Keywords: fractal geometry, fractional derivative, fractional Fourier transform, fractional power of a matrix, self similarity, complex partial differential equation, broadband ultrasound, frequency-dependent attenuation, time domain. 1. Backgrounds The rational behind this model is schematically illustrated below: Fractal geometry (irregular soft tissues) → Fractional Fourier transform (freque...

Whenever ranking data are collected, such as in elections, surveys, and database searches, it is frequently the case that partial rankings are available instead of, or sometimes in addition to, full rankings. Statistical methods for partial rankings have been discussed in the literature. However, there has been relatively little published on their Fourier analysis, perhaps because the abstract ...

The search for a canonical set of eigenvectors of the discrete Fourier transform has been ongoing for more than three decades. The goal is to find an orthogonal basis of eigenvectors which would approximate Hermite functions – the eigenfunctions of the continuous Fourier transform. This eigenbasis should also have some degree of analytical tractability and should allow for efficient numerical c...

We present a simple proof of the resolvent estimates of elliptic Fourier multipliers on the Euclidean space, and apply them to the analysis of time-global and spatially-local smoothing estimates of a class of dispersive equations. For this purpose we study in detail the properties of the restriction of Fourier transform on the unit cotangent sphere associated with the symbols of multipliers.

1 Some Basic Analysis 3 1.1 Function spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Linear operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.4 Fourier inversion on S and L . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.5 Sobolev spa...

Motivated by problems in signal recovery, we will investigate the distribution of the energy of the Fourier transform of a positive function on a finite group. In particular, we are able to bound from below the fraction of energy contained in various subsets of the Fourier transform of a positive function defined on a finite group. Applications to signal recovery for positive functions, as well...