Model subspaces techniques to study Fourier expansions in L^2 spaces associated to singular measures

Jorge Antezana; Maria Guadalupe Garcia

arxiv: 1907.08876 · v1 · pith:OCF4GTX7new · submitted 2019-07-20 · 🧮 math.CA · math.FA

Model subspaces techniques to study Fourier expansions in L² spaces associated to singular measures

Jorge Antezana , Maria Guadalupe Garcia This is my paper

Pith reviewed 2026-05-24 18:21 UTC · model grok-4.3

classification 🧮 math.CA math.FA

keywords model subspacessingular measuresKaczmarz algorithmParseval framesHardy spaceFourier expansionsreproducing kernelsinner functions

0 comments

The pith

For singular measures on the circle, the Kaczmarz Parseval frame on monomials coincides with the boundary values of a frame from a model subspace of the Hardy space.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper connects Fourier series expansions in L squared spaces built from singular measures to the theory of model subspaces in the Hardy space. Because the monomials form an effective sequence, the Kaczmarz algorithm produces a Parseval frame for these spaces. The main result identifies this frame with the boundary values of the projected monomials in the model subspace defined by an inner function. The work further characterizes the measures that can reproduce kernels lying inside model subspaces, with a complete description when the kernel is itself a reproducing kernel for such a subspace.

Core claim

The Parseval frame associated via the Kaczmarz algorithm to the monomials in L2(T,μ) is identified with the boundary values of the frame P_φ(z^n) for the model subspace H(φ)= H² ⊖ φ H², where P_φ is the orthogonal projection onto H(φ). The set of measures μ which reproduce a kernel contained in a model subspace is characterized when the kernel is the reproducing kernel of a model subspace.

What carries the argument

The model subspace H(φ) = H² ⊖ φ H² together with the orthogonal projection P_φ applied to the monomials, whose boundary values identify the Kaczmarz Parseval frame.

If this is right

Fourier expansions in L2(T,μ) can be analyzed using the projection and inner-function structure of the model subspace.
Positive kernels in the Hardy space arise directly from the study of these expansions for singular μ.
When the kernel is a model-subspace reproducing kernel, the supporting measures μ are completely determined by the characterization.
The identification transfers frame properties from the model subspace setting to the L2(mu) Fourier expansions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The boundary-value identification suggests that completeness or basis properties of the frame can be read off from the inner function φ.
The kernel-reproduction characterization may restrict which singular measures arise as the support of kernels inside a fixed model subspace.

Load-bearing premise

The monomials form an effective sequence in L2(T,μ) for singular probability measures, so that the Kaczmarz algorithm yields a Parseval frame.

What would settle it

A concrete singular measure μ for which the Kaczmarz frame on the monomials fails to equal the boundary values of P_φ(z^n) would disprove the identification.

read the original abstract

Let $\mu$ be a probability measure on $\mathbb{T}$ that is singular with respect to the Haar measure. In this paper we study Fourier expansions in $L^2(\mathbb{T},\mu)$ using techniques from the theory of model subspaces of the Hardy space. Since the sequence of monomials $\{z^n\}_{n\in \mathbb{N}}$ is effective in $L^2(\mathbb{T},\mu)$, it has a Parseval frame associated via the Kaczmarz algorithm. Our first main goal is to identify the aforementioned frame with boundary values of the frame $P_\varphi(z^n)$ for the model subspace $\mathcal{H}(\varphi)= H^2 \ominus \varphi H^2$, where $P_\varphi$ is the orthogonal projection from the Hardy space $H^2$ onto $\mathcal{H}(\varphi)$. The study of Fourier expansions in $L^2(\mathbb{T},\mu)$ also leads to consider positive kernels in the Hardy space. Our second main goal is to study the set of measures $\mu$ which reproduce a kernel contained in a model subspace. We completely characterize this set when the kernel is the reproducing kernel of a model subspace, and we study the consequences of this characterization.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper identifies the Kaczmarz frame from monomials on singular μ with model-subspace projections and characterizes reproducing measures for kernels in H(φ), but rests on an unexamined density claim for non-negative powers.

read the letter

The core advance is the explicit identification of the Parseval frame produced by the Kaczmarz algorithm on {z^n} in L²(T, μ) with the boundary values of the projected frame P_φ(z^n) inside the model space H(φ). The second part gives a complete description of the measures μ whose associated kernels sit inside a model subspace, at least when the kernel is itself a reproducing kernel for such a space. Both results sit inside the intersection of model-subspace theory and harmonic analysis on singular measures, and the authors carry the identification through cleanly once the setup is granted. The writing is direct and the goals are stated without unnecessary decoration. The main soft spot is the opening premise that {z^n : n ≥ 0} is effective in L²(T, μ) for every singular probability measure μ. The abstract treats this as given rather than derived, yet the stress-test observation is fair: for some singular measures the closed span of the non-negative powers need not be all of L²(μ), so the Kaczmarz construction would not automatically yield a Parseval frame in the stated generality. If the paper supplies a reference or a short argument for this density, the concern disappears; if not, the identification holds only on the subclass of μ where the premise is true. The kernel-characterization part looks less dependent on that assumption and stands on its own. This is a specialized note aimed at people already working with model subspaces or with frames on singular measures. It is not broad enough to interest a general harmonic-analysis audience, but the concrete identification and the characterization are the sort of incremental, usable results that a referee in the area can check and improve. I would send it to review rather than desk-reject.

