# 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,

O.Univ.-Prof. Dr. Ulrich Langer,
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:

1. 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).
2. 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).
3. 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}.
4. 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.