On the Invalidity of Fourier Series Expansions of Fractional Order

The purpose of this short paper is to show the invalidity of a Fourier series expansion of fractional order as derived by G. Jumarie in a series of papers. In his work the exponential functions eHe showed that any smooth periodic function f with period M

Approximation of Bivariate Functions via Smooth Extensions

For a smooth bivariate function defined on a general domain with arbitrary shape, it is difficult to do Fourier approximation or wavelet approximation. In order to solve these problems, in this paper, we give an extension of the bivariate function on a general domain with arbitrary shape to a smooth, periodic function in the whole space or to a smooth, compactly supported function in the whole space. These smooth extensions have simple and clear representations which are determined by this bivariate function and some polynomials. After that, we expand the smooth, periodic function into a Fourier series or a periodic wavelet series or we expand the smooth, compactly supported function into a wavelet series. Since our extensions are smooth, the obtained Fourier coefficients or wavelet coefficients decay very fast. Since our extension tools are polynomials, the moment theorem shows that a lot of wavelet coefficients vanish. From this, with the help of well-known approximation theorems, using our extension methods, the Fourier approximation and the wavelet approximation of the bivariate function on the general domain with small error are obtained.

Inertia Driven Dispersion Between Patterned Surfaces

Understanding and controlling stirring in micro-systems is necessary for the design of efficient passive micro-mixer. In this study, we focus on the dispersion of passive tracers injected in flows in between two rough surfaces under weak inertia influence (small but non-zero Reynolds number). The flow is induced by a constant applied pressure gradient between two cross-sections of the channel and the velocity field is calculated thanks to an extension of the lubrication approximation taking into account the first order inertial corrections. Tracers trajectories are obtained by integrating numerically the quasi-analytic velocity field. Our purpose is to examine the flow structure for various surface patterns and various Reynolds number. We focus on a simplified aperture field which is a smooth periodic function. This study puts forward interesting behavior of streamlines and show the dispersion of passive tracers in various geometries.

A global theory of flexes of periodic functions

For a real valued periodic smooth function u on R, n ≥ 0, one defines the osculating polynomial φs (of order 2n + 1) at a point s ∈ R to be the unique trigonometric polynomial of degree n, whose value and first 2n derivatives at s coincide with those of u at s. We will say that a point s is a clean maximal flex (resp. clean minimal flex) of the function u on S1 if and only if φs ≥ u (resp. φs ≤ u) and the preimage (φ - u)-1(0) is connected. We prove that any smooth periodic function u has at least n + 1 clean maximal flexes of order 2n + 1 and at least n + 1 clean minimal flexes of order 2n + 1. The assertion is clearly reminiscent of Morse theory and generalizes the classical four vertex theorem for convex plane curves.