Rearrangements of the Haar system which preserve \BMO

In this paper general rearrangements of the Haar system in BMO are considered. Several, necessary and suficient, conditions for the boundednes of the induced permutation operator are given. Using analytic families of operators extensions to the case of $L^p$ are obtained

Two Remarks on Marcinkiewicz decompositions by Holomorphic Martingales

The real part of H^\infty(\bT) is not dense in L^\infty_{\tR}(\bT). The John-Nirenberg theorem in combination with the Helson-Szeg\"o theorem and the Hunt Muckenhaupt Wheeden theorem has been used to determine whether f\in L^\infty_{\tR}(\bT) can be approximated by \Re H^\infty(\bT) or not: \dist(f,\Re H^\infty)=0 if and only if for every \e>0 there exists \l_0>0 so that for \l>\l_0 and any interval I\sbe \bT. |\{x\in I:|\tilde f-(\tilde f)_I|>\l\}|\le |I|e^{-\l/ \e}, where $\tilde f$ denotes the Hilbert transform of $f$. See [G] p. 259. This result is contrasted by the following \begin{theor} Let f\in L^\infty_{\tR} and \e>0. Then there is a function g\in H^\infty(\bT) and a set E\sb \bT so that |\bT\sm E|<\e and f=\Re g\quad\mbox{ on } E. \end{theor} This theorem is best regarded as a corollary to Men'shov's correction theorem. For the classical proof of Men'shov's theorem see [Ba, Ch VI \S 1-\S4]. Simple proofs of Men'shov's theorem -- together with significant extensions -- have been obtained by S.V. Khruschev in [Kh] and S.V. Kislyakov in [K1], [K2] and [K3]. In [S] C. Sundberg used \bar\pa-techniques (in particular [G, Theorem VIII.1. gave a proof of Theorem 1 that does not mention Men'shov's theorem. The purpose of this paper is to use a Marcinkiewicz decomposition on Holomorphic Martingales to give another proof of Theorem 1. In this way we avoid uniformly convergent Fourier series as well as \bar\pa-techniques

The Banach space H1(X,d,μ)H^1(X,d,\mu), II

In this paper we give the isomorphic classification of atomic $H^1(X,d,\mu)$, where $(X,d,\mu)$ is a space of homogeneous type, hereby completing a line of investigation opened by the work of Bernard Maurey [Ma1], [Ma2], [Ma3] and continued by Lennard Carleson [C] and Przemyslaw Wojtaszczyk [Woj1], [Wpj2]

A Decomposition for Hardy Martingales III

We prove Davis decompositions for vector valued Hardy martingales and illustrate their use. This paper continues our previous work on Davis and Garsia inequalities for scalar Hardy martingales

Jean Bourgain's analytic partition of unity via holomorphic martingales

Using stopping time arguments on holomorphic martingales we present a soft way of constructing J. Bourgain's analytic partitions of unity. Applications to Marcinkiewicz interploation in weighted Hardy spaces are discussed

Permutations of the Haar system

General permutations acting on the Haar system are investigated. We give a necessary and sufficient condition for permutations to induce an isomorphism on dyadic BMO. Extensions of this characterization to Lipschitz spaces \lip, (0 are obtained. When specialized to permutations which act on one level of the Haar system only, our approach leads to a short straightforward proof of a result due to E.M.Semyonov and B.Stoeckert

p-Summing Multiplication Operators, dyadic Hardy Spaces and atomic Decomposition

We constructively determine the Pietsch measure of the 2-summing multiplication operator $$\mathcal{M}_u:\ell^{\infty} \rightarrow H^p, \quad (\varphi_I) \mapsto \sum \varphi_Ix_Ih_I.$$ Our construction of the Pietsch measure for the multiplication operator $\mathcal{M}_u$ involves the Haar coefficients of $u$ and its atomic decomposition.Comment: 23 page

Interpolatory Estimates, Riesz Transforms and Wavelet Projections

We prove that directional wavelet projections and Riesz transforms are related by interpolatory estimates. The exponents of interpolation depend on the H\"older estimates of the wavelet system. This paper complements and continues previous work on Haar projections

The Theory of Diffraction Tomography

Tomography is the three-dimensional reconstruction of an object from images taken at different angles. The term classical tomography is used, when the imaging beam travels in straight lines through the object. This assumption is valid for light with short wavelengths, for example in x-ray tomography. For classical tomography, a commonly used reconstruction method is the filtered back-projection algorithm which yields fast and stable object reconstructions. In the context of single-cell imaging, the back-projection algorithm has been used to investigate the cell structure or to quantify the refractive index distribution within single cells using light from the visible spectrum. Nevertheless, these approaches, commonly summarized as optical projection tomography, do not take into account diffraction. Diffraction tomography with the Rytov approximation resolves this issue. The explicit incorporation of the wave nature of light results in an enhanced reconstruction of the object's refractive index distribution. Here, we present a full literature review of diffraction tomography. We derive the theory starting from the wave equation and discuss its validity with the focus on applications for refractive index tomography. Furthermore, we derive the back-propagation algorithm, the diffraction-tomographic pendant to the back-projection algorithm, and describe its implementation in three dimensions. Finally, we showcase the application of the back-propagation algorithm to computer-generated scattering data. This review unifies the different notations in literature and gives a detailed description of the back-propagation algorithm, serving as a reliable basis for future work in the field of diffraction tomography.Comment: 59 pages, 10 figures, 4 table

Scanning Fluorescence Correlation Spectroscopy (SFCS) with a Scan Path Perpendicular to the Membrane Plane

Scanning fluorescence correlation spectroscopy (SFCS) with a scan path perpendicular to the membrane plane was introduced to measure diffusion and interactions of fluorescent components in free standing biomembranes. Using a confocal laser scanning microscope (CLSM) the open detection volume is moved laterally with kHz frequency through the membrane and the photon events are continuously recorded and stored in a file. While the accessory hardware requirements for a conventional CLSM are minimal, data evaluation can pose a bottleneck. The photon events must be assigned to each scan, in which the maximum signal intensities have to be detected, binned, and aligned between the scans, in order to derive the membrane related intensity fluctuations of one spot. Finally, this time-dependent signal must be correlated and evaluated by well known FCS model functions. Here we provide two platform independent, open source software tools (PyScanFCS and PyCorrFit) that allow to perform all of these steps and to establish perpendicular SFCS in its one- or two-focus as well as its single- or dual-colour modality.Comment: 16 pages, 4 figure