Approximation of Subharmonic Functions

Yulmukhametov, R. S.

doi:10.1070/sm1985v052n02abeh002896

Cited by 47 publications

(49 citation statements)

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“…Azarin's result was refined substantially in [3]. Among other things, in that paper it was proved that for every subharmonic function u of finite order there exists an entire function f such that u(z) − ln |f (z)| = O(ln |z|), z / ∈ E, |z| −→ ∞, outside of an exceptional set E of finite measure.…”

Section: For Z ∈ C We Have the Upper Estimate Ln |L(z)| ≤ |U(z)| +mentioning

confidence: 99%

“…Among other things, in that paper it was proved that for every subharmonic function u of finite order there exists an entire function f such that u(z) − ln |f (z)| = O(ln |z|), z / ∈ E, |z| −→ ∞, outside of an exceptional set E of finite measure. It can easily be shown that the sine type functions introduced above share the properties of entire functions constructed in [3].…”

Section: For Z ∈ C We Have the Upper Estimate Ln |L(z)| ≤ |U(z)| +mentioning

confidence: 99%

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Entire functions of sine type and their applications

Башмаков

Путинцева

Yulmukhametov

2011

St. Petersburg Math. J.

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Abstract. For subharmonic functions that depend only on the real part of z, new constructions of "sine type functions" are presented. This term is reserved for entire functions whose deviation from a given function is majorized, everywhere except some collection of disks, by a certain constant. It is shown that the system of exponentials constructed by the zeros of a sine type function for some convex function is complete and minimal in a certain weighted Hilbert space on an interval of the real line.

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Section: For Z ∈ C We Have the Upper Estimate Ln |L(z)| ≤ |U(z)| +mentioning

confidence: 99%

Section: For Z ∈ C We Have the Upper Estimate Ln |L(z)| ≤ |U(z)| +mentioning

confidence: 99%

Entire functions of sine type and their applications

Башмаков

Путинцева

Yulmukhametov

2011

St. Petersburg Math. J.

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“…The idea that the behaviour of a general 5-subharmonic function U can be captured by another of the special form log \g\ with g meromorphic goes back several decades (asurvey is in [6], additional interesting references are [18], [12], [9], among others).…”

Section: Approximation By a Meromorphic Functionmentioning

confidence: 99%

“…Let \z\ = r and note that (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19) shows that z e BCCj, 10r/I(r)). Thus l0g í^) " C(A) " l0g|Z " OI " l0g í^) + C ' so the proof of Lemma 3-7 shows the expression in the first line of (3-17) is uniformly bounded.…”

Section: Then \Hj(z)\mentioning

confidence: 99%

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Asymptotic values of meromorphic functions of finite order

Pire¹,

Drasin²,

Granados³

2010

Indiana Univ. Math. J.

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ABSTRACT. The asymptotic valúes of a meromorphic function (of any order) defined in the complex plañe form a Suslinanalytic set. Moreover, given an analytic set A* we construct a meromorphic function of finite order and minimal growth having A* as its precise set of asymptotic valúes. 1NTRODUCTIONA nonconstant meromorphic function f(z) in the plañe has the asymptotic valué a if there is a curve y tending to oo such that f(z) ->• a as z ->• oo, z G y. Let As (f) be the set of asymptotic valúes of /; for example, As (e z ) ={ 0 , oo} . A classical result of Mazurkiewicz [13] asserts that As(f) is an analytic set in the sense of Suslin [3,16].Recall that the order of / is given by \osT(r,f)where T(r,f) is the Nevanlinna characteristic (when / is entire, T(r,f) may be replaced by logM(r, /), with M(r, f) the máximum modulus function).Heins [11] showed that given an analytic set A*, there is a meromorphic function / with As (f) = A* and, if oo e A*, then A* = As (f) for some entire function /. In general, Heins's function has infinite order. For example, if (1.1)A:=A* \ {oo} = A* nC, and card (A) = oo with A bounded, Heins produces a Riemann surface with infinitely many 'logarithmic branch points' over w = oo, so by Ahlfors's theorem A = oo. Note that A , as the intersection of two analytic sets, is analytic.

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The Beurling–Malliavin Multiplier Theorem and Its Analogs for the de Branges Spaces

Belov

Havin

2014

Operator Theory

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Let ω be a non-negative function on R. We are looking for a non-zero f from a given space of entire functions X satisfyingThe classical Beurling-Malliavin Multiplier Theorem corresponds to (a) and the classical Paley-Wiener space as X. We survey recent results for the case when X is a de Branges space H(E). Numerous answers mainly depend on the behaviour of the phase function of the generating function E.This survey article consists of two parts. The first (Section 1) is devoted to the Beurling-Malliavin Multiplier Theorem (the BM-theorem):where Ω is a non-negative Lipschitz function on R, then for any σ > 0 there exists a non-zero function f ∈ L 2 (R) such that its Fourier transform vanishes on R \ [−σ, σ] and |f | ≤ e −Ω .The term "multiplier" is explained in Subsection 1.4.2. This deep and difficult result has important connections with problems of harmonic and complex analysis. Published in 1962 (see [10]) it remains topical even nowadays. The second part (Sections 2 and 3) of this article describes recent analogs of the BM-theorem related to the de Branges spaces of entire functions.

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Approximation of Subharmonic Functions

Cited by 47 publications

References 2 publications

Entire functions of sine type and their applications

Entire functions of sine type and their applications

Asymptotic values of meromorphic functions of finite order

The Beurling–Malliavin Multiplier Theorem and Its Analogs for the de Branges Spaces

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