Filtering in a single time domain or in a single frequency domain has recently been generalized to filtering in a single fractional Fourier domain. In this correspondence, we further generalize this to repeated filtering in consecutive fractional Fourier domains and discuss its applications to signal restoration through an illustrative example.

Introduction Whereas ordinary functions on a locally compact group map the group elements into the complex numbers, a stochastic process can be understood as a mapping into a Hilbert space. The idea of a generalized function is to reduce the knowledge about the function to that of certain averages. It leads to the concept of generalized functions as continuous linear functionals on spaces of te...

We study a canonical basis for spaces of weakly holomorphic modular forms of weights 12, 16, 18, 20, 22, and 26 on the full modular group. We prove a relation between the Fourier coefficients of modular forms in this canonical basis and a generalized Ramanujan τ -function, and use this to prove that these Fourier coefficients are often highly divisible by 2.

Estimates for the decay of Fourier transforms of measures have extensive applications in numerous problems in harmonic analysis and convexity including the distribution of lattice points in convex domains, irregularities of distribution, generalized Radon transforms and others. Here we prove that the spherical L-average decay rate of the Fourier transform of the Lebesgue measure on an arbitrary...

The method of partial Fourier transform is used to find explicit formulas for two remarkable fundamental solutions for a generalized Tricomi operator. These fundamental solutions reflect clearly the mixed type of the Tricomi operator. In proving these results, we establish explicit formulas for Fourier transforms of some functions involving Bessel functions.

We propose a generalized entropy maximization procedure, which takes into account the generalized averaging procedures and information gain definitions underlying the generalized entropies. This novel generalized procedure is then applied to Rényi and Tsallis entropies. The generalized entropy maximization procedure for Rényi entropies results in the exponential stationary distribution asymptot...

A new generalized function space in which all Gelfand-Shilov classes S ′0 α (α > 1) of analytic functionals are embedded is introduced. This space of ultrafunctionals does not possess a natural nontrivial topology and cannot be obtained via duality from any test function space. A canonical isomorphism between the spaces of hyperfunctions and ultrafunctionals on R is constructed that extends the...

A method is described for validating fast Fourier transforms (FFTs) based on the use of simple input fiinctions whose discrete Fourier transforms can be evaluated in closed form. Explicit analytical results are developed for one-dimensional and two-dimensional discrete Fourier transforms. The analytical results are easily generalized to higher dimensions. The results offer a means for validatin...

In a series of recent papers (Adcock, Hansen and Poon, 2013, Appl. Comput. Harm. Anal. 45(5):3132–3167), (Adcock, Gataric and Hansen, 2014, SIAM J. Imaging Sci. 7(3):1690–1723) and (Adcock, Hansen, Kutyniok and Ma, 2015, SIAM J. Math. Anal. 47(2):1196–1233), it was shown that one can optimally recover the wavelet coefficients of an unknown compactly supported function from pointwise evaluations...