Inner functions and zero sets for ℓ^{𝑝}_{𝐴}
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Bibliographic record
Abstract
In this paper we characterize the zero sets of functions from <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script l Subscript upper A Superscript p"> <mml:semantics> <mml:msubsup> <mml:mi> ℓ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>A</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>p</mml:mi> </mml:mrow> </mml:msubsup> <mml:annotation encoding="application/x-tex">\ell ^{p}_{A}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (the analytic functions on the open unit disk <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper D"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">D</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {D}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> whose Taylor coefficients form an <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script l Superscript p"> <mml:semantics> <mml:msup> <mml:mi> ℓ </mml:mi> <mml:mi>p</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">\ell ^p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> sequence) by developing a concept of an “inner function” modeled by Beurling’s discussion of the Hilbert space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script l Subscript upper A Superscript 2"> <mml:semantics> <mml:msubsup> <mml:mi> ℓ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>A</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> </mml:mrow> </mml:msubsup> <mml:annotation encoding="application/x-tex">\ell ^{2}_{A}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , the classical Hardy space. The zero set criterion is used to construct families of zero sets which are not covered by classical results. In particular, we give an alternative proof of a result of Vinogradov [Dokl. Akad. Nauk SSSR 160 (1965), pp. 263–266] which says that when <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p greater-than 2"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>></mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">p > 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , there are zero sets for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script l Subscript upper A Superscript p"> <mml:semantics> <mml:msubsup> <mml:mi> ℓ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>A</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>p</mml:mi> </mml:mrow> </mml:msubsup> <mml:annotation encoding="application/x-tex">\ell ^{p}_{A}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which are not Blaschke sequences.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.005 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it