We prove that the multiparameter (product) space BMO of functions of bounded mean oscillation can be written as the intersection of finitely many dyadic product BMO spaces, with equivalent norms, generalizing the one-...
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We prove that the multiparameter (product) space BMO of functions of bounded mean oscillation can be written as the intersection of finitely many dyadic product BMO spaces, with equivalent norms, generalizing the one-parameter result of T. Mei. We establish the analogous dyadic structure theorems for the space VMO of functions of vanishing mean oscillation, for A, weights, for reverse-Holder weights and for doubling weights. We survey several definitions of VMO and prove their equivalences, in the continuous, dyadic, one-parameter and product cases. In particular, we introduce the space of dyadic product VMO functions. We show that the weighted product Hardy space H-omega(1) is the sum of finitely many translates of dyadic weighted H-omega(1), for each A(infinity). weight omega, and that the weighted strong maximal function is pointwise comparable to the sum of finitely many dyadic weighted strong maximal functions, for each doubling weight omega. Our results hold in both the compact and non-compact cases.
In earlier work the second author introduced the tool of pointwise directed families of characteristic functions to establish that certain continuous function spaces [X -> Q] were continuous domains. In this paper ...
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In earlier work the second author introduced the tool of pointwise directed families of characteristic functions to establish that certain continuous function spaces [X -> Q] were continuous domains. In this paper we extend those earlier results, while significantly simplifying and refining the machinery involved. Our main result asserts that the function space [X -> Q] is a continuous dcpo if X is locally compact and coherent and Q is a retract of a bifinite domain (RB-domain) with perpendicular to.
We analyse the Gottlieb groups of function spaces. Our results lead to explicit decompositions of the Gottlieb groups of many function spaces map(X, Y)-including the (iterated) free loop space of Y-directly in terms o...
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We analyse the Gottlieb groups of function spaces. Our results lead to explicit decompositions of the Gottlieb groups of many function spaces map(X, Y)-including the (iterated) free loop space of Y-directly in terms of the Gottlieb groups of Y. More generally, we give explicit decompositions of the generalised Gottlieb groups of map(X, Y) directly in terms of generalised Gottlieb groups of Y. Particular cases of our results relate to the torus homotopy groups of Fox. We draw some consequences for the classification of T-spaces and G-spaces. For X, Y finite and Y simply connected, we give a formula for the ranks of the Gottlieb groups of map(X, Y) in terms of the Betti numbers of X and the ranks of the Gottlieb groups of Y. Under these hypotheses, the Gottlieb groups of map(X, Y) are finite groups in all but finitely many degrees.
The primary purpose of this thesis is to determine when a function space is equivalent to an algebra, that is, when it is closed with respect to pointwise multiplication. Firstly, the theory of some function spaces, n...
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The primary purpose of this thesis is to determine when a function space is equivalent to an algebra, that is, when it is closed with respect to pointwise multiplication. Firstly, the theory of some function spaces, namely Lebesgue Lp spaces, the class of Banach function spaces, rearrangement-invariant Banach function spaces, Morrey spaces, Campanato spaces, and weak−L∞ , is introduced. Secondly, a general necessary condition, as well as a general sufficient condition, for a function space to be equivalent to an algebra is given. In each of these two conditions, a crucial role is played by the space L∞ . Furthermore, as a corollary, a characterisation when a Banach function space is equivalent to an algebra is obtained. Thereafter, a few examples illustrating possible usage of these results are presented. After that, a special case when a Banach function space is rearrangement invariant is dealt with. Lastly, the matter of equivalence to an algebra is addressed for the function spaces introduced before. 1
We aim at constructing a smooth basis for isogeometric function spaces on domains of reduced geometric regularity. In this context an isogeometric function is the composition of a piecewise rational function with the ...
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ISBN:
(纸本)9783319228044;9783319228037
We aim at constructing a smooth basis for isogeometric function spaces on domains of reduced geometric regularity. In this context an isogeometric function is the composition of a piecewise rational function with the inverse of a piecewise rational geometry parameterization. We consider two types of singular parameterizations, domains where a part of the boundary is mapped onto one point and domains where the two parameter lines at a corner of the parameter domain are collinear in the physical domain. We locally map a singular tensor-product patch of arbitrary degree onto a triangular patch, thus splitting the parameterization into a singular bilinear mapping and a regular mapping on a triangular domain. This construction yields an isogeometric function space of prescribed smoothness. Generalizations to higher dimensions are also possible and are briefly discussed in the final section.
