Title: Exponential separation of analytic self-conformal sets on the real line
Abstract. In a recent article, Rapaport showed that there is no dimension drop for exponentially separated analytic IFSs on the real line. We show that the set of exponentially separated IFSs in the space of analytic IFSs contains an open and dense set in the C2 topology. Moreover, we provide sufficient conditions for the IFS to be exponentially separated, thereby allowing us to construct explicit examples that are exponentially separated. The key technical tool is the introduction of the dual IFS, which we believe has significant interest in its own right. As an application, we also characterise when an analytic IFS can be conjugated to a self-similar IFS. This is a joint work with István Kolossváry and Sascha Troscheit.
Title: Fractal Interpolation and Quaternions
Abstract. In recent years, quaternionic methodologies have found their way into many applications, one of which is digital signal processing. The main idea is to use the multidimensionality of the quaternions to model signals with multiple channels or images with multiple color values, and to use the underlying algebraic structure of quaternions to operate on these signals. On the other hand, fractal interpolation has been employed successfully in numerous applied settings over the last decades. This talk focuses on fractal interpolation in the quaternionic setting. Properties and main results are presented.
Title: Harmonic analysis, frames, and algorithms for fractal IFS L2 spaces via infinite products of projections
Abstract. We present new results at the crossroads of spectral theory for operators in Hilbert space, optimization, large sparse systems, and the geometry of fractals arising from iterated function systems (IFS), together with new developments in fractal harmonic analysis. In particular, we introduce a new recursive iteration scheme that takes as input a prescribed sequence of selfadjoint projections. Applications include random Kaczmarz recursions, their limits, and associated error estimates.