2 research outputs found
Approximating fractional derivative of the Gaussian function and Dawson's integral
A new method for approximating fractional derivatives of the Gaussian
function and Dawson's integral are presented. Unlike previous approaches, which
are dominantly based on some discretization of Riemann-Liouville integral using
polynomial or discrete Fourier basis, we take an alternative approach which is
based on expressing the Riemann-Liouville definition of the fractional integral
for the semi-infinite axis in terms of a moment problem. As a result,
fractional derivatives of the Gaussian function and Dawson's integral are
expressed as a weighted sum of complex scaled Gaussian and Dawson's integral.
Error bounds for the approximation are provided. Another distinct feature of
the proposed method compared to the previous approaches, it can be extended to
approximate partial derivative with respect to the order of the fractional
derivative which may be used in PDE constraint optimization problems
Sampling for approximating -limited functions
-limited functions are multivariate generalization of band-limited
functions whose Fourier transforms are supported within a compact region
. In this work, we generalize sampling and
interpolation theorems for band-limited functions to -limited functions.
More precisely, we investigated the following question: "For a function
compactly supported within a region similar to , does there exist an
-limited function that agrees with the function over its support for a
desired accuracy?". Starting with the Fourier domain definition of an
-limited function, we write the equivalent convolution and a discrete
Fourier transform representations for -limited functions through
approximation of the convolution kernel using a discrete subset of Fourier
basis. The accuracy of the approximation of the convolution kernel determines
the accuracy of the discrete Fourier representation. Construction of the
discretization can be achieved using the tools from approximation theory as
demonstrated in the appendices. The main contribution of this work is proving
the equivalence between the discretization of the Fourier and convolution
representations of -limited functions. Here discrete convolution
representation is restricted to shifts over a compactly supported region
similar to . We show that discrete shifts for the convolution representation
are equivalent to the spectral parameters used in discretization of the Fourier
representation of the convolution kernel. This result is a generalization of
the cardinal theorem of interpolation of band-limited functions. The error
corresponding to discrete convolution representation is also bounded by the
approximation of the convolution kernel using discretized Fourier basis