ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΡΗΤΗΣ - ΤΜΗΜΑ ΜΑΘΗΜΑΤΙΚΩΝ
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Δύο ομιλίες στην ανάλυση

Prof. Joseph Rosenblatt
University of Illinois at Urbana-Champaign

Οκτώβριος 2003

Ο καθηγητής Rosenblatt επισκέπτεται το Τμήμα Μαθηματικών για μια εβδομάδα αρχές Οκτωβρίου, 2003. Θα δώσει δύο ομιλίες (δείτε τη τη σελίδα του Σεμιναρίου Ανάλυσης για το πότε και πού) με τα ακόλουθα θέματα.

1. Convergence Along Subsequences in Ergodic Theory and Harmonic Analysis
   Many sequences of operators $(T_n)$ on $L_p$ spaces, that average functions in one way or another, naturally yield norm convergent sequences that may fail sometimes to be pointwise convergent. We will examine examples of how the pointwise convergence on all of $L_p$ can be achieved when one restricts to a subsequence $(T_{m_n})$ of the original sequence. We will describe completely some cases of this phenomenon and give a broad view of what has been done, and what needs yet to be done, in this area of analysis.

2. Searching for a Correct Factor
   Given a sequence of operators $(T_n)$ on $L_p$, such that $(T_nf)$ fails to converge almost everywhere for some functions $f \in L_p$, we would like to know how bad the divergence can be by finding the optimal constants $Q_n$ such $\frac 1{Q_n} T_nf$ converges almost everywhere for all $f \in L_p$. The search for this correct factor in ergodic theory and harmonic analysis has lead to interesting discoveries about large deviations, the growth of sums of equidistributed functions, etc. We will survey these results and describe some of the particular behavior in detail.