640 research outputs found
Generating and zeta functions, structure, spectral and analytic properties of the moments of Minkowski question mark function
In this paper we are interested in moments of Minkowski question mark
function ?(x). It appears that, to certain extent, the results are analogous to
the results obtained for objects associated with Maass wave forms: period
functions, L-series, distributions, spectral properties. These objects can be
naturally defined for ?(x) as well. Despite the fact that there are various
nice results about the nature of ?(x), these investigations are mainly
motivated from the perspective of metric number theory, Hausdorff dimension,
singularity and generalizations. In this work it is shown that analytic and
spectral properties of various integral transforms of ?(x) do reveal
significant information about the question mark function. We prove asymptotic
and structural results about the moments, calculate certain integrals involving
?(x), define an associated zeta function, generating functions, Fourier series,
and establish intrinsic relations among these objects. At the end of the paper
it is shown that certain object associated with ?(x) establish a bridge between
realms of imaginary and real quadratic irrationals.Comment: 34 pages, 4 figures (submitted 01/2008). Minor revisions and typos. A
graph of dyadic zeta function on the critical line was added. Theorem 3 was
strengthene
On the Singular Spectrum of the Radiation Operator for Multiple and Extended Observation Domains
The problem of studying how spatial
diversity impacts on the spectrum (singular values) of the
radiation operator is addressed. This topic is of great importance
because of its connection with the so-called number of degrees of
freedom concept which in turn is a key parameter in inverse source
problems as well as to the problem of transmitting information by
waves from a source domain to an observation domain. The case of a bounded rectilinear source with the
radiated field observed over multiple bounded rectilinear domains
parallel to the source is considered. Then, the analysis is
generalized to two-dimensional extended observation domains.
Analytical arguments are developed to estimate the pertinent
singular value behavior. This allows highlighting the way
observation domain features affect spectrum behavior. Numerical
examples are shown to support the analytical results
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