# Johannes Kepler Symposium on Mathematics

As part of the Johannes Kepler symposium on mathematics
**Dr. Markus
Passenbrunner**, Institut für Analysis,
will give a public talk (followed by a discussion) on **Wed, Oct. 30, 2019**
at **17:15 o'clock** at **HS 12**
on the topic of
"Martingale Properties of Spline Sequences"
. The organziers of the symposium,

Univ.-Prof. Dr. Gerhard Larcher

A.Univ.-Prof. Dr. Jürgen Maaß, and

die ÖMG (Österreichische Mathematische Gesellschaft),

hereby cordially invite you.

Series B - Mathematical Colloquium:

The intention is to present new mathematical results for an audience interested in general mathematics.

### Martingale Properties of Spline Sequences

Given a sequence of increasing $\sigma$-algebras $(\mathscr F_n)$ (i.e. a filtration) on a probability space, a \emph{martingale} is a sequence of integrable random variables $(X_n)$ so that (for all $n$), the conditional expectation $\mathbb E_n X_{n+1}$ of $X_{n+1}$ with respect to $\mathscr F_n$ is precisely $X_n$. Martingales are an important concept in Probability Theory and other branches of Mathematics and Physics. A very prominent example of a martingale is Brownian motion (if one allows a continuous time variable $t$ instead of $n$).

In the case that each random variable $X_n$ is measurable with respect to a $\sigma$-algebra $\mathscr F_n$ on the unit interval $I$ that is generated by a finite partition of $I$ into intervals of positive length, the conditional expectation operator $\mathbb E_n$ is the orthogonal projection operator onto the space of functions that are constant on the atoms of $\mathscr F_n$, i.e., onto a space of piecewise constant functions.

We now replace the space of piecewise constant functions by the spline space $S_n^{(d)}$ of $d-1$ times continuously differentiable functions on $I$ that are polynomials of degree $d$ on atoms of $\mathscr F_n$. Moreover, we denote by $P_n=P_n^{(d)}$ the orthogonal projection operator onto $S_n^{(d)}$ with respect to the standard $L^2$ inner product on $I$. Similarly to the definition of a martingale---by simply replacing $\mathbb E_n$ by $P_n$---we say that a sequence of integrable functions $(g_n)$ is a \emph{$d$-martingale spline sequence}, if $P_n g_{n+1} = g_n$ for all $n$.

Many results for martingales in fact transfer to martingale spline sequences, \emph{independently of the underlying

filtration $(\mathscr F_n)$}. The starting point of this development was A.~Shadrin's solution \cite{Shadrin2001} of C.~de Boor's conjecture \cite{deBoor1973} showing that the operators $P_n$ are uniformly bounded in $L^1$ independently of the filtration. Important results about martingale spline sequences---that hold without any assumption on the filtration $(\mathscr F_n)$ whatsoever---include the following:

- The spline version of J.~Doob's maximal function inequalities implying in particular the almost everywhere convergence of the sequence $(P_n f)$ for $f\in L^1$, see \cite{PassenbrunnerShadrin2014} (and \cite{Passenbrunner2017} for its periodic version).
- The spline version of D.~Burkholder's theorem about the equivalence of martingale square and martingale maximal function in $L^p$ for finite $p>1$, see \cite{Passenbrunner2014} (and \cite{KeryanPassenbrunner2019} for its periodic version).
- The spline version of the Martingale Convergence Theorem in its most general form even covering Banach space valued martingales for Banach spaces that have the Radon-Nikod\'{y}m property \cite{MuellerPassenbrunner2017}.
- A characterization of the Radon-Nikod\'{y}m property of Banach spaces in terms of spline sequences \cite{Passenbrunner2019}.

In this talk we discuss the aforementioned theorems and additional results of a similar spirit.