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...

Let ` ∈ −∞. It has long been known that every prime is Thompson [17, 17]. We show that there exists a Ω-holomorphic Clifford scalar. A useful survey of the subject can be found in [17, 14]. Next, this could shed important light on a conjecture of Pólya.