2023/05/02 by Dzyubenko, German, Kopotun, Kirill A.
#41A05 #41A10 #41A15 #41A25 #41A29 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics
paper · doi:10.48550/arxiv.2305.01745
Given k∈ N, a nonnegative function f∈ Cr[a,b], r≥ 0, an arbitrary finite collection of points \αi\i∈ J ⊂ [a,b], and a corresponding collection of nonnegative integers \mi\i∈ J with 0≤ mi ≤ r, i∈ J, is it true that, for sufficiently large n∈ N, there exists a polynomial Pn of degree n such that (i) |f(x)-Pn(x)| ≤ c ρnr(x) ωk(f(r), ρn(x); [a,b]), x∈ [a,b], where ρn (x):= n-1 √(1-x2) +n-2 and ωk is the classical k-th modulus of smoothness, (ii) P(ν)(αi)=f(ν)(αi), for all 0≤ ν≤ mi and all i∈ J, and (iii) either P ≥ f on [a,b] (onesided approximation), or P ≥ 0 on [a,b] (positive approximation)? We provide \em precise answers not only to this question, but also to similar questions for more general \em intertwining and \em copositive polynomial approximation. It turns out that many of these answers are quite unexpected. We also show that, in general, similar questions for q-monotone approximation with q≥ 1 have negative answers, i.e., q-monotone approximation with general interpolatory constraints is impossible if q≥ 1.