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Analysis in Banach spaces
Pernecká, Eva ; Hájek, Petr (advisor) ; Johanis, Michal (referee) ; Godefroy, Gilles (referee)
The thesis consists of two papers and one preprint. The two papers are de- voted to the approximation properties of Lipschitz-free spaces. In the first pa- per we prove that the Lipschitz-free space over a doubling metric space has the bounded approximation property. In particular, the Lipschitz-free space over a closed subset of Rn has the bounded approximation property. We also show that the Lipschitz-free spaces over ℓ1 and over ℓn 1 admit a monotone finite-dimensional Schauder decomposition. In the second paper we improve this work and obtain even a Schauder basis in the Lipschitz-free spaces over ℓ1 and ℓn 1 . The topic of the preprint is rigidity of ℓ∞ and ℓn ∞ with respect to uniformly differentiable map- pings. Our main result is a non-linear analogy of the classical result on rigidity of ℓ∞ with respect to non-weakly compact linear operators by Rosenthal, and it generalises the theorem on non-complementability of c0 in ℓ∞ due to Phillips. 1
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Descriptive set properties of collections of exceptional sets in Harmonic analysis
Kovařík, Vojtěch ; Vlasák, Václav (advisor) ; Zelený, Miroslav (referee)
We study families of small sets which appear in Harmonic analysis. We focus on the systems H(N) , N ∈ N, U and U0. In particular we compare their sizes via comparing the polars of these classes, i.e. the families of measures annihilating all sets from given class. Lyons showed that in this sense, the family N∈N H(N) is smaller than U0. The main goal of this thesis is the study of the question whether this also holds when the system U0 is replaced by the much smaller system U. To this end we define a new system H(∞) and systems of sets of type N where N ∈ N∪{∞}. We then prove some of their properties, which might be useful in solving the studied question. 1
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On convergence of a series
Procházka, Antonín ; Zelený, Miroslav (advisor) ; Kaplický, Petr (referee)
in English This text is devoted to the series, whose n-th term is defined by (−1)n |sin n| /n. The goal of this work is to prove convergence of this series. The solution uses standard convergence tests, the theory of Fourier Series and findings about approximation of number π. 1
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Semiconvex functions and their differences
Kryštof, Václav ; Zajíček, Luděk (advisor) ; Zelený, Miroslav (referee)
Semiconvex functions generalize convex functions on a normed linear space. In this thesis we deal with semiconvex functions of one real varia- ble, so with a special case. In the first part we especially study properties of semiconvex functions from one-side derivative point of view and give charac- terization of locally semiconvex functions on an open interval by right-hand derivative. The second part of the thesis concerns functions which are a dif- ference of two semiconvex functions. Also in this case we give (by right-hand derivative) characterization of functions which are a difference of two locally semiconvex functions on an open interval. This is related to a generalized va- riation which we introduce and which enables to characterize those functions which are a difference of two semiconvex functions on an open interval with modulus Mω, where M ∈ (0, ∞). 1
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Monotonicity of functions which can be expressed using elementary functions
Peltan, Libor ; Bárta, Tomáš (advisor) ; Pyrih, Pavel (referee)
For certain types of functions expressible with formula (equivalently: functions from classes closed to arithmetic operations) under stated assumptions, we prove monotonicity at some neighbourhood of +∞. They are: formulas containing exp, log, sin, arctan, etc. with constrainted domain of these functions; power series with cofinite many coefficients positive; various classes of functions expressible with formulas with the requirement of preserving monotony in summation, or multiplication, or the monotony resulting from having a finite number of zero points; and finally formulas with square root. 1
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Differentiability of the inverse mapping
Konopecký, František ; Hencl, Stanislav (advisor)
Primary objective of the thesis is proof of the statement that if for ∈ ℕ a ≥ 1 a bilipschitz mapping belongs to +1, loc ∩ ,∞ loc then also its inverse −1 belongs to +1, loc . We prove a similar statement also for spaces loc . For this purpose we construct a new ordering of -th partial derivatives to generalized Jacobian matrix. Thanks to this matrix we are able to differentiate matrices in an applicable way. Generalized Jacobian matrix is projected so that there still holds the Chain rule and, in some way, also rules for matrices product differentiation. 1
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Spaces of functions with fractional derivatives on interval
Lopata, Jan ; Kaplický, Petr (advisor) ; Hencl, Stanislav (referee)
In literature we can find a variety of ways to introduce Sobolev space W1,1 on bounded and open interval. In this thesis we will put them in context. We will show that completion of set of function with continuous first derivative, the space of functions with weak derivative and space of absolutely continuous functions are isometrically isomorphic. Furthemore, we will demonstrate that the Sobolev space W1,∞ is isometrically isomorphic to space of Lipschitz functions. We will also show several trivial and nontrivial embeddings for Besov spaces. Finnaly, we will examine the question, whether functions from Besov space are, given some parameters, included in set of continuous functions. 1
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Convergence of Fourier series in Lp spaces
Michálek, Martin ; Zelený, Miroslav (advisor) ; Spurný, Jiří (referee)
The main question of this thesis is whether the partial sums of Fourier series converge in some sense to the function from which the series was derived. In our case we will analyze the convergence of Fourier series of Lebesgue integrable functions and the convergence will be meant in the sense of Lp spaces for p ∈ [1, ∞). The case p = 2 could be concluded from properties of orthogonal basis in Hilbert spaces. Our intention is to analyze the problem especially for the other p ∈ [1, ∞). Therefore we need to use some results from the theory of Banach (particularly Lp ) spaces.
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Symbolické reprezentace kompaktních prostorů
Kazda, Alexandr
Title: Symbolic representations of compact spaces Author: Alexandr Kazda Department: Department of Algebra Supervisor: Prof. RNDr. Petr Kůrka, CSc. Supervisor's e-mail address: kurka@cts.cuni.cz Abstract: The thesis concerns itself with Möbius number systems. These systems represent points using sequences of Möbius transformations. We are mainly inter- ested in representing the unit circle (which is equivalent to representing R ∪ {∞}). The main aim of the thesis is to improve already known tools for proving that a given subshift-iterative system pair is in fact a Möbius number system. We also study the existence problem: How to describe iterative systems resp. subshifts for which there exists a subshift resp. iterative system such that the resulting pair forms a Möbius number system. While we were unable to provide a complete answer to this question, we present both positive and negative partial results. As Möbius number systems are also subshifts, we can ask when a given Möbius number system is sofic. We give this problem a short treatment at the end of our thesis. Keywords: Möbius transformation, numeral system, subshift
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Differentiability of the inverse mapping
Konopecký, František ; Hencl, Stanislav (advisor) ; Honzík, Petr (referee)
Primary objective of the thesis is proof of the statement that if for ∈ ℕ a ≥ 1 a bilipschitz mapping belongs to +1, loc ∩ ,∞ loc then also its inverse −1 belongs to +1, loc . We prove a similar statement also for spaces loc . For this purpose we construct a new ordering of -th partial derivatives to generalized Jacobian matrix. Thanks to this matrix we are able to differentiate matrices in an applicable way. Generalized Jacobian matrix is projected so that there still holds the Chain rule and, in some way, also rules for matrices product differentiation. 1
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