On the coefficient formula for de Branges-Rovnyak norms

@misc{ransford2026coefficient,
  title = {{On the coefficient formula for de Branges-Rovnyak norms}},
  author = {Thomas Ransford},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2605.30114},
  eprint = {2605.30114},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.30114},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - On the coefficient formula for de Branges-Rovnyak norms
AU  - Thomas Ransford
PY  - 2026
DA  - 2026/09/04
PB  - arXiv
DO  - 10.48550/arxiv.2605.30114
UR  - https://arxiv.org/abs/2605.30114
AB  - Let ℋ(b) be the de Branges-Rovnyak space associated to a non-extreme point b of the unit ball of H^∞, and let ϕ=b/a, where a is the Pythagorean mate of b. It is known that, if f is a function holomorphic on a neighbourhood of the closed unit disk, then it belongs to ℋ(b), and its norm in ℋ(b) can be expressed in terms of the Taylor coefficients of f and ϕ via the formula |f|_ℋ(b)^2=∑_m≥0|f̂(m)|^2 +∑_m≥0|∑_n≥0ϕ̂(n)̅f̂(m+n)|^2. However, the formula can break down for some other f∈ℋ(b). In this article we extend the scope of the formula to all f∈ H^2 for which the right-hand side is finite, provided that either ϕ∈ H^2 or ϕ is rational. If merely ϕ∈ H^p for some p∈(0,2], then the formula still holds provided that, in addition, ∑_m≥0m^2/p−1|f̂(m)|^2<∞. We also establish a limit-form of the formula that is valid for all non-extreme b and all f∈ℋ(b).
ER  -