Narrow Results

Let be a metric space with doubling measure, L a nonnegative self-adjoint operator in satisfying the Davies–Gaffney estimate, ω a concave function on (0, ∞) of strictly lower type p_ω∈(0, 1] and ρ(t) = t^-1/ω^-1(t^-1) for all t∈(0, ∞). The authors introduce the Orlicz–Hardy space via the Lusin area function associated to the heat semigroup, and the BMO-type space . The authors then establish the duality between and ; as a corollary, the authors obtain the ρ-Carleson measure characterization of the space . Characterizations of , including the atomic and molecular characterizations and the Lusin area function characterization associated to the Poisson semigroup, are also presented. Let and L = -Δ+V be a Schrödinger operator, where is a nonnegative potential. As applications, the authors show that the Riesz transform ∇L^-1/2 is bounded from H_{ω, L}(ℝⁿ) to L(ω). Moreover, if there exist q₁, q₂∈(0, ∞) such that q₁<1<q₂ and [ω(t^q₂)]^q₁ is a convex function on (0, ∞), then several characterizations of the Orlicz–Hardy space H_{ω, L}(ℝⁿ), in terms of the Lusin-area functions, the non-tangential maximal functions, the radial maximal functions, the atoms and the molecules, are obtained. All these results are new even when ω(t) = t^p for all t ∈ (0, ∞) and p ∈ (0, 1).

Let λ > 0, p ∈ ((2λ + 1)/(2λ + 2), 1], and be the Bessel operator. In this paper, the authors establish the characterizations of atomic Hardy spaces H^p((0,∞),dm_λ) associated with △_λ in terms of the radial maximal function, the nontangential maximal function, the grand maximal function, the Littlewood–Paley g-function and the Lusin-area function, where dm_λ(x) ≡ x^2λ dx. As an application, the authors further obtain the Riesz transform characterization of these Hardy spaces.

Let $L$ be a divergence form inhomogeneous higher order elliptic operator with complex bounded measurable coefficients. In this paper, for all $p\in (0,\infty )$ and $L$ satisfying a weak ellipticity condition, the authors introduce the local Hardy spaces $h_L^p({ℝ}^n)$ associated with $L$, which coincide with Goldberg’s local Hardy spaces $h^p({ℝ}^n)$ for all $p\in (0,\infty )$ when $L\equiv -\mathrm{\Delta }$ (the Laplace operator). The authors also establish a real-variable theory of $h_L^p({ℝ}^n)$, which includes their characterizations in terms of the local molecules, the square functions or the maximal functions, the complex interpolation and dual spaces. These real-variable characterizations on the local Hardy spaces are new even when $L\equiv -\mathrm{\text{div}}(A\nabla )$ (the divergence form homogeneous second-order elliptic operator). Moreover, the authors show that $h_L^p({ℝ}^n)$ coincides with the Hardy space $H_{L+\delta }^p({ℝ}^n)$ associated with the operator $L+\delta$ for all $p\in (0,\infty )$, where $\delta$ is some positive constant depending on the ellipticity and the off-diagonal estimates of $L$. As an application, the authors establish some mapping properties for the local Riesz transforms ${\nabla }^k{(L+\delta )}^{-1/2}$ on $H_{L+\delta }^p({ℝ}^n)$, where $k\in \{0,\dots ,m\}$ and $p\in (0,2]$.

Let $\lambda \in (0,\infty )$ and ${△}_{\lambda }:=-\frac{d^2}{d{x}^2}-\frac{2\lambda }{x}\frac{d}{dx}$ be the Bessel operator on ${ℝ}_{+}:=(0,\infty )$. In this paper, the authors show that $b\in \mathrm{\text{BMO}}({ℝ}_{+},d{m}_{\lambda })$ (or $\mathrm{\text{CMO}}({ℝ}_{+},d{m}_{\lambda })$, respectively) if and only if the Riesz transform commutator $[b,R_{{△}_{\lambda }}]$ is bounded (or compact, respectively) on Morrey spaces $L^{p,\kappa }({ℝ}_{+},d{m}_{\lambda })$, where $d{m}_{\lambda }(x):=x^{2\lambda }dx$, $p\in (1,\infty )$ and $\kappa \in (0,1)$. A weak factorization theorem for functions belonging to the Hardy space $H^1({ℝ}_{+},d{m}_{\lambda })$ in the sense of Coifman–Rochberg–Weiss in Bessel setting, via $R_{{△}_{\lambda }}$ and its adjoint, is also obtained.

In this paper, the authors establish a general (two-weight) boundedness criterion for a pair of functions, $(F,f)$, on ${ℝ}^n$ in the scale of weighted Lebesgue spaces, weighted Lorentz spaces, (Lorentz–)Morrey spaces, and variable Lebesgue spaces. As applications, the authors give a unified approach to prove the (two-weight) boundedness of Calderón–Zygmund operators, Littlewood–Paley $g$-functions, Lusin area functions, Littlewood–Paley $g_{\lambda }^{\ast }$-functions, and fractional integral operators, in the aforementioned function spaces. Moreover, via applying the above (two-weight) boundedness criterion, the authors further obtain the (two-weight) boundedness of Riesz transforms, Littlewood–Paley $g$-functions, and fractional integral operators associated with second-order divergence elliptic operators with complex bounded measurable coefficients on ${ℝ}^n$ in the aforementioned function spaces.

