Anisotropic Mixed-Norm Hardy Spaces Cleanthous, G.; Georgiadis, A. G.; Nielsen, M.
The Journal of geometric analysis,
10/2017, Letnik:
27, Številka:
4
Journal Article
Recenzirano
We introduce and explore Hardy spaces defined by mixed Lebesgue norms and anisotropic dilations. We prove that the definitions of these spaces in terms of smooth, non-tangential, auxiliary, grand, ...and Poisson maximal operators coincide. We also study the relation between anisotropic mixed-norm Hardy spaces and mixed-norm Lebesgue spaces.
Smooth molecular decompositions for holomorphic Besov and Triebel–Lizorkin spaces on the unit disk of the complex plane are constructed. The decompositions are used to obtain a boundedness result for ...Fourier multipliers. As further applications, we provide equivalent norms for the spaces under consideration, we consider the implications of the results on Hardy and Hardy–Sobolev spaces, and we study boundedness of coefficient multipliers.
The family of anisotropic decomposition spaces of modulation and Triebel–Lizorkin type on
R
n
is a large family of smoothness spaces that include classical Besov, Triebel–Lizorkin, modulation and
α
...-modulation spaces. The decomposition space approach allows for a unified treatment of such smoothness spaces in both the isotropic and an anisotropic setting. We derive a boundedness result for Fourier multipliers on anisotropic decomposition spaces of modulation and Triebel–Lizorkin type. As an application, we obtain equivalent quasi-norm characterizations for this class of decomposition spaces.
Gaussian random fields defined over compact two-point homogeneous spaces are considered and Sobolev regularity and Hölder continuity are explored through spectral representations. It is shown how ...spectral properties of the covariance function associated to a given Gaussian random field are crucial to determine such regularities and geometric properties. Furthermore, fast approximations of random fields on compact two-point homogeneous spaces are derived by truncation of the series expansion, and a suitable bound for the error involved in such an approximation is provided.
Abstract We are studying the problem of estimating density in a wide range of metric spaces, including the Euclidean space, the sphere, the ball, and various Riemannian manifolds. Our framework ...involves a metric space with a doubling measure and a self-adjoint operator, whose heat kernel exhibits Gaussian behaviour. We begin by reviewing the construction of kernel density estimators and the related background information. As a novel result, we present a pointwise kernel density estimation for probability density functions that belong to general Hölder spaces. The study is accompanied by an application in Seismology. Precisely, we analyze a globally-indexed dataset of earthquake occurrence and compare the out-of-sample performance of several approximated kernel density estimators indexed on the sphere.
We consider the problem of estimating the density of observations taking values in classical or nonclassical spaces such as manifolds and more general metric spaces. Our setting is quite general but ...also sufficiently rich in allowing the development of smooth functional calculus with well localized spectral kernels, Besov regularity spaces, and wavelet type systems. Kernel and both linear and nonlinear wavelet density estimators are introduced and studied. Convergence rates for these estimators are established, which are analogous to the existing results in the classical setting of real-valued variables.
In evoked potential testing, tolerance limits are used to define parameters measured as normal or abnormal. These tolerance limits are calculated using means and standard deviations and depend on the ...samples having a Normal or Gaussian distribution. It is important to check samples for Normal distribution before calculating tolerance limits. When looking at interside differences during evoked potential testing, we have found that absolute values are less accurate than using the values obtained by subtracting values obtained from the right side from those on the left or vice versa.
Mixed-norm \alpha-modulation spaces Cleanthous, Galatia; Georgiadis, Athanasios G.
Transactions of the American Mathematical Society,
05/2020, Letnik:
373, Številka:
5
Journal Article
Recenzirano
Mixed-norm \alpha -modulation spaces are introduced and their properties are explored. Precisely we define \alpha -modulation spaces with mixed Lebesgue norms, based on mixed-norm bounded admissible ...partitions of unity ( \vec {p}-BAPU). We prove that the definition is independent of the \vec {p}-BAPU and that the lifting property holds for the newly introduced spaces. We provide embedding theorems between the new spaces, we find their duals, and we study the boundedness of Fourier multipliers under general mixed-norm conditions. In addition we obtain a discrete decomposition based on frame systems whose construction combines the operations used to construct Gabor and wavelet systems. Some properties of mixed Lebesgue spaces, of independent interest, are proved as well.
