Constructive description of Hölder-like classes on an arc in R3 by means of harmonic functions

Constructive Description of Function Classes on Surfaces in R^3 and R^4

“…A proof of this Lemma is in [6, chapter 6]; it also can be proved similarly to the arguments in [1].…”

“…After that, consider the function f 0 defined by relations (11)-(13). Apply to f 0 the arguments similar to those from [1] for the space R 4 instead of R 3 . This gives the desired relations (4)-(6).…”

“…It turned out that classes of functions on a flat curve has its own peculiarity [2,3,5]. Up to the work [1], there were no constructive description of function classes on a curve in R 3 with using polynomials, rational functions, harmonic functions, or any other functions used for approximation. In [1] functions harmonic in a neighbourhood of the curve were used; the approximation speed or estimation of their gradient depended on this neighbourhood.…”

“…Up to the work [1], there were no constructive description of function classes on a curve in R 3 with using polynomials, rational functions, harmonic functions, or any other functions used for approximation. In [1] functions harmonic in a neighbourhood of the curve were used; the approximation speed or estimation of their gradient depended on this neighbourhood. A curve on a plane can be treated as a partial case of a spatial curve; then functional classes on this curve can be described not only in terms of approximation speed by analytical polynomials, but also in terms of functions harmonic in threedimensional neighbourhoods of the curve.…”

“…In the proof of the Theorem, we will use the continuation of f on R 4 called pseudo-harmonic, similarly to continuation of a function from a curve in [1].…”