20 research outputs found
The Lebesgue Constant for the Periodic Franklin System
We identify the torus with the unit interval and let
, and . Then we define the
(partially equally spaced) knots t_{j}=\{[c]{ll}% \frac{j}{2n}, &
\text{for}j=0,...,2\nu, \frac{j-\nu}{n}, & \text{for}j=2\nu+1,...,N-1.]
Furthermore, given $n,\nu$ we let $V_{n,\nu}$ be the space of piecewise linear
continuous functions on the torus with knots $\{t_j:0\leq j\leq N-1\}$.
Finally, let $P_{n,\nu}$ be the orthogonal projection operator of
$L^{2}([0,1))$ onto $V_{n,\nu}.$ The main result is
\[\lim_{n\rightarrow\infty,\nu=1}\|P_{n,\nu}:L^\infty\rightarrow
L^\infty\|=\sup_{n\in\mathbb{N},0\leq \nu\leq n}\|P_{n,\nu}:L^\infty\rightarrow
L^\infty\|=2+\frac{33-18\sqrt{3}}{13}. This shows in particular that the
Lebesgue constant of the classical Franklin orthonormal system on the torus is
.Comment: Mathematica Notebook for creating Table 1 on page 21 is attache
Adaptive deterministic dyadic grids on spaces of homogeneous type
In the context of spaces of homogeneous type, we develop a method to
deterministically construct dyadic grids, specifically adapted to a given
combinatorial situation. This method is used to estimate vector-valued
operators rearranging martingale difference sequences such as the Haar system.Comment: 17 pages, 2 figure
Probabilistic estimates for tensor products of random vectors
We prove some probabilistic estimates for tensor products of random vectors.
As an application we obtain embeddings of certain matrix spaces into .Comment: 15 page
On the Distribution of Random variables corresponding to Musielak-Orlicz norms
Given a normalized Orlicz function we provide an easy formula for a
distribution such that, if is a random variable distributed accordingly and
are independent copies of , then the expected value of the
p-norm of the vector is of the order (up to
constants dependent on p only). In case we need the function to be 2-concave and as an application immediately obtain an
embedding of the corresponding Orlicz spaces into . We also provide a
general result replacing the -norm by an arbitrary -norm. This
complements some deep results obtained by Gordon, Litvak, Sch\"utt, and Werner.
We also prove a result in the spirit of their work which is of a simpler form
and easier to apply. All results are true in the more general setting of
Musielak-Orlicz spaces