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Function Spaces of Logarithmic Smoothness: Embeddings and Characterizations
In this paper we present a comprehensive treatment of function spaces with logarithmic smoothness (Besov, Sobolev, Triebel-Lizorkin).
We establish the following results:
The key tools behind our results
are limiting interpolation techniques and new characterizations of Besov and Sobolev norms in terms of the behavior of the Fourier
transforms for functions such that their Fourier transforms are of monotone type or lacunary series.
eBook
Embeddings of Decomposition Spaces
Many smoothness spaces in harmonic analysis are decomposition spaces. In this paper we ask: Given two such spaces, is there an
embedding between the two?
A decomposition space
We establish readily verifiable criteria which ensure the
existence of a continuous inclusion (“an embedding”)
In a nutshell, in order to apply the embedding results presented in this
article, no knowledge of Fourier analysis is required; instead, one only has to study the geometric properties of the involved
coverings, so that one can decide the finiteness of certain sequence space norms defined in terms of the coverings.
These
sufficient criteria are quite sharp: For almost arbitrary coverings and certain ranges of
We also prove a
The resulting embedding theory is illustrated by applications
to
eBook
Decoupling on the Wiener Space, Related Besov Spaces, and Applications to BSDEs
We introduce a decoupling method on the Wiener space to define a wide class of anisotropic Besov spaces. The decoupling method is
based on a general distributional approach and not restricted to the Wiener space.
The class of Besov spaces we introduce
contains the traditional isotropic Besov spaces obtained by the real interpolation method, but also new spaces that are designed to
investigate backwards stochastic differential equations (BSDEs). As examples we discuss the Besov regularity (in the sense of our
spaces) of forward diffusions and local times. It is shown that among our newly introduced Besov spaces there are spaces that
characterize quantitative properties of directional derivatives in the Malliavin sense without computing or accessing these Malliavin
derivatives explicitly.
Regarding BSDEs, we deduce regularity properties of the solution processes from the Besov regularity of
the initial data, in particular upper bounds for their
Among other tools, we use methods from harmonic analysis. As a
by-product, we improve the asymptotic behaviour of the multiplicative constant in a generalized Fefferman inequality and verify the
optimality of the bound we established.
eBook
Functional Analysis, Harmonic Analysis, and Image Processing
This volume is dedicated to the memory of Björn Jawerth. It contains original research contributions and surveys in several of the areas of mathematics to which Björn made important contributions. Those areas include harmonic analysis, image processing, and functional analysis, which are of course interrelated in many significant and productive ways.Among the contributors are some of the world's leading experts in these areas. With its combination of research papers and surveys, this book may become an important reference and research tool.This book should be of interest to advanced graduate students and professional researchers in the areas of functional analysis, harmonic analysis, image processing, and approximation theory. It combines articles presenting new research with insightful surveys written by foremost experts.
eBook
Can Recurrence Quantification Analysis Be Useful in the Interpretation of Airborne Turbulence Measurements?
In airborne data or model outputs, clouds are often defined using information about Liquid Water Content (LWC). Unfortunately LWC is not enough to retrieve information about the dynamical boundary of the cloud, that is, volume of turbulent air around the cloud. In this work, we propose an algorithmic approach to this problem based on a method used in time series analysis of dynamical systems, namely Recurrence Plot (RP) and Recurrence Quantification Analysis (RQA). We construct RPs using time series of turbulence kinetic energy, vertical velocity and temperature fluctuations as variables important for cloud dynamics. Then, by studying time series of laminarity (LAM), a variable which is calculated using RPs, we distinguish between turbulent and non‐turbulent segments along a horizontal flight leg. By selecting a single threshold of this quantity, we are able to reduce the number of subjective variables and their thresholds used in the definition of the dynamical cloud boundary. Plain Language Summary Cloud is defined by the presence of liquid (or solid) water in the atmosphere. By agreeing on a certain threshold of liquid water content, one can define a cloud boundary. Cloud processes result in disturbances in the airflow around the cloud (turbulence). Its presence should allow the cloud dynamical boundary to be defined in a similar manner. However, a single simple variable, which could be used in an analogous way is not well defined. In this study we propose a method to define such a variable to distinguish between turbulent and non‐turbulent volumes around the cloud (i.e. determining dynamic cloud interface), adopting a method used in the study of dynamical systems. Based on recorded temperature and air velocity fluctuations characteristic for turbulent transport and mixing, a threshold of a variable named “laminarity” is used to define the dynamic cloud boundary on the aircraft trajectory. Key Points A vector characterizing turbulence along the aircraft track based on velocity and temperature fluctuations is constructed A time series of this vector allows the construction of a Recurrence Plot, a tool used in studies of dynamical systems A quantity named laminarity, derived from the Recurrence Plot, is used to detect turbulent and non‐turbulent regions
Journal Article