arXiv subjects

Jan Vybíral

Publications and source records attributed to Jan Vybíral.

24 records · Page 2Linked to original sources

arXiv2012

Non-smooth atomic decompositions, traces on Lipschitz domains, and pointwise multipliers in function spaces

We provide non-smooth atomic decompositions for Besov spaces $\Bd(\rn)$, $s>0$, $0<p,q\leq \infty$, defined via differences. The results are used to compute the trace of Besov spaces on the boundary $Γ$ of bounded Lipschitz domains $Ω$ with smoothness $s$ restricted to $0<s<1$ and no further restrictions on the parameters $p,q$. We conclude with some more applications in terms of pointwise multipliers.

math.FA↗

arXiv2012

Average best $m$-term approximation

We introduce the concept of average best $m$-term approximation widths with respect to a probability measure on the unit ball of $\ell_p^n$. We estimate these quantities for the embedding $id:\ell_p^n\to\ell_q^n$ with $0<p\le q\le \infty$ for the normalized cone and surface measure. Furthermore, we consider certain tensor product weights and show that a typical vector with respect to such a measure exhibits a strong compressible (i.e. nearly sparse) structure.

math.FA↗

arXiv2011

Homogeneity property of Besov and Triebel-Lizorkin spaces

We consider the classical Besov and Triebel-Lizorkin spaces defined via differences and prove a homogeneity property for functions with bounded support in the frame of these spaces. As the proof is based on compact embeddings between the studied function spaces we present also some results on the entropy numbers of these embeddings. Moreover, we derive some applications in terms of pointwise multipliers.

math.FA↗

arXiv2010

A variant of the Johnson-Lindenstrauss lemma for circulant matrices

We continue our study of the Johnson-Lindenstrauss lemma and its connection to circulant matrices started in \cite{HV}. We reduce the bound on $k$ from $k=O(ε^{-2}\log^3n)$ proven there to $k=O(ε^{-2}\log^2n)$. Our technique differs essentially from the one used in \cite{HV}. We employ the discrete Fourier transform and singular value decomposition to deal with the dependency caused by the circulant structure.

math.FA↗

arXiv2010

Johnson-Lindenstrauss lemma for circulant matrices

We prove a variant of a Johnson-Lindenstrauss lemma for matrices with circulant structure. This approach allows to minimise the randomness used, is easy to implement and provides good running times. The price to be paid is the higher dimension of the target space $k=O(ε^{-2}\log^3n)$ instead of the classical bound $k=O(ε^{-2}\log n)$.

math.FA↗

arXiv2008

On sharp embeddings of Besov and Triebel-Lizorkin spaces in the subcritical case

We discuss the growth envelopes of Fourier-analytically defined Besov and Triebel-Lizorkin spaces $B^s_{p,q}(\R^n)$ and $F^s_{p,q}(\R^n)$ for $s=σ_p=n\max(\frac 1p-1,0)$. These results may be also reformulated as optimal embeddings into the scale of Lorentz spaces $L_{p,q}(\R^n)$. We close several open problems outlined already by H. Triebel in [H. Triebel, The structure of functions, Birkhäuser, Basel, 2001.] and explicitly formulated by D. D. Haroske in [D. D. Haroske, Envelopes and sharp embeddings of function spaces, Chapman & Hall / CRC, Boca Raton, 2007.].

math.FA↗