We present several results giving some estimates for the error in best polynomial approximation of holomorphic functions on compact subsets of R N . We base our approach on the Bernstein-Walsh-Siciak theorem, which states in terms of the Siciak extremal function how fast a holomorphic function, defined in an appropriate neighborhood of a compact L-regular set K ⊂ C N , can be approximated on K by complex polynomials. Our purpose is among others to state a result for an arbitrary compact subset of R N (not necessarily L-regular) and to replace the Siciak extremal function (which can hardly ever be computed, especially if N > 1) simply by the distance function to K .Keywords Polynomial approximation · The Bernstein-Walsh-Siciak theorem · Subanalytic set · Degree of approximation · Extremal function · Pluricomplex Green function · (HCP) property Mathematics Subject Classification 32E30 · 41A10 · 41A25 · 32B20 · 03C64 · 32U35
IntroductionWe identify R N with the set {z ∈ C N : Im(z ν ) = 0 for ν = 1, . . . , N }. Throughout the paper, N := {1, 2, 3, . . .}. Suppose that U ⊂ C N is a nonempty open set. We will denote by H ∞ (U ) the Banach space of all bounded and holomorphic functions in U (with the norm U ). The problem of approximation of holomorphic functions (of several variables) was studied by many authors -see, for example, [4][5][6][7][8][9][10]16,32,33,37] and the huge bibliography therein. Our aim is among others to prove Theorems 1.1 and 1.2. Theorem 1.1 is the basis for Theorem 1.2, which directly concerns the polynomial approximation of holomorphic functions. Theorem 1.1 There exists a constant ε N > 0 (depending only on N ∈ N) such that, for each compact set K ⊂ R N containing at least two distinct points,for all z ∈ C N . 1
Theorem 1.2 Let K ⊂ R N be a compact set containing at least two distinct points.For each λ > 0, set K λ := {z ∈ C N : dist(z; K ) < λ}. Assume that 0 < υ < ς(K ). 2 Then, there exists a function ϑ : (0, +∞) −→ (0, +∞) (depending on K and υ) such that, for each λ ∈ (0, +∞), each f ∈ H ∞ (K λ ), and each n ∈ N,The proof of Theorem 1.2 relies on Theorem 3.1, which is usually called the Bernstein-Walsh-Siciak theorem. Theorem 3.1 is a very precise version of the OkaWeil theorem, but when we want to apply this theorem directly, we encounter a significant inconvenience. Namely, it uses the sets of the form {z ∈ C N : K (z) < R}, where K is the Siciak extremal function and R > 1. The problem is that the function K can hardly ever be computed (even for very simple sets). The advantage of our approach is that the sets {z ∈ C N : K (z) < R} are replaced by the natural sets K λ = {z ∈ C N : dist(z; K ) < λ}. It should be stressed, however, that there is also a disadvantage of such an approach. This is underlined in Remark 6.6. As we explain in the example following Corollary 5.5, for the set K := [−1, 1] and the family of functions f λ : K λ w −→ 1/(w − iλ) ∈ C with λ ∈ (0, +∞), the estimate of Corollary 4.1 (which is very closely connected with Theorem 1.2) is asymptotically exact a...