arXiv subjects
Publications and source records attributed to Markus Passenbrunner.
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.
Given any natural number $k$ and any dense point sequence $(t_n)$, we prove that the corresponding orthonormal spline system of order $k$ is an unconditional basis in reflexive $L^p$.
We present a new method for proving a certain geometric-decay inequality for entries of inverses of B-spline Gram matrices, which is given in [Passenbrunner,Shadrin 2013, arXiv:1308.4824].
The main result of this paper is a proof that, for any $f \in L_1[a,b]$, a sequence of its orthogonal projections $(P_{Δ_n}(f))$ onto splines of order $k$ with arbitrary knots $Δ_n$, converges almost everywhere provided that the mesh diameter $|Δ_n|$ tends to zero, namely \[ f \in L_1[a,b] \Rightarrow P_{Δ_n}(f,x) \to f(x) \quad \mbox{a.e.} \quad (|Δ_n|\to 0)\,. \] This extends the earlier result that, for $f \in L_p$, we have convergence $P_{Δ_n}(f) \to f$ in the $L_p$-norm for $1 \le p \le \infty$.}
Given a normalized Orlicz function $M$ we provide an easy formula for a distribution such that, if $X$ is a random variable distributed accordingly and $X_1,...,X_n$ are independent copies of $X$, then the expected value of the p-norm of the vector $(x_iX_i)_{i=1}^n$ is of the order $\| x \|_M$ (up to constants dependent on p only). In case $p=2$ we need the function $t\mapsto tM'(t) - M(t)$ to be 2-concave and as an application immediately obtain an embedding of the corresponding Orlicz spaces into $L_1[0,1]$. We also provide a general result replacing the $\ell_p$-norm by an arbitrary $N$-norm. This complements some deep results obtained by Gordon, Litvak, Schütt, 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.
We identify the torus with the unit interval $[0,1)$ and let $n,ν\in\mathbb{N}$, $1\leq ν\leq n-1$ and $N:=n+ν$. Then we define the (partially equally spaced) knots \[ t_{j}=\{[c]{ll}% \frac{j}{2n}, & \text{for}j=0,...,2ν, \frac{j-ν}{n}, & \text{for}j=2ν+1,...,N-1.] Furthermore, given $n,ν$ we let $V_{n,ν}$ be the space of piecewise linear continuous functions on the torus with knots $\{t_j:0\leq j\leq N-1\}$. Finally, let $P_{n,ν}$ be the orthogonal projection operator of $L^{2}([0,1))$ onto $V_{n,ν}.$ The main result is \[\lim_{n\rightarrow\infty,ν=1}\|P_{n,ν}:L^\infty\rightarrow L^\infty\|=\sup_{n\in\mathbb{N},0\leq ν\leq n}\|P_{n,ν}:L^\infty\rightarrow L^\infty\|=2+\frac{33-18\sqrt{3}}{13}.\] This shows in particular that the Lebesgue constant of the classical Franklin orthonormal system on the torus is $2+\frac{33-18\sqrt{3}}{13}$.
Let $t_{i}=\frac{i}{n}$ for $i=0,...,n$ be equally spaces knots in the unit interval $[0,1].$ Let $\mathcal{S}_{n}$ be the space of piecewise linear continuous functions on $[0,1]$ with knots $π_{n}=\{t_{i}:0\leq i\leq n\}.$ Then we have the orthogonal projection $P_{n}$ of $L^{2}([0,1])$ onto $\mathcal{S}_{n}.$ In Section 1 we collect a few preliminary facts about the solutions of the recurrence $f_{k-1}-4f_{k}+f_{k+1}=0$ that we need in Section 2 to show that the sequence $% a_{n}=\Vert P_{n}\Vert_{1}$ of $L^{1}-$norms of these projection operators is strictly increasing.
Let (X,d,μ) be a space of homogeneous type and E a UMD Banach space. Under the assumption mu({x})=0 for all x in X, we prove a representation theorem for singular integral operators on (X,d,mu) as a series of simple shifts and rearrangements plus two paraproducts. This gives a T(1) Theorem in this setting.