can be considered as a Fourier transform of the distribution
F (x) =
X
j
f(x
j
)δ(x − x
j
).
Suppose now that ϕ is a “bump function”, e.g., a Gaussian or a B-spline. From
the Fourier transform
Z
∞
−∞
F ∗ ϕ(x)e
−2πixξ
dξ =
b
F (ξ)ϕ(ξ) =
b
f(ξ)ϕ(ξ),
the relation
b
f(ξ) =
1
ϕ(ξ)
Z
∞
−∞
X
j
f(x
j
)ϕ(x − x
j
)e
−2πixξ
dξ
follows. What makes this formulation attractive is that if ϕ chosen sufficiently
regular (with compact numerical support) then the integral above can be approxi-
mated by the trapezoidal rule, which allows a rapid approximation of the integral
by means of a fast Fourier transform (FFT) for the case when the points ξ are
chosen at equally spaced points. The approximation is thus done in three steps: 1)
convolution in the space domain; 2) FFT; and 3) division in the frequency domain.
There is an apparent loss of precision at the third step at points where ϕ is close
or equal to zero. However, it turns out that approximation to prescribed precision
for ξ in a certain range can be achieved by proper oversampling and choice of ϕ.
The algorithm sketched out above would be the Fourier transform correspon-
dence of (2) and there are similar constructions using convolutions in one domain
followed by division in the dual one for sums of the form (1), (3) and (4). Crucial
for the construction of fast algorithms is that the function ϕ has small (numerical)
support, in order to keep down the computational time for numerical evaluation
of the convolutions. It turns out that the (numerical) width of ϕ is proportional to
−log , where denotes the computational precision, cf. Section 2.
In this paper, we will choose ϕ to be of Gaussian type, and make explicit use
of this fact to generalize USFFT algorithms to deal with fast Laplace transforms,
in a similar spirit to the explicit use of the structure of Gaussians for the USFFT
style algorithms developed in [5, 6]. The extra cost for this generalization is that
the (numerical) width of ϕ has to be increased slightly depending on the range of
allowed decay parameters a.
The computation of the sums (1) and (2) corresponds to the application of
complex Vandermonde matrices and its adjoints. Such are useful, for instance
when trying to represent data using complex exponentials, cf. [7]. Another direct
3
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