The only quadrature operator of order two on L₂(ℝ²) which covaries with orthogonal transforms, in particular rotations is (up to the sign) the Riesz transform. This property was used for the construction of monogenic wavelets and curvelets. Recently, shearlets were applied for various signal processing tasks. Unfortunately, the Riesz transform does not correspond with the shear operation. In this paper, we propose a novel quadrature operator called linearized Riesz transform which is related to the shear operator. We prove properties of this transform and analyze its performance vs. the usual Riesz transform numerically. Furthermore, we demonstrate the relation between the corresponding optical filters. Based on the linearized Riesz transform we introduce finite discrete quasi-monogenic shearlets and prove that they form a tight frame. Numerical experiments show the good fit of the directional information given by the shearlets and the orientation obtained from the quasi-monogenic shearlet coefficients. Finally, we provide experiments on the directional analysis of textures using our quasi-monogenic shearlets.

In this paper, we unify the Riesz transform with the Teager–Kaiser energy operator (TKEO) and monogenic multi-resolution analysis in comparison to conventional gradient and Riesz transform-based approaches. We show that benefits of our approach maintain the following aspects of different known estimators: the stability of mixed (higher) order of derivatives, like given in TKEO, the all-pass filter characteristics featuring the Riesz transform and the scale-based analysis. Furthermore, we demonstrate the ability of the considered technique for extracting structural defects at different scales, estimating orientation or being used for 2D demodulation of amplitude- and phase modulated signals, like interferometric fringe patterns. Additionally, the abilities of orientation estimation of 3D data is demonstrated. From the viewpoint of mathematical analysis, we will emphasize the connection to monogenic higher dimensional signals, Clifford frames, and monogenic multi-resolution signal analysis. Furthermore, we will discuss higher order Riesz transform.

Let M be an n dimensional complete Riemannian manifold satisfying the doubling volume property and an on-diagonal heat kernel estimate. The necessary-sufficient condition for the Sobolev inequality ‖f‖_q ≤ C_n,,ν,p,q(‖∇ f‖_p + ‖f‖_p) (2 ≤ p < q < ∞) is given.

In this paper, we propose some recent works on data analysis and synthesis based on Empirical Mode Decomposition (EMD). Firstly, a direct 2D extension of original Huang EMD algorithm with application to texture analysis, and fractional Brownian motion synthesis. Secondly, an analytical version of EMD based on PDE in 1D-space is presented.

We proposed an extension in 2D-case of the so-called "sifting process" used in the original Huang's EMD. The 2D-sifting process is performed in two steps: extrema detection (by neighboring window or morphological operators) and surface interpolation by splines (thin plate splines or multigrid B-splines). We propose a multiscale segmentation approach by using the zero-crossings from each 2D-intrinsic mode function (IMF) obtained by 2D-EMD.

We apply the Hilbert–Huang transform (which consists of two parts: (a) Empirical mode decomposition, and (b) the Hilbert spectral analysis) to texture analysis. We analyze each 2D-IMF obtained by 2D-EMD by studying local properties (amplitude, phase, isotropy, and orientation) extracted from the monogenic signal of each one of them. The monogenic signal proposed by Felsberg et al. is a 2D-generalization of the analytic signal, where the Riesz transform replaces the Hilbert transform. These local properties are obtained by the structure multivector such as proposed by Felsberg and Sommer.

We present numerical simulations of fractional Brownian textures. Recent works published by Flandrin et al. relate that, in the case of fractional Gaussian noise (fGn), EMD acts essentially as a dyadic filter bank that can be compared to wavelet decompositions. Moreover, in the context of fGn identification, Flandrin et al. show that variance progression across IMFs is related to Hurst exponent H through a scaling law. Starting with these results, we proposed an algorithm to generate fGn, and fractional Brownian motion (fBm) of Hurst exponent H from IMFs obtained from EMD of a White noise, i.e., ordinary Gaussian noise (fGn with H = 1/2).

Deléchelle et al. proposed an analytical approach (formulated as a partial differential equation (PDE)) for sifting process. This PDE-based approach is applied on signals. The analytical approach has a behavior similar to that of the EMD proposed by Huang.

The Hilbert–Huang-Transform (HHT) has proven to be an appropriate multiscale analysis technique specifically for nonlinear and nonstationary time series on non-equidistant grids. It is empirically adapted to the data: first, an additive decomposition of the data (empirical mode decomposition, EMD) into certain multiscale components is computed, denoted as intrinsic mode functions. Second, to each of these components, the Hilbert transform is applied. The resulting Hilbert spectrum of the modes provides a localized time-frequency spectrum and instantaneous (time-dependent) frequencies.

For the first step, the empirical decomposition of the data, a different method based on local means has been developed by Chen et al. (2006). In this paper, we extend their method to multivariate data sets in arbitrary space dimensions. We place special emphasis on deriving a method which is numerically fast also in higher dimensions. Our method works in a coarse-to-fine fashion and is based on adaptive (tensor-product) spline-wavelets. We provide some numerical comparisons to a method based on linear finite elements and one based on thin-plate-splines to demonstrate the performance of our method, both with respect to the quality of the approximation as well as the numerical efficiency.

Second, for a generalization of the Hilbert transform to the multivariate case, we consider the Riesz transformation and an embedding into Clifford-algebra valued functions, from which instantaneous amplitudes, phases and orientations can be derived. We conclude with some numerical examples.