Maximal function and Multiplier Theorem for Weighted Space on the Unit Sphere

For a family of weight functions invariant under a finite reflection group, the boundedness of a maximal function on the unit sphere is established and used to prove a multiplier theorem for the orthogonal expansions with respect to the weight function on the unit sphere. Similar results are also established for the weighted space on the unit ball and on the standard simplex.Comment: 24 pages, to appear in J. Funct. Analysi

Thermal and Non-thermal radiation from pulsars: hints of physics

Thermal and non-thermal radiation from pulsars carries significant information from surface and would have profound implications on the state of dense matter in compact stars. For the non-thermal radio emission, subpulse drifting phenomena suggest the existence of Ruderman-Sutherland-like gap-sparking and strong binding of particles on pulsar polar caps. While conventional neutron star models can hardly provide such a high binding energy, the strong self-bound surface of quark-cluster stars can naturally solve this problem. As for the thermal one, the featureless X-ray spectra of pulsars may indicate a bare surface without atmosphere, and the ultrarelativistic fireball of gamma-ray bursts and supernovae would also require strong self-bound surfaces. Recent achievements in measuring pulsar mass and mass-radius relation further indicate a stiff equation of state and a self-bound surface. Therefore, we conjecture that matters inside pulsar-like compact stars could be in a quark-cluster phase. The surface of quark-cluster stars is chromatically confined and could initially be bare. Such a surface can not only explain above features, but may also promote a successful core-collapse supernova, and the hydro-cyclotron oscillation of the electron sea above the surface could be responsible for those absorption features detected in the X-ray spectrum.Comment: 4 pages, contribution to the ERPM conferences, Zielona Gora, April 201

Polynomial Approximation in Sobolev Spaces on the Unit Sphere and the Unit Ball

This work is a continuation of the recent study by the authors on approximation theory over the sphere and the ball. The main results define new Sobolev spaces on these domains and study polynomial approximations for functions in these spaces, including simultaneous approximation by polynomials and relation between best approximation to a function and to its derivatives.Comment: 16 page

Moduli of Smoothness and Approximation on the Unit Sphere and the Unit Ball

A new modulus of smoothness based on the Euler angles is introduced on the unit sphere and is shown to satisfy all the usual characteristic properties of moduli of smoothness, including direct and inverse theorem for the best approximation by polynomials and its equivalence to a $K$-functional, defined via partial derivatives in Euler angles. The set of results on the moduli on the sphere serves as a basis for defining new moduli of smoothness and their corresponding $K$-functionals on the unit ball, which are used to characterize the best approximation by polynomials on the ball.Comment: 63 pages, to appear in Advances in Mat