On Uniform Complex Approximation by Lacunary Polynomials

Martirosyan, V. A.

doi:10.1070/sm1984v048n02abeh002685

Cited by 6 publications

(3 citation statements)

References 6 publications

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“…with constantsM, γ > 0, a set S, and a sequence {z m } as in the lemma (without loss of generality we may suppose that the sets z m · S are pairwise disjoint). This case was considered in Lemma 8 of [10] for sequences Q having density d(Q) < 1.…”

Section: A Liouville-type Resultsmentioning

confidence: 98%

A Liouville-type result for lacunary power series and converse results for universal holomorphic functions

Martirosian¹,

Müller²

2006

Analysis

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A Liouville-type result is proved for holomorphic functions having lacunary power series with restricted growth for a subsequence of the partial sums. As an application, converse results concerning existence of universal holomorphic functions are given. A Liouville-type resultLiouville's theorem states that an entire function ϕ reduces to a polynomial of degree at most [γ ] (where [γ ] is the integral part of the real number γ ) if |ϕ(z)| ≤ M|z| γ for some M > 0. The conclusion of Liouville's theorem holds more generally if ∞ ν=0 ϕ ν z ν is a (formal) power series with partial sums s n (z) such that for some subsequence {k m } of the positive integers and some M > 0 we have |s k m (z)| ≤ M|z| γ for z ∈ E m , where {E m } is an exhaustion of the complex plane. Let in the sequel Q always be a subset of the nonnegative integers. We shall prove a Liouville-type result for lacunary power series ϕ(z) = ν∈Q ϕ ν z ν , (1.1) in which we replace the sets E m by appropriate smaller subsets of the plane which depend on Q. We first recall some notions about entire functions. If Q = {q j : j = 1, 2, . . . } then f(w) := ∞ j=1 1 − w 2 q 2 j AMS 2000 subject classifications: 30B10

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Section: A Liouville-type Resultsmentioning

confidence: 98%

A Liouville-type result for lacunary power series and converse results for universal holomorphic functions

Martirosian¹,

Müller²

2006

Analysis

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“…The sector in Martirosian's result may, however, be curvilinear. We remark further that in [13] an example of a set Λ with d(Λ) = 1 and a compact set K with 0 ∈ ∂ K such that P Λ (K ) = A(K ) is given.…”

Section: Consequences and Interpretationsmentioning

confidence: 98%

A lacunary version of Mergelian’s approximation theorem

Gharibyan

Luh

Müller

2010

Journal of Approximation Theory

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Let K be a compact plane set having connected complement. Then Mergelian's theorem states that the linear span of the monomials z n , i.e. the polynomials, are dense in the Banach space A(K ) of all functions continuous on K and holomorphic in the interior of K endowed with the sup-norm. We consider the question under which conditions the linear span of z n , with n running through a sequence of nonnegative integers having upper density one, is dense in A(K ) or appropriate subspaces.

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“…For the latter property, there is an extensive amount of sufficient conditions in the literature (see e.g. [DK77], [Mar84], [LMM98a], [LMM98b], [LMM02]).…”

Section: Corollarymentioning

confidence: 99%

Integral transforms and probability distributions involving a generalized hypergeometric function

Khan

Usman

Aman

et al. 2021

Georgian Mathematical Journal

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Various extensions of the beta function together with their associated extended hypergeometric and confluent hypergeometric functions have been introduced and investigated. In this paper, using the very recently contrived extended beta function, we aim to introduce an extension F v p , q ; λ ; σ , τ u {{}_{u}F_{v}^{p,q;\lambda;\sigma,\tau}} of the generalized hypergeometric function F v u {{}_{u}F_{v}} and investigate certain classes of transforms and several identities of a generalized probability distribution involving this extension. In fact, we present some interesting formulas of Jacobi, Gegenbauer, pathway, Laplace, and Legendre transforms of this extension multiplied by a polynomial. We also introduce a generalized probability distribution to investigate its several related properties. Further, we consider some special cases of our main results with an argument about the derived process of a known result.

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On Uniform Complex Approximation by Lacunary Polynomials

Cited by 6 publications

References 6 publications

A Liouville-type result for lacunary power series and converse results for universal holomorphic functions

A Liouville-type result for lacunary power series and converse results for universal holomorphic functions

A lacunary version of Mergelian’s approximation theorem

Integral transforms and probability distributions involving a generalized hypergeometric function

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