Strongly annular functions with small Taylor coefficients

Bonar, Daniel D.; Carroll, F. W.; Piranian, George

doi:10.1007/bf01215130

Cited by 5 publications

(7 citation statements)

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“…(1) D. D. Bonar, F. Carroll, and G. Piranian [13,Theorem 1] proved that there exist positive numbers A 1 and A 2 such that for all k and n the coefficients b n 1/2 (k) satisfy the inequality…”

Section: 31mentioning

confidence: 99%

“…That is why Bonar asked in [11,Question 6.9] whether every strongly annular function is a sum of a bounded function and the sum of a lacunary Taylor series. In 1977 Bonar, Carroll, and Piranian [13] constructed a strongly annular function…”

Section: 31mentioning

confidence: 99%

“…Strongly annular functions. Using the ideas from [13] and [12] and estimates on the asymptotics of b n λ (k) obtained in our paper, we construct strongly annular functions f such that (a) f belongs to q \ p for any given 2 ≤ p < q or (b) f belongs to 2 ϕ \ 2 where 2 ϕ is the set of sequences (a n ) such that…”

Section: 31mentioning

confidence: 99%

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On the Fourier coefficients of powers of a Blaschke factor and strongly annular fonctions

Borichev,

Fouchet,

Zarouf

2021

Preprint

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We compute asymptotic formulas for the k th Fourier coefficients of b n λ , where b λ (z) = z−λ 1−λz is the Blaschke factor associated to λ ∈ D, k ∈ [0, ∞) and n is a large integer. We distinguish several regions of different asymptotic behavior of those coefficients in terms of k and n.Airy-type behavior is happening near the k-transition points n(1−λ)/(1+λ) and n(1+λ)/(1−λ). The asymptotic formulas for the k th Fourier coefficients of b n λ are derived using standard tools of asymptotic analysis of Laplace-type integrals. More precisely, the integral defining the k th Fourier coefficient of b n λ is perfectly suited for an application of the method of stationary phase when k ∈ (n(1 − λ)/(1 + λ), n(1 + λ)/(1 − λ)) and requires the use of the method of the steepest descent when k / ∈ [n(1−λ)/(1+λ), n(1+λ)/(1−λ)]. Uniform versions of those standard methods are required when k approaches one of the boundaries n(1 − λ)/(1 + λ), n(1 + λ)/(1 − λ). As an application, we construct strongly annular functions with Taylor coefficients satisfying sharp summation properties.

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Section: 31mentioning

confidence: 99%

Section: 31mentioning

confidence: 99%

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On the Fourier coefficients of powers of a Blaschke factor and strongly annular fonctions

Borichev,

Fouchet,

Zarouf

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…R. W. Howell [8] has shown that under the topology of locally uniform convergence, a residual subset of the space of functions 2 e n z" (e n = ± 1) consists of strongly annular functions. Also, Bonar, Carroll, and Piranian [2] have proved that if the sequence {k n } increases rapidly enough, then the function n = l I I ' is strongly annular, although the sequence of its Taylor coefficients at the origin tends to 0.…”

Section: Strongly Annular Functionsmentioning

confidence: 99%

“…In Section 6, we saw that on the circle \z\ = r n = l-\/h n , the («+l)st term of the series in (1) has modulus less than 3. We shall now examine the modulus of the same term on the slightly larger circle \z\ = r n + \/h,, 2 . By elementary computations, By Lemma 1, we may assume that h n+l > 3h n 2 , so that on the circle \z\ = r n + l/h n 2 the (« + l)st term in (1) has modulus greater than exp /?".…”

Section: One-term Dominancementioning

confidence: 99%