditioning growth of the majority of currently available quasi-Helmholtz decomposition methods (such as Loop-Tree/Star, Loop-Rearranged Trees, etc.) that cure the EFIE low-frequency breakdown. It is shown that these methods have a conditioning that grows polynomially with the number of unknowns. To solve this problem, in this work we present a quasi-Helmholtz decomposition method that leads to an EFIE whose conditioning grows only logaritmically with the number of unknowns. This result is obtained by properly regularizing both the solenoidal and non-solenoidal part of the EFIE. The regularization is obtained by introducing a new set of loop hierarchical basis functions. Numerical tests are provided to confirm the results obtained by the theory.
We consider trees with root at infinity endowed with flow measures, which are nondoubling measure... more We consider trees with root at infinity endowed with flow measures, which are nondoubling measures of at least exponential growth and which do not satisfy the isoperimetric inequality. In this setting, we develop a Calderón–Zygmund theory and we define BMO and Hardy spaces, proving a number of desired results extending the corresponding theory as known in more classical settings.
A. We consider an infinite homogeneous tree V endowed with the usual metric d defined Hebisch and... more A. We consider an infinite homogeneous tree V endowed with the usual metric d defined Hebisch and Steger [8] developed a new Calderón-Zygmund theory which can be applied also to nondoubling metric measure spaces and showed that such a theory can be applied to the space (V , d, µ). In particular they proved that there exists a family of appropriate sets in V , which are called Calderón-Zygmund sets, which replace the family of balls in the classical Calderón-Zygmund theory. We mention also that some properties of the space (V , d, µ) were investigated in more detail in [1]. Key words and phrases. Hardy spaces; homogeneous trees; exponential growth. The second and third authors are members of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). .
Advances in Microlocal and Time-Frequency Analysis, 2020
A. We consider an infinite homogeneous tree V endowed with the usual metric d defined Hebisch and... more A. We consider an infinite homogeneous tree V endowed with the usual metric d defined Hebisch and Steger [8] developed a new Calderón-Zygmund theory which can be applied also to nondoubling metric measure spaces and showed that such a theory can be applied to the space (V , d, µ). In particular they proved that there exists a family of appropriate sets in V , which are called Calderón-Zygmund sets, which replace the family of balls in the classical Calderón-Zygmund theory. We mention also that some properties of the space (V , d, µ) were investigated in more detail in [1]. Key words and phrases. Hardy spaces; homogeneous trees; exponential growth. The second and third authors are members of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). .
The Wavelet Element Method (WEM) combines biorthogonal wavelet systems with the philosophy of Spe... more The Wavelet Element Method (WEM) combines biorthogonal wavelet systems with the philosophy of Spectral Element Methods in order to obtain a biorthogonal wavelet system on fairly general bounded domains in some Rn. The domain of interest is split into subdomains which are mapped to a simple reference domain, heren-dimensional cubes. Thus, one has to construct appropriate biorthogonal wavelets on the
We consider the (extended) metaplectic representation of the semidirect product $\mathcal{G}={\ma... more We consider the (extended) metaplectic representation of the semidirect product $\mathcal{G}={\mathbb H}^d\rtimes Sp(d,{\mathbb R})$ between the Heisenberg group and the symplectic group. Subgroups $H=\Sigma \rtimes D$, with $\Sigma$ being a $d\times d$ symmetric matrix and $D$ a closed subgroup of $GL(d,{\mathbb R})$, are our main concern. We shall give a general setting for the reproducibility of such groups which include and assemble the ones for the single examples treated in [5]. As a byproduct, the extended metaplectic representation restricted to some classes of such subgroups is either the Schrödinger representation of ${\mathbb R}^{2d}$ or the wavelet representation of ${\mathbb R}^d\rtimes D$, with $D$ closed subgroup of $GL(d,{\mathbb R})$. Finally, we shall provide new examples of reproducing groups of the type $H=\Sigma\rtimes D$, in dimension $d=2$.
An anisotropic functional setting for convection-diffusion problems
Abstract A new functional framework for consistently stabilized discrete approximations to convec... more Abstract A new functional framework for consistently stabilized discrete approximations to convection-diffusion problems was recently proposed. The key ideas are the evaluation of the residual in an inner product of the type H –1/2 and the realization of this inner product via explicitely computable multilevel decompositions of function spaces. Here we improve such approach, by taking into account the anisotropic nature of the convection-diffusion operator. We derive uniform (in the diffusion parameter) anisotropic estimates for both the exact and the discrete solutions and we study the convergence of the approximation. To this end we develop a functional framework involving anisotropic Sobolev spaces which depend on the velocity field.
The Wavelet Element Method (WEM) is a construction of multiresolution systems and biorthogonal wa... more The Wavelet Element Method (WEM) is a construction of multiresolution systems and biorthogonal wavelets on fairly general domains. Domain decomposition and mapping to a reference domain allow the use of tensor products of scaling functions and wavelets on the unit interval. Appropriate matching conditions across the interelement boundaries yield a globally continuous biorthogonal wavelet basis. In this paper, we detail the general construction for two{dimensional domains and show how to use the WEM for the numerical solution of elliptic PDE's in an L{shaped domain.
The goal of this paper is to provide wavelet characterizations for anisotropic Besov spaces. Depe... more The goal of this paper is to provide wavelet characterizations for anisotropic Besov spaces. Depending on the anisotropy, appropriate biorthogonal tensor product bases are introduced and Jackson and Bernstein estimates are proved for two-parameter families of finite-dimensional spaces. These estimates lead to characterizations for anisotropic Besov spaces by anisotropy-dependent linear approximation spaces and lead further on to interpolation and embedding results. Finally, wavelet characterizations for anisotropic Besov spaces with respect to L pspaces with 0 < p < ∞ are derived.
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