For a convex body $K$ in $\mathbb{R}^n$, we introduce and study the extremal general affine surface areas, defined by \[ {\rm IS}_{\varphi}(K):=\sup_{K^\prime\subset K}{\rm as}_{\varphi}(K),\quad os}_{\psi}(K):=\inf_{K^\prime\supset as}_{\psi}(K) \] where ${\rm as}_{\varphi}(K)$ as}_{\psi}(K)$ are $L_\varphi$ $L_\psi$ area of $K$, respectively. We prove that there exist bodies achieve supremum ...

In this paper we establish a connection between the associated variety of a representation and the existence of certain degenerate Whittaker functionals, for both smooth and K-finite vectors, for all quasi-split real reductive groups, thereby generalizing results of Kostant, Matumoto and others.

We present a characterization of the approximation errors of the Post-Widder and the Gamma operators in Lp[0, ∞), 1 ≤ p ≤ ∞, with a weight x γ for any real γ. Two types of characteristics are used – weighted K-functionals of the approximated function itself and the classical fixed step moduli of smoothness taken on a simple modification of it.

We establish the various relationships that exist among the integral transform Ᏺ α,β F , the convolution product (F * G) α , and the first variation δF for a class of functionals defined on K[0,T ], the space of complex-valued continuous functions on [0,T ] which vanish at zero. 1. Introduction and definitions. In a unifying paper [10], Lee defined an integral transform Ᏺ α,β of analytic functi...

We give an alternative proof of a recent result of Klartag on the existence of almost subgaussian linear functionals on convex bodies. If K is a convex body in Rn with volume one and center of mass at the origin, there exists x 6= 0 such that |{y ∈ K : |〈y, x〉| > t‖〈·, x〉‖1}| 6 exp(−ct 2/ log2(t+ 1)) for all t > 1, where c > 0 is an absolute constant. The proof is based on the study of the Lq–c...

For the infinite triangular arrays of points whose rows consist of (i) the nth roots of unity, (ii) the extrema of Chebyshev polynomials Tn(x) on [—1,1], and (iii) the zeros of Tn(x), we consider the corresponding sequences of divided difference functionals {In}f in the successive rows of these arrays. We investigate the totality of such functionals as well as the convergence of the generalized...

Rate of approximation of combinations of averages on the spheres is shown to be equivalent to K-functionals yielding higher degree of smoothness. Results relating combinations of averages on rims of caps of spheres are also achieved. © 2004 Elsevier Inc. All rights reserved. MSC: 41A17; 41A63; 26B33; 46E35