Sfb 288 Differential Geometry and Quantum Physics

Abstract for Sfb Preprint No. 409

Wavelet Analysis on Manifolds I

Dieter Klusch

The purpose of this note is to describe a new Cauchy wavelet analysisof selfadjoint operators in Hilbert spaces and its applications tofundamental problems of Global Analysis. We use the wavelet synthesisintegral operator to introduce (i) a new wavelet calculus for power andexponential functions and (ii) a new class of generalized (fractal) zetaand eta functions. This extends the Seeley--Grubb functional calculus[GS96] and the abstract setting of the recentBrüning--Lesch spectral theory [BL99].

In the applications we describe how Cauchy wavelet analysis works in thetheoryelliptic differential operators on manifolds.We show that the Duistermaat--Guillemin--Weinstein ave traceinvariants [Gui96] are actually ractal zeta function invariants. We use the pseudodifferential resolvent analysis of Grubb and Seeley[GS95], [G99] to determine the full singularity structures of thefractal zeta and eta functions and the resulting heat traceexpansions for pseudodifferential boundary problems for generalelliptic systems of order $dgeq 1$ on compact manifolds. Thisgeneralizes recent fundamental trace expansion results of Grubb [G99],and - for the important special case of first order well-posedboundary problems for Dirac-type operators - of course theprominent results of Atiyah-Patodi-Singer [APS75], of Grubb andSeeley [GS95], [GS96], of Booss and Wojciechowski [BW93], of Müller[Mü94] and of Brüning and Lesch [BL97], [BL99].

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