Referee Report

1 major / 2 minor

Summary. The paper studies Fourier expansions in L²(𝕋, μ) for singular probability measures μ using model subspaces of the Hardy space. It claims to identify the Parseval frame associated to the monomials {z^n} via the Kaczmarz algorithm with the boundary values of the frame {P_φ(z^n)} in the model subspace ℋ(φ) = H² ⊖ φ H². It further characterizes the set of measures μ that reproduce a kernel contained in a model subspace, providing a complete characterization when the kernel is the reproducing kernel of a model subspace.

Significance. If the central identification holds, the work establishes a concrete link between the Kaczmarz algorithm in L² spaces associated to singular measures and the theory of model subspaces, which could facilitate the study of frames and expansions in these settings. The characterization of reproducing measures for kernels in model subspaces contributes to the analysis of positive kernels in Hardy spaces. The manuscript employs standard tools of the field (model subspaces, orthogonal projections, reproducing kernels) in a direct manner.

major comments (1)

[Abstract] Abstract and opening paragraphs: the effectiveness of the sequence {z^n}_{n∈ℕ} in L²(𝕋, μ) for arbitrary singular μ is asserted as a premise (the abstract states “Since the sequence of monomials {z^n} is effective…”). This premise is load-bearing for the first main result, because the Kaczmarz algorithm produces a Parseval frame only when the sequence is effective. For some singular measures the non-negative powers alone are not dense in L²(μ), so the claimed identification with boundary values of P_φ(z^n) does not hold in the stated generality without additional justification or restrictions on μ.

minor comments (2)

Notation for the model subspace ℋ(φ) and the projection P_φ should be introduced with a brief reminder of the standard definition before the main statements.
The second main goal (characterization of reproducing measures) would benefit from an explicit statement of the kernel in question before the characterization theorem.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying this important qualification needed in the statement of the results. We address the major comment below.

read point-by-point responses

Referee: [Abstract] Abstract and opening paragraphs: the effectiveness of the sequence {z^n}_{n∈ℕ} in L²(𝕋, μ) for arbitrary singular μ is asserted as a premise (the abstract states “Since the sequence of monomials {z^n} is effective…”). This premise is load-bearing for the first main result, because the Kaczmarz algorithm produces a Parseval frame only when the sequence is effective. For some singular measures the non-negative powers alone are not dense in L²(μ), so the claimed identification with boundary values of P_φ(z^n) does not hold in the stated generality without additional justification or restrictions on μ.

Authors: We agree that the effectiveness of {z^n} does not hold for every singular measure μ; the manuscript asserts the premise without qualification, which is incorrect in full generality. We will revise the abstract and opening paragraphs to state explicitly that we consider singular probability measures μ for which the sequence {z^n}_{n∈ℕ} is effective in L²(𝕋, μ). Under this restriction the Kaczmarz algorithm produces the Parseval frame, and the identification with the boundary values of {P_φ(z^n)} in ℋ(φ) holds as claimed. The second main result on reproducing measures for kernels in model subspaces is independent of this assumption and requires no change. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper states the effectiveness of the monomials {z^n} in L²(T,μ) for singular μ as a premise ('Since the sequence of monomials {z^n} is effective...'), then identifies the resulting Kaczmarz Parseval frame with boundary values of P_φ(z^n) in the model subspace H(φ). No equations, self-citations, fitted parameters, or ansatzes are shown that reduce this identification to a tautology or to the input premise by construction. The derivation is self-contained once the stated premise is accepted, with the model-subspace identification providing independent content from the theory of H(φ).

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are stated or derivable from the provided text.

pith-pipeline@v0.9.0 · 5751 in / 1047 out tokens · 19378 ms · 2026-05-24T18:21:44.335601+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Since the sequence of monomials {z^n} is effective in L²(T,μ), it has a Parseval frame associated via the Kaczmarz algorithm... identify... with boundary values of the frame P_φ(z^n) for the model subspace H(φ)
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

ϕ(z) = (ψ(z)−1)/(ψ(z)+1) ... ϕ is an inner function... μ_α Clark measures
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 3.1... gn(ζ) = lim P_φ(z^n)(w)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

30 extracted references · 30 canonical work pages · 1 internal anchor

[1]

A. B. Aleksandrov, Isometric embeddings of coinvariant subspaces of the Shift oper- ator, J. Math. Sci. 92 (1998) 3543–3549

work page 1998
[2]

A. B. Aleksandrov, On the existence of angular boundary v alues of pseudocontinuable functions, Zap. Nauchn. Semin. POMI 222 (1995) 5-17; transl ation in J. Math. Sci. 87 (1997) 3781–3787

work page 1995
[3]

