Uniformly Computable Aspects of Inner Functions

AbstractThe theory of inner functions plays an important role in the study of bounded analytic functions. Inner functions are also very useful in applied mathematics. Two foundational results in this theory are Frostman's Theorem and the Factorization Theorem. We give a uniformly computable version of Frostman's Theorem. We then claim that the Factorization Theorem is not uniformly computably true. We then claim that for an inner function u, the Blaschke sum of u provides the exact amount of information necessary to compute the factorization of u. Along the way, we discuss some uniform computability results for Blaschke products. These results play a key role in the analysis of factorization. We also give some computability results concerning zeros and singularities of analytic functions. We use Type-Two Effectivity as our foundation

Bit complexity of computing solutions for symmetric hyperbolic systems of PDEs (Extended abstract)

© 2018, Springer International Publishing AG, part of Springer Nature. We establish upper bounds of bit complexity of computing solution operators for symmetric hyperbolic systems of PDEs. Here we continue the research started in our papers of 2009 and 2017, where computability, in the rigorous sense of computable analysis, has been established for solution operators of Cauchy and dissipative boundary-value problems for such systems

Computing Solution Operators of Boundary-value Problems for Some Linear Hyperbolic Systems of PDEs

We discuss possibilities of application of Numerical Analysis methods to proving computability, in the sense of the TTE approach, of solution operators of boundary-value problems for systems of PDEs. We prove computability of the solution operator for a symmetric hyperbolic system with computable real coefficients and dissipative boundary conditions, and of the Cauchy problem for the same system (we also prove computable dependence on the coefficients) in a cube $Q\subseteq\mathbb R^m$. Such systems describe a wide variety of physical processes (e.g. elasticity, acoustics, Maxwell equations). Moreover, many boundary-value problems for the wave equation also can be reduced to this case, thus we partially answer a question raised in Weihrauch and Zhong (2002). Compared with most of other existing methods of proving computability for PDEs, this method does not require existence of explicit solution formulas and is thus applicable to a broader class of (systems of) equations.Comment: 31 page

Effective representations of the space of linear bounded operators

[EN] Representations of topological spaces by infinite sequences of symbols are used in computable analysis to describe computations in topological spaces with the help of Turing machines. From the computer science point of view such representations can be considered as data structures of topological spaces. Formally, a representation of a topological space is a surjective mapping from Cantor space onto the corresponding space. Typically, one is interested in admissible, i.e. topologically well-behaved representations which are continuous and characterized by a certain maximality condition. We discuss a number of representations of the space of linear bounded operators on a Banach space. Since the operator norm topology of the operator space is nonseparable in typical cases, the operator space cannot be represented admissibly with respect to this topology. However, other topologies, like the compact open topology and the Fell topology (on the operator graph) give rise to a number of promising representations of operator spaces which can partially replace the operator norm topology. These representations reflect the information which is included in certain data structures for operators, such as programs or enumerations of graphs. We investigate the sublattice of these representations with respect to continuous and computable reducibility. Certain additional conditions, such as finite dimensionality, let some classes of representations collapse, and thus, change the corresponding graph. Altogether, a precise picture of possible data structures for operator spaces and their mutual relation can be drawn.Work partially supported by DFG Grant BR 1807/4-1Brattka, V. (2003). Effective representations of the space of linear bounded operators. Applied General Topology. 4(1):115-131. https://doi.org/10.4995/agt.2003.20141151314

A computable spectral theorem

Computing the spectral decomposition of a normal matrix is among the most frequent tasks to numerical mathematics. A vast range of methods are employed to do so, but all of them suffer from instabilities when applied to degenerate matrices, i.e., those having multiple eigenvalues. We investigate the spectral representation’s effectivity properties on the sound formal basis of computable analysis. It turns out that in general the eigenvectors cannot be computed from a given matrix. If however the size of the matrix ’ spectrum (=number of different eigenvalues) is known in advance, it can be diagonalized effectively. Thus, in principle the spectral decomposition can be computed under remarkably weak non-degeneracy conditions