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1

Approximation of entire transcendental functions of several complex variables in some Banach spaces for slow growth

Kumar D.

Journal of Applied Analysis

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2017

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Vol. 23, nr 1

9--20

EN

In the present paper, the coefficients characterizations of generalized type Tm(f; α, α) of entire transcendental functions f of several complex variables m (m ≥ 2) for slow growth have been obtained in terms of the sequence of best polynomial approximations of f in the Hardy Banach spaces Hq(Um) and in the Banach spaces Bm(p, q, λ). The presented work is the extension and refinement of the corresponding assertions made by Vakarchuk and Zhir [20-25], Gol’dberg [4] and Sheremeta [17, 18] to the multidimensional case.

2

On the Łojasiewicz exponent of the gradient of holomorphic functions

Bogusławska M.

Bulletin of the Polish Academy of Sciences. Mathematics

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1999

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Vol. 47, no 4

337-343

EN

We show that the Łojasiewicz exponent L[sub o] (grad f) at zero of the gradient of a distinguished pseudo-polynomial f [...] is attained on the zero-set of f'[sub y].

3

Lemmass A and B for sub-Pfaffian sets

Hajto Z.

Bulletin of the Polish Academy of Sciences. Mathematics

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1999

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Vol. 47, no 4

325-336

EN

In this paper we prove two fundamental lemmas of sub-Pfaffian geometry which are counterparts of Lemmas A and B for subanalytic sets [4]. We use a generalized version of the Tangent Mapping Theorem [2], following our program announced in [11].

4

A model-theoretic version of the complement theorem

Nowak K.

Bulletin of the Polish Academy of Sciences. Mathematics

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1999

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Vol. 47, no 4

345-353

EN

This paper deals with an axiomatic theory T[sub an] and the expansion R[sub an] of the ordered field of reals, formed by attaching the restricted analytic functions. We show that the theory T[sub an] is model-complete, which may be regarded as a version of Gabrielov's complement theorem. Our proof is based on Robinson's test and it does not involve a partition technique. An immediate corollary is that T[sub an] coincides with the semantic theory Th(R[sub an]) of all sentences true in the structure R[sub an].

5

Hoelder continuity property of composite Julia sets

Kosek M.

Bulletin of the Polish Academy of Sciences. Mathematics

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1998

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Vol. 46, no 4

391-399

EN

In this paper we consider the composite Julia associated with a finite family of the proper polynomial mappings in [C^n]. We show its pluricomplex Green function is Hoelder continous. This yields in particular that the set preserves Markov's inequality.

6

Borne d'invariant metrique pour une famille noetherienne

El Khadiri A.

Bulletin of the Polish Academy of Sciences. Mathematics

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1998

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Vol. 46, no 3

265-275

EN

The motivation of this paper is a question asked by B. Teissier in [1]: if [fi] : M --> N is an analytic morphism between two real analytic manifolds M and N, and if K is a compact subanalytic set of M, then for every point x[sub 0] in [fi](K) there exists an open neighbourhood U of x[sub 0] in N and a constant [gamma] > 0 such that for all x in U and all (a, b) in the same connected component of [fi^-1(x) intersection of sets K], there exist a rectifiable curve in [fi-1(x) intersection of sets K joining a and b with length less than [gamma]. In this paper we prove the following statement: let [Omega] be an open set of [R^n], N a real analytic manifold, [fi] : [Omega] --> N a proper analytic morphism and [K is a subset of set Omega] an analytic subset of [Omega]. Then for every point y[sub 0] of N there an open neighbourhood U in N and a constant [eta] > 0 such that for all y in U there exists C[sub y] > 0 satisfying the following: for every points a, b of the same connected component of [fi^-1(y) intersection of sets K] there exists an analytic rectifiable curve [sigma] in [fi^-1(y) intersection of sets] K joining a and b with [absolute value of sigma is less than or equal to] C[sub y] [absolute value a - b^eta], where [absolute value of sigma] is the length of [sigma] and [absolute value a - b] is the euclidean distance between a and b.