An independent derivation of solutions to the Camassa-Holm equation in terms of multi-dimensional theta functions is presented using an approach based on Fay’s identities. Reality and smoothness conditions are studied for these solutions from the point of view of the topology of the underlying real hyperelliptic surface. The solutions are studied numerically for concrete examples, also in the l...

We discuss various properties of the new modulus of smoothness ω k,r (f, t)p := sup 0<h6t ‖W kh(·)∆khφ(·)(f , ·)‖Lp [−1,1], where φ(x) := √ 1− x2 and Wδ(x) = ( (1−x−δφ(x)/2)(1+x−δφ(x)/2) )1/2 . Related moduli with more general weights are also considered.

We investigate interpolatory multiscale transformations for functions between manifolds which are based on interpolatory subdivision rules. We characterize the HölderZygmund smoothness of a function between manifolds in terms of the coe cient decay w.r.t. this multiscale transform.

Via a random construction we establish necessary conditions for L(l) inequalities for certain families of operators arising in harmonic analysis. In particular we consider dilates of a convolution kernel with compactly supported Fourier transform, vector maximal functions acting on classes of entire functions of exponential type, and a characterization of Sobolev spaces by square functions and ...

and Applied Analysis 3 Since 1967, Durrmeyer introduced Bernstein-Durrmeyer operators, and there are many paperswhich studied theirproperties 1–7 . In 1991, Zhang studied the characterization of convergence forMn,1 f ;x with Jacobi weights. In 1992, Zhou 5 considered multivariate Bernstein-Durrmeyer operators Mn,d f ;x and obtained a characterization of convergence. In 2002, Xuan et al. studied...

One purpose of this article is to establish results on the rate of almost everywhere convergence of approximation processes of convolution type in L(R), where instead of a particular rate (like t, μ > 0, t → 0+) fractional moduli of smoothness are employed. An essential tool is a modified K-functional. Away from saturation orders these results are nearly optimal. A second purpose is to illustra...

We examined in which way gradual changes in the geometric structure of the illumination affect the perceived glossiness of a surface. The test stimuli were computer-generated three-dimensional scenes with a single test object that was illuminated by three point light sources, whose relative positions in space were systematically varied. In the first experiment, the subjects were asked to adjust...