Iterated Function Systems and Łojasiewicz–Siciak Condition of Green’s Function

Białas-Cież, Leokadia; Kosek, Marta

doi:10.1007/s11118-010-9188-7

Cited by 12 publications

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References 13 publications

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“…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,…”

Section: Theorem 12 Let K ⊂ R N Be a Compact Set Containing At Leastmentioning

confidence: 99%

“…3) and diamK stands for the diameter of K . 2 The constant ς(K ) is a certain constant depending on K and is precisely defined in Sect. 4 via Theorem 1.1. g λ : K λ w −→ 1/(w − λ − 1) ∈ C (cf.…”

Section: Theorem 12 Let K ⊂ R N Be a Compact Set Containing At Leastmentioning

confidence: 99%

“…Białas-Cież and Kosek claim in [2] that so far very few examples of sets with socalled Łojasiewicz-Siciak property are known. Their paper is devoted to the problem of delivering some new examples of such sets (which are connected with iterated function systems).…”

Section: Lemma 32 Assume That K ⊂ R N Is a Compact Set Containing Atmentioning

confidence: 99%

“…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.…”

mentioning

confidence: 99%

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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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Section: Theorem 12 Let K ⊂ R N Be a Compact Set Containing At Leastmentioning

confidence: 99%

Section: Theorem 12 Let K ⊂ R N Be a Compact Set Containing At Leastmentioning

confidence: 99%

Section: Lemma 32 Assume That K ⊂ R N Is a Compact Set Containing Atmentioning

confidence: 99%

mentioning

confidence: 99%

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

Pierzchała

2014

Constr Approx

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“…Białas-Cież and M. Kosek claim in [3] that so far very few examples of sets with the (ŁS) property are known. Their paper is devoted to the problem of delivering some new examples of such sets (which are connected with iterated function systems).…”

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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The minimum sets and free boundaries of strictly plurisubharmonic functions

Dinew

2016

Calc. Var.

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We study the minimum sets of plurisubharmonic functions with strictly positive Monge-Ampère densities. We investigate the relationship between their Hausdorff dimension and the regularity of the function. Under suitable assumptions we prove that the minimum set cannot contain analytic subvarieties of large dimension. In the planar case we analyze the influence on the regularity of the right hand side and consider the corresponding free boundary problem with irregular data. We provide sharp examples for the Hausdorff dimension of the minimum set and the related free boundary. We also draw several analogues with the corresponding real results.

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Iterated Function Systems and Łojasiewicz–Siciak Condition of Green’s Function

Cited by 12 publications

References 13 publications

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

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

The Łojasiewicz–Siciak Condition of the Pluricomplex Green Function

The minimum sets and free boundaries of strictly plurisubharmonic functions

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