Orders of orthoprojection diameters of classes of periodic functions of one and several variables. (Russian) Zbl 0659.42008

Galeev, Eh. M.

Mat. Zametki 43, No. 2, 197-211 (1988).

Let \(x^{(k)}(t)\), \(t\in R\), be the derivative of fractional order k in the sense of formal differentiation of the Fourier series of a \(2\pi\)- periodic function x(t), and let \(\tilde W^ k_ p=\{x:\) \(\| x^{(k)}\|_ p\leq 1\}\). Let \(1<p^ i<\infty\), \(k^ i\in R\) for \(i=1,2,...,m\). The order of the orthoprojective diameter \(d^{\perp}_ N(\cap^{m}_{i=1}\tilde W^{k^ i}_{p^ i},L_ q)\) is established. The results are generalized also to the multidimensional case.

Reviewer: J.Musielak

MSC:

42A75	Classical almost periodic functions, mean periodic functions
46E35	Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems

Keywords:

formal differentiation of the Fourier series; orthoprojective diameter