A rational polytope is the convex hull of a finite set of points in Rd with rational coordinates. Given a rational polytope P ⊆ Rd, Ehrhart proved that, for t ∈ Z>0, the function #(tP ∩ Zd) agrees with a quasi-polynomial LP(t), called the Ehrhart quasi-polynomial. The Ehrhart quasi-polynomial can be regarded as a discrete version of the volume of a polytope. We use that analogy to derive a new ...

We study the behavior of best simultaneous (lq , Lp)-approximation by rational functions on an interval, when the measure tends to zero. In addition, we consider the case of polynomial approximation on a finite union of intervals. We also get an interpolation result.

Lp−Lq–boundedness of the map f → w(x) ∫ b(x) a(x) k(x, y)f(y)v(y)dy is described by different types of criteria expressed in terms of given parameters 0 < p, q < ∞, strictly increasing boundaries a(x) and b(x), locally integrable weight functions v, w and a positive continuous kernel k(x, y) satisfying some growth conditions.

Let A = (aj,k)j,k≥1 be a non-negative matrix. In this paper, we characterize those A for which ‖A‖lp,lq are determined by their actions on non-negative decreasing sequences, where one of p and q is 1 or ∞. The conditions forcing on A are sufficient and they are also necessary for nonnegative finite matrices.

Let 0 < T: LP(Y, v) -+ Lq(X, ) be a positive linear operator and let HITIP ,q denote its operator norm. In this paper a method is given to compute 1Tllp, q exactly or to bound 11Tllp q from above. As an application the exact norm 11VIlp,q of the Volterra operator Vf(x) = fo f(t)dt is computed.

We consider second order parabolic and elliptic systems with leading coefficients having the property of vanishing mean oscillation (VMO) in the spatial variables. An Lq −Lp theory is established for systems both in divergence and non-divergence form. Higher order parabolic and elliptic systems are also discussed briefly.

We give a characterization of the weights u(·) and v(·) for which the fractional maximal operator Ms is bounded from the weighted Lebesgue spaces Lp(lr, vdx) into Lq(lr, udx) whenever 0 ≤ s < n, 1 < p, r < ∞, and 1 ≤ q < ∞.

We determine the asymptotic behaviour (as k →∞, up to multiplicative constants not depending on k) of the entropy numbers ek (Dσ : lp → lq), 1 ≤ p ≤ q ≤ ∞, of diagonal operators generated by logarithmically decreasing sequences σ = (σn). This complements earlier results by Carl [2] who investigated the case of power-like decay of the diagonal. 2000 Mathematics Subject Classification: 47B06, 46B...

We obtain estimates on the order of best approximation by polynomials and ridge functions in the spaces 11 Lq of classes of s-monotone radial functions which belong to the space Lp , 1 q p ∞. © 2007 Published by Elsevier Inc. 13

New conditions of Lp[0,∞)−Lq[0,∞) boundedness and compactness for the operator f → w(x) R b(x) a(x) k(x, y)f(y)v(y)dy with locally integrable weight functions v, w and a positive continuous kernel k(x, y) from Oinarov’s type class are obtained for 1 < p, q <∞.