About twenty years ago the measure of smoothness ω φ (f, t) was introduced and related to the rate of polynomial approximation. In this article we survey developments about this and related concepts since that time. MSC: 41A10, 41A17, 41A25, 41A27, 41A30, 41A36, 41A40, 41A50, 41A63, 26A15, 26B35, 26B05, 42C05, 26A51, 26A33, 46E35 keywords: Moduli of smoothness, K-functionals, realization functi...

About twenty years ago the measure of smoothness ω φ (f, t) was introduced and related to the rate of polynomial approximation. In this article we survey developments about this and related concepts since that time. MSC: 41A10, 41A17, 41A25, 41A27, 41A30, 41A36, 41A40, 41A50, 41A63, 26A15, 26B35, 26B05, 42C05, 26A51, 26A33, 46E35 keywords: Moduli of smoothness, K-functionals, realization functi...

In this paper, we study the asymptotics of singularly perturbed phase-transition functionals form ℱ k (u) = 1/ε ∫ A f (

The first such linear functional seems to have been constructed in [1], for the purpose of demonstrating the linear independence over an interval of all B-splines which do not vanish identically on that interval. Since then, such linear functionals have been constructed in various ways and for a variety of jobs [2] – [7], [11], [13], [16], some of which are listed in Section 2. In particular, i...

The problem of optimal estimation of the linear functionals Aξ = ∑∞ k=0 a(k)ξ(−k) and AN ξ = ∑N k=0 a(k)ξ(−k) which depend on the unknown values of a stochastic sequence ξ(k) with stationary nth increments is considered. Estimates are based on observations of the sequence ξ(k) + η(k) at points of time k = 0,−1,−2, . . ., where the sequence η(k) is stationary and uncorrelated with the sequence ξ...

The existence of infinitely many solutions is established for a class of nonlinear functionals involving the p-biharmonic operator with nonhomoge- neous Neumann boundary conditions. Using a recent critical-point theorem for nonsmooth functionals and under appropriate behavior of the nonlinear term and nonhomogeneous Neumann boundary conditions, we obtain the result.

The Chern–Simons functionals built from various connections determined by the initial data hμν , χμν on a 3-manifold Σ are investigated. First it is shown that for asymptotically flat data sets the logarithmic fall off for hμν and rχμν is the necessary and sufficient condition of the existence of these functionals. The functional Y(k,l), built in the vector bundle corresponding to the irreducib...

We consider in this paper a class of variational models introduced for image decomposition into cartoon and texture in [16] (see also [9]), of the form infu { |u|BV + λ‖K ∗ (f − u)‖qLp } where K is a real analytic integration kernel. We analyse and characterize the extremals of these functionals and list some of their properties.