Explicit traces of functions from Sobolev spaces and quasi-optimal linear interpolators

Abstract

Abstract. Let Λ ⊂ R be a strictly increasing sequence. For r = 1,2 , we give a simple explicit expression for an equivalent norm on the trace spaces We also construct an interpolating spline of low degree having optimal norm up to a constant factor. This spline and the equivalent trace norm are very easy to compute. We also conjecture, what is the expression for the equivalent trace norm for any r 1 and give some partial results, which, in particular, confirm this conjecture.Mathematics subject classification (2010): Primary 46E35, Secondary 41A15, 58C25.

Sobolev functions on closed subsets of the real line

“…(Note that, according to the notational convention adopted in (1.2), it must be understood that in (1.7) We prove Theorem 1.3 in Section 5. For a special case of Theorem 1.3 for m = 2 and strictly increasing sequences {x i } i∈Z with x i+1 − x i ≤ const, and also for other results related to characterization of the trace space W m p (R)| E , see Estévez [5]. In Section 6 we also give another characterization of the trace space W m p (R)| E expressed in terms of L p -norms of certain kinds of "sharp maximal functions" which are defined as follows.…”

“…For a special case of Theorem 1.3 for m = 2 and strictly increasing sequences {x i } i∈Z with x i+1 − x i ≤ const, and other results related to characterization of the trace space W m p (R)| E see Estévez [22].…”