Discrete Fourier analysis with lattices on planar domains

A discrete Fourier analysis associated with translation lattices is developed recently by the authors. It permits two lattices, one determining the integral domain and the other determining the family of exponential functions. Possible choices of lattices are discussed in the case of lattices that tile \RR^2 and several new results on cubature and interpolation by trigonometric, as well as algebraic, polynomials are obtained

Discrete Fourier analysis, Cubature and Interpolation on a Hexagon and a Triangle

Several problems of trigonometric approximation on a hexagon and a triangle are studied using the discrete Fourier transform and orthogonal polynomials of two variables. A discrete Fourier analysis on the regular hexagon is developed in detail, from which the analysis on the triangle is deduced. The results include cubature formulas and interpolation on these domains. In particular, a trigonometric Lagrange interpolation on a triangle is shown to satisfy an explicit compact formula, which is equivalent to the polynomial interpolation on a planer region bounded by Steiner's hypocycloid. The Lebesgue constant of the interpolation is shown to be in the order of $(\log n)^2$. Furthermore, a Gauss cubature is established on the hypocycloid.Comment: 29 page

Discrete Fourier Analysis and Chebyshev Polynomials with G2G_2 Group

The discrete Fourier analysis on the $30^{\degree}$-$60^{\degree}$-$90^{\degree}$ triangle is deduced from the corresponding results on the regular hexagon by considering functions invariant under the group $G_2$, which leads to the definition of four families generalized Chebyshev polynomials. The study of these polynomials leads to a Sturm-Liouville eigenvalue problem that contains two parameters, whose solutions are analogues of the Jacobi polynomials. Under a concept of $m$-degree and by introducing a new ordering among monomials, these polynomials are shown to share properties of the ordinary orthogonal polynomials. In particular, their common zeros generate cubature rules of Gauss type

The clustering of galaxies with pseudo bulge and classical bulge in the local Universe

We investigate the clustering properties and close neighbour counts for galaxies with different types of bulges and stellar masses. We select samples of "classical" and "pseudo" bulges, as well as "bulge-less" disk galaxies, based on the bulge/disk decomposition catalog of SDSS galaxies provided by Simard et al. (2011). For a given galaxy sample we estimate: the projected two-point cross-correlation function with respect to a spectroscopic reference sample, w_p(r_p), and the average background-subtracted neighbour count within a projected separation using a photometric reference sample, N_neighbour(<r_p). We compare the results with the measurements of control samples matched in color, concentration and redshift. We find that, when limited to a certain stellar mass range and matched in color and concentration, all the samples present similar clustering amplitudes and neighbour counts on scales above ~0.1h^{-1}Mpc. This indicates that neither the presence of a central bulge, nor the bulge type is related to intermediate-to-large scale environments. On smaller scales, in contrast, pseudo-bulge and pure-disk galaxies similarly show strong excess in close neighbour count when compared to control galaxies, at all masses probed. For classical bulges, small-scale excess is also observed but only for M_stars < 10^{10} M_sun; at higher masses, their neighbour counts are similar to that of control galaxies at all scales. These results imply strong connections between galactic bulges and galaxy-galaxy interactions in the local Universe, although it is unclear how they are physically linked in the current theory of galaxy formation.Comment: 14 pages, 16 figures, accepted for publication in MNRA