4,034 research outputs found
Left-Inverses of Fractional Laplacian and Sparse Stochastic Processes
The fractional Laplacian commutes with the primary
coordination transformations in the Euclidean space \RR^d: dilation,
translation and rotation, and has tight link to splines, fractals and stable
Levy processes. For , its inverse is the classical Riesz potential
which is dilation-invariant and translation-invariant. In this work,
we investigate the functional properties (continuity, decay and invertibility)
of an extended class of differential operators that share those invariance
properties. In particular, we extend the definition of the classical Riesz
potential to any non-integer number larger than and
show that it is the unique left-inverse of the fractional Laplacian
which is dilation-invariant and
translation-invariant. We observe that, for any and
, there exists a Schwartz function such that is not -integrable. We then introduce the new unique left-inverse
of the fractional Laplacian with the
property that is dilation-invariant (but not
translation-invariant) and that is -integrable for any
Schwartz function . We finally apply that linear operator
with to solve the stochastic partial differential equation
with white Poisson noise as its driving term
.Comment: Advances in Computational Mathematics, accepte
New Series Expansions of the Gauss Hypergeometric Function
The Gauss hypergeometric function can be computed by using
the power series in powers of . With
these expansions is not completely computable for all
complex values of . As pointed out in Gil, {\it et al.} [2007, \S2.3], the
points are always excluded from the domains of convergence
of these expansions. B\"uhring [1987] has given a power series expansion that
allows computation at and near these points. But, when is an integer, the
coefficients of that expansion become indeterminate and its computation
requires a nontrivial limiting process. Moreover, the convergence becomes
slower and slower in that case. In this paper we obtain new expansions of the
Gauss hypergeometric function in terms of rational functions of for which
the points are well inside their domains of convergence . In
addition, these expansion are well defined when is an integer and no
limits are needed in that case. Numerical computations show that these
expansions converge faster than B\"uhring's expansion for in the
neighborhood of the points , especially when is close to
an integer number.Comment: 18 pages, 6 figures, 4 tables. In Advances in Computational
Mathematics, 2012 Second version with corrected typos in equations (18) and
(19
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