## Nikol’skij-Besov classes on compact symmetric spaces of rank 1.(English. Russian original)Zbl 1011.46033

Dokl. Math. 55, No. 2, 266-268 (1997); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 353, No. 6, 723-725 (1997).
From the introduction: In this work, we study the Nikol’skij-Besov spaces $$B^r_{p,\theta} (M)$$ for compact symmetric spaces of rank 1 so-called (CROSS) $$M$$. First, we define the spaces $$B^r_{p,\theta} (M)$$ in terms of averaged differences of functions along geodesics; then we obtain equivalent descriptions of these spaces with the help of the best approximations by spherical polynomials on $$M$$ and provide various equivalent normalizations of these spaces. Similar results for the sphere were obtained by S. M. Nikol’skij and P. I. Lizorkin.
See also the author’s paper in Tr. Petrozavodsk. Gos. Univ. Ser. Mat. 3, 153-172 (1996; Zbl 1042.46018).

### MSC:

 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)

### Keywords:

CROSS; Nikolskij-Besov spaces

Zbl 1042.46018