Approximation on Banach spaces of functions on the sphere

AbstractMany approximation results were proved on Lp(Sd-1), 1⩽p⩽∞ where Sd-1 is the unit sphere in Rd. We will show here that most of these results extend to Banach spaces on the sphere for which operation by a d×d orthogonal matrix is a continuous isometry

Jackson theorem in Lp,0<p<1, for functions on the sphere

AbstractThe best approximation of functions in Lp(Sd−1),0<p<1 by spherical harmonic polynomials is shown to be bounded by a modulus of smoothness recently introduced by the second author

Bernstein-type polynomials on several intervals

We construct the analogues of Bernstein polynomials on the set Js of s finitely many intervals. Two cases are considered: first when there are no restrictions on Js, and then when Js has a so-called T-polynomial. On such sets we define approximating operators resembling the classic Bernstein polynomials. Reproducing and interpolation properties as well as estimates for the rate of convergence are given. © Springer International Publishing AG 2017

Towards a unified theory of Sobolev inequalities

We discuss our work on pointwise inequalities for the gradient which are connected with the isoperimetric profile associated to a given geometry. We show how they can be used to unify certain aspects of the theory of Sobolev inequalities. In particular, we discuss our recent papers on fractional order inequalities, Coulhon type inequalities, transference and dimensionless inequalities and our forthcoming work on sharp higher order Sobolev inequalities that can be obtained by iteration.Comment: 39 pages, made some changes to section 1

Random Field Models for Relaxor Ferroelectric Behavior

Heat bath Monte Carlo simulations have been used to study a four-state clock model with a type of random field on simple cubic lattices. The model has the standard nonrandom two-spin exchange term with coupling energy $J$ and a random field which consists of adding an energy $D$ to one of the four spin states, chosen randomly at each site. This Ashkin-Teller-like model does not separate; the two random-field Ising model components are coupled. When $D / J = 3$, the ground states of the model remain fully aligned. When $D / J \ge 4$, a different type of ground state is found, in which the occupation of two of the four spin states is close to 50%, and the other two are nearly absent. This means that one of the Ising components is almost completely ordered, while the other one has only short-range correlations. A large peak in the structure factor $S (k)$ appears at small $k$ for temperatures well above the transition to long-range order, and the appearance of this peak is associated with slow, "glassy" dynamics. The phase transition into the state where one Ising component is long-range ordered appears to be first order, but the latent heat is very small.Comment: 7 pages + 12 eps figures, to appear in Phys Rev