Markov's inequality andC ∞ functions on sets with polynomial cusps

Pawłucki, Wiesław; Pleśniak, W.

doi:10.1007/bf01458617

Cited by 84 publications

(62 citation statements)

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“…Proof By Theorems 4.1 and 6.4 in [18], there exists κ = κ(λ 0 ) > 0, = (λ 0 ) > 0 such that (1 + t κ ). Now, it is enough to apply Theorem 6.1.…”

Section: Theorem 63 Assume That a Nonempty Compact Set K ⊂ R N Is Fatmentioning

confidence: 89%

Approximation of Holomorphic Functions on Compact Subsets of $$\mathbb {R}^N$$ R N

Pierzchała

2014

Constr Approx

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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...

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“…Proof By Theorems 4.1 and 6.4 in [18], there exists κ = κ(λ 0 ) > 0, = (λ 0 ) > 0 such that (1 + t κ ). Now, it is enough to apply Theorem 6.1.…”

Section: Theorem 63 Assume That a Nonempty Compact Set K ⊂ R N Is Fatmentioning

confidence: 89%

Approximation of Holomorphic Functions on Compact Subsets of $$\mathbb {R}^N$$ R N

Pierzchała

2014

Constr Approx

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“…The (HCP) property finds applications in the theory of polynomial inequalities (for example, Markov's inequality) and was investigated in [9,13,14,16,28]. Recently, L. Gendre introduced another condition, called the Łojasiewicz-Siciak condition, or (ŁS) for short (cf.…”

Section: R Pierzchała (B)mentioning

confidence: 99%

The Łojasiewicz–Siciak Condition of the Pluricomplex Green Function

Pierzchała

2013

Potential Anal

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The aim of this paper is to address a problem raised originally by L. Gendre, later by W. Pleśniak and recently by L. Białas-Cież and M. Kosek. This problem concerns the pluricomplex Green function and consists in finding new examples of sets with so-called Łojasiewicz-Siciak ((ŁS) for short) property. So far, the known examples of such sets are rather of particular nature. We prove that each compact subset of R N , treated as a subset of C N , satisfies the Łojasiewicz-Siciak condition. We also give a sufficient geometric criterion for a semialgebraic set in R 2 , but treated as a subset of C, to satisfy this condition. This criterion applies more generally to a set in C definable in a polynomially bounded o-minimal structure.

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“…Moreover, for such a set, the formula for the Siciak extremal function is known (see [17], p. 37), which allows one to compute the Markov's exponent (see [20], p. 129). (For other examples of Markov's sets, see [23].) Finally, let us consider the following: For any r > 0, put K r = {x ∈ C : |x| < r}.…”

Section: Construction Ofmentioning

confidence: 99%