3,750 research outputs found
Bounded time computation on metric spaces and Banach spaces
We extend the framework by Kawamura and Cook for investigating computational
complexity for operators occurring in analysis. This model is based on
second-order complexity theory for functions on the Baire space, which is
lifted to metric spaces by means of representations. Time is measured in terms
of the length of the input encodings and the required output precision. We
propose the notions of a complete representation and of a regular
representation. We show that complete representations ensure that any
computable function has a time bound. Regular representations generalize
Kawamura and Cook's more restrictive notion of a second-order representation,
while still guaranteeing fast computability of the length of the encodings.
Applying these notions, we investigate the relationship between purely metric
properties of a metric space and the existence of a representation such that
the metric is computable within bounded time. We show that a bound on the
running time of the metric can be straightforwardly translated into size bounds
of compact subsets of the metric space. Conversely, for compact spaces and for
Banach spaces we construct a family of admissible, complete, regular
representations that allow for fast computation of the metric and provide short
encodings. Here it is necessary to trade the time bound off against the length
of encodings
Computable Jordan Decomposition of Linear Continuous Functionals on
By the Riesz representation theorem using the Riemann-Stieltjes integral,
linear continuous functionals on the set of continuous functions from the unit
interval into the reals can either be characterized by functions of bounded
variation from the unit interval into the reals, or by signed measures on the
Borel-subsets. Each of these objects has an (even minimal) Jordan decomposition
into non-negative or non-decreasing objects. Using the representation approach
to computable analysis, a computable version of the Riesz representation
theorem has been proved by Jafarikhah, Lu and Weihrauch. In this article we
extend this result. We study the computable relation between three Banach
spaces, the space of linear continuous functionals with operator norm, the
space of (normalized) functions of bounded variation with total variation norm,
and the space of bounded signed Borel measures with variation norm. We
introduce natural representations for defining computability. We prove that the
canonical linear bijections between these spaces and their inverses are
computable. We also prove that Jordan decomposition is computable on each of
these spaces
- …