  We present a fundamentally new regularization method for the solution of the
Fredholm integral equation of the first kind, in which we incorporate solutions
corresponding to a range of Tikhonov regularizers into the end result. This
method identifies solutions within a much larger function space, spanned by
this set of regularized solutions, than is available to conventional
regularizaton methods. Each of these solutions is regularized to a different
extent. In effect, we combine the stability of solutions with greater degrees
of regularization with the resolution of those that are less regularized. In
contrast, current methods involve selection of a single, or in some cases
several, regularization parameters that define an optimal degree of
regularization. Because the identified solution is within the span of a set of
differently-regularized solutions, we call this method \textit{span of
regularizations}, or SpanReg. We demonstrate the performance of SpanReg through
a non-negative least squares analysis employing a Gaussian basis, and
demonstrate the improved recovery of bimodal Gaussian distribution functions as
compared to conventional methods. We also demonstrate that this method exhibits
decreased dependence of the end result on the optimality of regularization
parameter selection. We further illustrate the method with an application to
myelin water fraction mapping in the human brain from experimental magnetic
resonance imaging relaxometry data. We expect SpanReg to be widely applicable
as an effective new method for regularization of inverse problems.
