  In the present paper we consider the problem of Laplace deconvolution with
noisy discrete non-equally spaced observations on a finite time interval. We
propose a new method for Laplace deconvolution which is based on expansions of
the convolution kernel, the unknown function and the observed signal over
Laguerre functions basis (which acts as a surrogate eigenfunction basis of the
Laplace convolution operator) using regression setting. The expansion results
in a small system of linear equations with the matrix of the system being
triangular and Toeplitz. Due to this triangular structure, there is a common
number $m$ of terms in the function expansions to control, which is realized
via complexity penalty. The advantage of this methodology is that it leads to
very fast computations, produces no boundary effects due to extension at zero
and cut-off at $T$ and provides an estimator with the risk within a logarithmic
factor of the oracle risk. We emphasize that, in the present paper, we consider
the true observational model with possibly nonequispaced observations which are
available on a finite interval of length $T$ which appears in many different
contexts, and account for the bias associated with this model (which is not
present when $T\rightarrow\infty$). The study is motivated by perfusion imaging
using a short injection of contrast agent, a procedure which is applied for
medical assessment of micro-circulation within tissues such as cancerous
tumors. Presence of a tuning parameter $a$ allows to choose the most
advantageous time units, so that both the kernel and the unknown right hand
side of the equation are well represented for the deconvolution. The
methodology is illustrated by an extensive simulation study and a real data
example which confirms that the proposed technique is fast, efficient,
accurate, usable from a practical point of view and very competitive.
