12,271 research outputs found
Reconstruction of density functions by sk-splines
Reconstruction of density functions and their characteristic functions by
radial basis functions with scattered data points is a popular topic in the
theory of pricing of basket options. Such functions are usually entire or admit
an analytic extension into an appropriate tube and "bell-shaped" with rapidly
decaying tails. Unfortunately, the domain of such functions is not compact
which creates various technical difficulties. We solve interpolation problem on
an infinite rectangular grid for a wide range of kernel functions and calculate
explicitly their Fourier transform to obtain representations for the respective
density functions
Non-unitarisable representations and random forests
We establish a connection between Dixmier's unitarisability problem and the
expected degree of random forests on a group. As a consequence, a residually
finite group is non-unitarisable if its first L2-Betti number is non-zero or if
it is finitely generated with non-trivial cost. Our criterion also applies to
torsion groups constructed by D. Osin, thus providing the first examples of
non-unitarisable groups not containing a non-Abelian free subgroup
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