908 research outputs found
Positive-part moments via the Fourier-Laplace transform
Integral expressions for positive-part moments E X_+^p (p>0) of random
variables X are presented, in terms of the Fourier-Laplace or Fourier
transforms of the distribution of X. A necessary and sufficient condition for
the validity of such an expression is given. This study was motivated by
extremal problems in probability and statistics, where one needs to evaluate
such positive-part moments.Comment: Accepted for publication in Journal of Theoretical Probability, with
Proposition 2.6 downgraded to Example 1. A couple of possible applications
are added. Other changes are mino
Kinematic Diffraction from a Mathematical Viewpoint
Mathematical diffraction theory is concerned with the analysis of the
diffraction image of a given structure and the corresponding inverse problem of
structure determination. In recent years, the understanding of systems with
continuous and mixed spectra has improved considerably. Simultaneously, their
relevance has grown in practice as well. In this context, the phenomenon of
homometry shows various unexpected new facets. This is particularly so for
systems with stochastic components. After the introduction to the mathematical
tools, we briefly discuss pure point spectra, based on the Poisson summation
formula for lattice Dirac combs. This provides an elegant approach to the
diffraction formulas of infinite crystals and quasicrystals. We continue by
considering classic deterministic examples with singular or absolutely
continuous diffraction spectra. In particular, we recall an isospectral family
of structures with continuously varying entropy. We close with a summary of
more recent results on the diffraction of dynamical systems of algebraic or
stochastic origin.Comment: 30 pages, invited revie
Uniform distribution and algorithmic randomness
A seminal theorem due to Weyl states that if (a_n) is any sequence of
distinct integers, then, for almost every real number x, the sequence (a_n x)
is uniformly distributed modulo one. In particular, for almost every x in the
unit interval, the sequence (a_n x) is uniformly distributed modulo one for
every computable sequence (a_n) of distinct integers. Call such an x "UD
random". Here it is shown that every Schnorr random real is UD random, but
there are Kurtz random reals that are not UD random. On the other hand, Weyl's
theorem still holds relative to a particular effectively closed null set, so
there are UD random reals that are not Kurtz random
Wavelet frame bijectivity on Lebesgue and Hardy spaces
We prove a sufficient condition for frame-type wavelet series in , the
Hardy space , and BMO. For example, functions in these spaces are shown to
have expansions in terms of the Mexican hat wavelet, thus giving a strong
answer to an old question of Meyer.
Bijectivity of the wavelet frame operator acting on Hardy space is
established with the help of new frequency-domain estimates on the
Calder\'on-Zygmund constants of the frame kernel.Comment: 23 pages, 7 figure
Computable Cyclic Functions
This dissertation concerns computable analysis where the idea of a representation of a set is of central importance. The key ideas introduced are those commenting on the computable relationship between two newly constructed representations, a representation of integrable cyclic functions, and the continuous cyclic function representation. Also, the computable relationship of an absolutely convergent Fourier series representation is considered. It is observed that the representation of integrable cyclic functions gives rise to a much larger set of computable functions than obtained by the continuous cyclic function representation and that integration remains a computable operation, but that basic evaluation of the function is not computable. Many other representations are acknowledged enhancing the picture of the partial order structure on the space of representations of cyclic functions. The paper can also be seen as a foundation for the study of Fourier analysis in a computable universe and concludes with an investigation into the computability of the Fourier transform
Computability and analysis: the legacy of Alan Turing
We discuss the legacy of Alan Turing and his impact on computability and
analysis.Comment: 49 page
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