On the Interpolation of Bounded Sequences by Bounded Functions

Earl, J. P.

doi:10.1112/jlms/2.part_3.544

Cited by 66 publications

(37 citation statements)

References 3 publications

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“…Further constructive and most interesting proofs for different cases were given by Earl [15], Jones (see [22] or Khavin's Appendix I of [26]), and Schuster and Seip (see [32] or § 6.2 of [13]). In view of these results, a Blaschke product whose sequence of zeros is uniformly separated is called an interpolating Blaschke product.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Interpolating Blaschke Products: Stolz and Tangential Approach Regions

2007

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Abstract. A result of D.J. Newman asserts that a uniformly separated sequence contained in a Stolz angle is a finite union of exponential sequences. We extend this by obtaining several equivalent characterizations of such sequences.If the zeros of a Blaschke product B lie in a Stolz angle then B ∈ A p for all p < 3/2 and it has been recently shown that this result cannot be improved. Also, Newman's result can be used to prove that if B is an interpolating Blaschke product whose zeros lie in a Stolz angle thenIn this paper we prove that if the zeros of an interpolating Blaschke product lie in a disc internally tangent to the unit circle then B ∈ ∩ 0

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Interpolating Blaschke Products: Stolz and Tangential Approach Regions

2007

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“…First we remark that the interpolating function in (ii), as constructed in the proof below, will actually be a Blaschke product with thin zero sequence. (This fact, though not really needed for our purposes, can be verified along the lines of Earl [3], whose method we borrow.) Therefore, our result extends an earlier theorem from [9], where interpolating -and thin -Blaschke products solving NevanlinnaPick problems were shown to exist under more restrictive hypotheses.…”

Section: Letmentioning

confidence: 99%

“…The only difference between Earl's original version and ours, as stated above, is that in [3] the noneuclidean radii τ j of the disks involved are all equal to a single constant τ . However, Earl's proof works in our situation as well, once obvious adjustments are made.…”

Section: N)mentioning

confidence: 99%

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Free interpolation by nonvanishing analytic functions

Dyakonov

Nicolau

2007

Trans. Amer. Math. Soc.

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Abstract. We are concerned with interpolation problems in H ∞ where the values prescribed and the function to be found are both zero-free. More precisely, given a sequence {z j } in the unit disk, we ask whether there exists a nontrivial minorant {ε j } (i.e., a sequence of positive numbers bounded by 1 and tending to 0) such that every interpolation problem f (z j ) = a j has a nonvanishing solution f ∈ H ∞ whenever 1 ≥ |a j | ≥ ε j for all j. The sequences {z j } with this property are completely characterized. Namely, we identify them as "thin" sequences, a class that arose earlier in Wolff's work on free interpolation in H ∞ ∩ VMO.

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“…Earl's interpolation theorem [4] states that for (w n ) in the unit ball of ∞ , the interpolation problem f (z n ) = w n , f ∈ H ∞ , can be solved by a multiple M · B of an interpolating Blaschke product B, whenever (z n ) is an H ∞ -interpolating sequence. In general, this solution has norm strictly bigger than one, that is |M | > 1.…”

Section: Bhmentioning

confidence: 99%

Thin interpolating sequences in the disk

Mortini¹

2009

Arch. Math.

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Abstract. Using a kind of summability procedure we give, within this survey, a new proof of a result by Sundberg and Wolff on large perturbations of thin interpolating sequences and present a detailed proof of the statement of Dyakonov and Nicolau that asymptotic interpolation problems in H ∞ can be solved by thin Blaschke products. Mathematics Subject Classification (2000). Primary 30D50; Secondary 46J15.

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On the Interpolation of Bounded Sequences by Bounded Functions

Cited by 66 publications

References 3 publications

Interpolating Blaschke Products: Stolz and Tangential Approach Regions

Interpolating Blaschke Products: Stolz and Tangential Approach Regions

Free interpolation by nonvanishing analytic functions

Thin interpolating sequences in the disk

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