Radial growth of harmonic functions in the unit ball

We study harmonic functions which admit a certain majorant in the unit ball in $\R^m$. We prove that when the majorant fulfills a doubling condition, the extremal growth or decay may occur only along small sets of radii, and we give precise estimates of these exceptional sets

Homotopy method for seismic modeling in strongly scattering acoustic media with density variation

The wave equation for acoustic media with variable density and velocity can be transformed into an integral equation of the Lippmann-Schwinger type; but for a 4-dimensional state vector involving the gradient of the pressure field as well as the pressure field itself. The Lippmann-Schwinger equation can in principle be solved exactly via matrix inversion, but the computational cost of matrix inversion scales like N^3, where N is the number of grid blocks. The computational cost can be significantly reduced if one solves the Lippmann-Schwinger equation iteratively. However, the popular Born series is only guaranteed to converge if the contrasts and the size of the model (relative to the wavelength) are relatively small. In this study, we have used the so-called homotopy analysis method to derive an iterative method of the Lippmann-Schwinger equation which is guaranteed to converge independent of the contrasts and size of the model. The computational cost of our convergent scattering series scales as N^2 times the number of iterations. Our algorithm, which is based on the homotopy analysis method, involves a convergence control operator that we select using a randomized matrix factorization. We illustrate the performance of the new convergent scattering series by seismic wave-field modelling in a strongly scattering salt model with variable density and velocity.acceptedVersio

Random harmonic functions in growth spaces and Bloch-type spaces

Let $h^\infty_v(\mathbf D)$ and $h^\infty_v(\mathbf B)$ be the spaces of harmonic functions in the unit disk and multi-dimensional unit ball which admit a two-sided radial majorant $v(r)$. We consider functions $v$ that fulfill a doubling condition. In the two-dimensional case let u (re^{i\ta},\xi) = \sum_{j=0}^\infty (a_{j0} \xi_{j0} r^j \cos j\theta +a_{j1} \xi_{j1} r^j \sin j\theta) where $\xi =\{\xi_{ji}\}$ is a sequence of random subnormal variables and $a_{ji}$ are real; in higher dimensions we consider series of spherical harmonics. We will obtain conditions on the coefficients $a_{ji}$ which imply that $u$ is in $h^\infty_v(\mathbf B)$ almost surely. Our estimate improves previous results by Bennett, Stegenga and Timoney, and we prove that the estimate is sharp. The results for growth spaces can easily be applied to Bloch-type spaces, and we obtain a similar characterization for these spaces, which generalizes results by Anderson, Clunie and Pommerenke and by Guo and Liu

Product of Hyperfunctions with Disjoint Support

We prove that if two hyperfunctions on the unit circle have disjoint support, then the convolution of their Fourier coefficients multiplied with a weight is zero when the weight goes to 1. We prove this by using the Fourier-Borel transform and the G-transform of analytic functionals. The proof is inspired by an article by Yngve Domar. In the end of his article he proves the existence of a translation-invariant subspace of a certain weighted l^p-space. This proof has similarities to our proof, so we compare them. We also look at other topics related to Domar's article, for example the existence of entire functions of order less than or equal to 1 under certain restrictions on the axes. We will see how the Beurling-Malliavin theorem gives some answers to this question. Finally, we prove that if two hyperfunctions on the real line have compact and disjoint support, then the convolution of their Fourier transforms multiplied with a weight is zero when the weight goes to 1

Radial growth of harmonic functions in the unit ball

Let $\Psi_v$ be the class of harmonic functions in the unit disk or unit ball in ${\mathsf R}^m$ which admit a radial majorant $v(r)$. We prove that a function in $\Psi_v$ may grow or decay as fast as $v$ only along a set of radii of measure zero. For the case when $v$ fulfills a doubling condition, we give precise estimates of these exceptional sets in terms of Hausdorff measures

Homotopy scattering series for seismic forward modelling with variable density and velocity

We have derived a convergent scattering series solution for the frequency-domain wave equation in acoustic media with variable density and velocity. The convergent scattering series solution is based on the homotopy analysis of a vectorial integral equation of the Lippmann–Schwinger type. By using the Green's function and partial integration, we have derived the vectorial integral equation of the Lippmann–Schwinger type that involves the pressure gradient field as well as the pressure field from the wave equation. The vectorial Lippmann–Schwinger equation can in principle be solved via matrix inversion, but the computational cost of matrix inversion scales like �3 , where � is the number of grid blocks. The computational cost can be significantly reduced if one solves the vectorial Lippmann–Schwinger equation iteratively. A simple iterative solution is the Born series, but it is only convergent when the scattering potential is sufficiently small. In this study, we have used the so-called homotopy analysis method to derive an iterative solution for the vectorial Lippmann–Schwinger equation which can be made convergent even in strongly scattering media. The computational cost of our convergent scattering series scales as �2 . Our algorithm, which is based on the homotopy analysis method, involves a convergence control operator that we select using hierarchical matrices. We use a three-layer model and a resampled version of the SEG/EAGE salt model to show the performance of the developed convergent scattering series

Homotopy method for seismic modeling in strongly scattering acoustic media with density variation

The wave equation for acoustic media with variable density and velocity can be transformed into an integral equation of the Lippmann-Schwinger type; but for a 4-dimensional state vector involving the gradient of the pressure field as well as the pressure field itself. The Lippmann-Schwinger equation can in principle be solved exactly via matrix inversion, but the computational cost of matrix inversion scales like N^3, where N is the number of grid blocks. The computational cost can be significantly reduced if one solves the Lippmann-Schwinger equation iteratively. However, the popular Born series is only guaranteed to converge if the contrasts and the size of the model (relative to the wavelength) are relatively small. In this study, we have used the so-called homotopy analysis method to derive an iterative method of the Lippmann-Schwinger equation which is guaranteed to converge independent of the contrasts and size of the model. The computational cost of our convergent scattering series scales as N^2 times the number of iterations. Our algorithm, which is based on the homotopy analysis method, involves a convergence control operator that we select using a randomized matrix factorization. We illustrate the performance of the new convergent scattering series by seismic wave-field modelling in a strongly scattering salt model with variable density and velocity