The Łojasiewicz–Siciak Condition of the Pluricomplex Green Function

Pierzchała, Rafał

doi:10.1007/s11118-013-9339-8

Cited by 7 publications

(8 citation statements)

References 19 publications

(13 reference statements)

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“…We note in [23] that a straightforward consequence of Lemma 3.2 is that each compact subset of R N satisfies the Łojasiewicz-Siciak condition with the exponent 1. However, this is insufficient for our purpose, and we will need Theorem 1.1 which is a more precise result.…”

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

confidence: 98%

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

confidence: 98%

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

Pierzchała

2014

Constr Approx

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“…If the set K ⊂ C has simple geometry -it is connected, does not disconnect the plane and it is locally a smooth curve except for a finite set of points at which branching may occur, and if the finite number of curves at the branching points meet at angles [24]). …”

Section: Remark 13mentioning

confidence: 99%

Differential Tests for Plurisubharmonic Functions and Koch Curves

Dinew

2018

Potential Anal

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We study minimum sets of singular plurisubharmonic functions and their relation to upper contact sets. In particular we develop an algorithm checking when a naturally parametrized curve is such a minimum set. The case of Koch curves is studied in detail. We also study the size of the set of upper non-contact points. We show that this set is always of Lebesgue measure zero thus answering an open problem in the viscosity approach to the complex Monge-Ampère equation. Finally, we prove that similarly to the case of convex functions, strictly plurisubharmonic lower tests yield existence of upper tests with a control on the opening.

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“…This notion was introduced by Gendre in [18] and was further studied by Białas-Cież and Kosek in [5] (see also [28] Proof The complement on the Riemann sphere of a connected compact set is simply connected and hence the Riemann mapping exists. Let z ∈ C\K be a point satisfying dist (z, K ) ≤ 1.…”

Section: Definition 12mentioning

confidence: 99%

“…This is because any such set satisfies the Łojasiewicz-Siciak condition with exponent 1 (see [28]) and is obviously porous.…”

Section: Corollary 27 Any Compact Regular Subset K Of the Real Line Imentioning

confidence: 99%

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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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The Łojasiewicz–Siciak Condition of the Pluricomplex Green Function

Cited by 7 publications

References 19 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

Differential Tests for Plurisubharmonic Functions and Koch Curves

The minimum sets and free boundaries of strictly plurisubharmonic functions

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