1 results
Constructions of some families of smooth Cauchy transforms
- Part of
-
- Journal:
- Canadian Journal of Mathematics / Volume 76 / Issue 1 / February 2024
- Published online by Cambridge University Press:
- 26 January 2023, pp. 319-344
- Print publication:
- February 2024
-
- Article
-
- You have access
- Open access
- HTML
- Export citation
-
For a given Beurling–Carleson subset E of the unit circle
$\mathbb {T}$ which has positive Lebesgue measure, we give explicit formulas for measurable functions supported on E such that their Cauchy transforms have smooth extensions from 
$\mathbb {D}$ to 
$\mathbb {T}$. The existence of such functions has been previously established by Khrushchev in 1978, in non-constructive ways by the use of duality arguments. We construct several families of such smooth Cauchy transforms and apply them in a few related problems in analysis: an irreducibility problem for the shift operator, an inner factor permanence problem. Our development leads to a self-contained duality proof of the density of smooth functions in a very large class of de Branges–Rovnyak spaces. This extends the previously known approximation results.