This book is the continuation of Local function spaces, Heat and Navier–Stokes Equations (EMS Tracts in Mathematics, volume 20, 2013) by the author. A new approach is presented to exhibit relations between Sobolev sp...
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ISBN:
(数字)9783037196502
ISBN:
(纸本)3037191503;9783037191507
This book is the continuation of Local function spaces, Heat and Navier–Stokes Equations (EMS Tracts in Mathematics, volume 20, 2013) by the author. A new approach is presented to exhibit relations between Sobolev spaces, Besov spaces, and Hölder–Zygmund spaces on the one hand and Morrey–Campanato spaces on the other. Morrey–Campanato spaces extend the notion of functions of bounded mean oscillation. These spaces play a crucial role in the theory of linear and nonlinear PDEs. Chapter 1 (Introduction) describes the main motivations and intentions of this book. Chapter 2 is a self-contained introduction to Morrey spaces. Chapter 3 deals with hybrid smoothness spaces (which are between local and global spaces) in Euclidean \(n\)-space based on the Morrey–Campanato refinement of the Lebesgue spaces. The presented approach, which relies on wavelet decompositions, is applied in Chapter 4 to linear and nonlinear heat equations in global and hybrid spaces. The obtained assertions about function spaces and nonlinear heat equations are used in Chapters 5 and 6 to study Navier–Stokes equations in hybrid and global spaces. This book is addressed to graduate students and mathematicians who have a working knowledge of basic elements of (global) function spaces and who are interested in applications to nonlinear PDEs with heat and Navier–Stokes equations as prototypes.
In this paper, we investigate sampling expansion for the linear canonical transform (LCT) in function spaces. First, some properties of the function spaces related to the LCT are obtained. Then, a sampling theorem for...
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In this paper, we investigate sampling expansion for the linear canonical transform (LCT) in function spaces. First, some properties of the function spaces related to the LCT are obtained. Then, a sampling theorem for the LCT in function spaces with a single-frame generator is derived by using the Zak Transform and its generalization to the LCT domain. Some examples are also presented.
The linear canonical transform (LCT) has proven to be a powerful tool in optics and signal processing. Most existing sampling theories of this transform were derived from the LCT band-limited signal viewpoint. However...
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The linear canonical transform (LCT) has proven to be a powerful tool in optics and signal processing. Most existing sampling theories of this transform were derived from the LCT band-limited signal viewpoint. However, in the real world, many analog signals encountered in practical engineering applications are non-bandlimited. The purpose of this paper is to derive sampling theorems of the LCT in function spaces for frames without bandlimiting constraints. We extend the notion of shift-invariant spaces to the LCT domain and then derive a sampling theorem of the LCT for regular sampling in function spaces with frames. Further, the theorem is modified to the shift sampling in function spaces by using the Zak transform. Sampling and reconstructing signals associated with the LCT are also discussed in the case of Riesz bases. Moreover, some examples and applications of the derived theory are presented. The validity of the theoretical derivations is demonstrated via simulations. (C) 2013 Elsevier B.V. All rights reserved.
We study sound-soft time-harmonic acoustic scattering by general scatterers, including fractal scatterers, in 2D and 3D space. For an arbitrary compact scatterer Gamma we reformulate the Dirichlet boundary value probl...
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We study sound-soft time-harmonic acoustic scattering by general scatterers, including fractal scatterers, in 2D and 3D space. For an arbitrary compact scatterer Gamma we reformulate the Dirichlet boundary value problem for the Helmholtz equation as a first kind integral equation (IE) on Gamma involving the Newton potential. The IE is well-posed, except possibly at a countable set of frequencies, and reduces to existing single-layer boundary IEs when Gamma is the boundary of a bounded Lipschitz open set, a screen, or a multi-screen. When Gamma is uniformly of d-dimensional Hausdorff dimension in a sense we make precise (a d-set), the operator in our equation is an integral operator on Gamma with respect to d-dimensional Hausdorff measure, with kernel the Helmholtz fundamental solution, and we propose a piecewise-constant Galerkin discretization of the IE, which converges in the limit of vanishing mesh width. When Gamma is the fractal attractor of an iterated function system of contracting similarities we prove convergence rates under assumptions on Gamma and the IE solution, and describe a fully discrete implementation using recently proposed quadrature rules for singular integrals on fractals. We present numerical results for a range of examples and make our software available as a Julia code.
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