J. Cima, A. Matheson, W. Ross, The Cauchy transform, Math ematical Surveys and Monographs Volumen 125, American Mathematical Society, Pr ovidence, RI, 2006, pp. 179–208

work page 2006
[4]

D. N. Clark, One dimensional perturbations of restricte d shifts, J. Anal. Math. 25 (1972) 169–191

work page 1972
[5]

Dai, Spectra of Cantor measures, Math

X.-R. Dai, Spectra of Cantor measures, Math. Ann. 366 (20 16), no. 3-4, 1621–1647

work page
[6]

Dai, X.-G

X.-R. Dai, X.-G. He, C.-K. Lai, Spectral property of Cant or measures with consecu- tive digits. Adv. Math. 242 (2013), 187–208

work page 2013
[7]

D. E. Dutkay, C.-K. Lai, Uniformity of measures with Four ier frames. Adv. Math. 252 (2014), 684–707

work page 2014
[8]

D. E. Dutkay, C.-K. Lai, Y.W ang, Fourier bases and Fourie r frames on self-aﬃne measures. Preprint (2016), arXiv:1602.04750

work page internal anchor Pith review Pith/arXiv arXiv 2016
[9]

Dutkay, D

D.E. Dutkay, D. Han, Q. Sun, On the spectra of a Cantor meas ure, Adv. Math. 221 (2009) 251–276

work page 2009
[10]

D. E. Dutkay, Palle E. T. Jorgensen, Aﬃne fractals as bou ndaries and their harmonic analysis, Proc. Amer. Math. Soc. 139 (2011) 3291–3305

work page 2011
[11]

R.Duﬃn, A.Schaeﬀer, A class of non-harmonic Fourier se ries, Trans. Amer. Math. Soc. 72 (1952) 341–366

work page 1952
[12]

Haller, R

R. Haller, R. Szwarc, Kaczmarz algorithm in Hilbert spa ce, Studia Math. 169 (2005), no. 2, 123–132

work page 2005
[13]

He, C.-K

X.-G. He, C.-K. Lai, K.-S. Lau, Exponential spectra in L2(µ). Appl. Comput. Har- mon. Anal. 34 (2013), no. 3, 327–338

work page 2013
[14]

J. E. Herr, E. S. W eber, Fourier series for singular meas ures, Axioms 6(2) (2017) 7

work page 2017
[15]

J. E. Herr, P. E. T. Jorgensen, E. S. W eber, Positive matr ices in the Hardy space with prescribed boundary representations via the Kaczmarz algothim, J. d’Analyse Mathematique, to appear

work page
[16]

Hruschev, N.K

S.V. Hruschev, N.K. Nikolskii, B.S. Pavlov, Unconditi onal bases of exponentials and of reproducing kernels. Complex analysis and spectral theo ry (Leningrad, 1979/1980) pp. 214–335 (in Russian). English translation in Lecture No tes in Math., 864, Springer, Berlin-New York, 1981

work page 1979
[17]

P. E. T. Jorgensen, S. Pedersen, Dense analytic subspac es in fractal L2-spaces. J. Anal.Math. 75 (1998), 185–228

work page 1998
[18]

Kaczmarz, Approximate solution of systems of linear equations, Bulletin Inter- national de l’Académie Plonaise des Sciences et des Lettres

S. Kaczmarz, Approximate solution of systems of linear equations, Bulletin Inter- national de l’Académie Plonaise des Sciences et des Lettres . Classe des Sciences Mathématiques et Naturelles. Série A. Sciences Mathématiq ues 35 (1937) 355-357 (in German). English translate in Internat. J. Control 57 (1 993) 1269–1271

work page 1937
[19]

Kwapie ´n, J Mycielski, On the Kaczmarz algorithm of approximation i n inﬁnitedi- mensional spaces, Studia Math

S. Kwapie ´n, J Mycielski, On the Kaczmarz algorithm of approximation i n inﬁnitedi- mensional spaces, Studia Math. 148 (2001) no. 1. 75–86

work page 2001
[20]

I. Laba, Y. W ang, Some properties of spectral measures. Appl. Comput. Harmon. Anal.20 (2006), no. 1, 149–157

work page 2006
[21]

Lai, On Fourier frame of absolutely continuous me asures

C.-K. Lai, On Fourier frame of absolutely continuous me asures. J. Funct. Anal. 261 (2011),no. 10, 2877–2889

work page 2011
[22]

A. W. Marcus, D. A. Spielman, N. Srivastava, Interlacin g families II: Mixed charac- teristic polynomials and the Kadison-Singer problem. Ann. of Math. (2) 182 (2015), no. 1, 327–350

work page 2015
[23]

Nikolski, Operators, functions, and systems: an eas y reading

N. Nikolski, Operators, functions, and systems: an eas y reading. Vol. 1. Hardy, Han- kel, and Toeplitz. Mathematical Surveys and Monographs, 92 . American Mathemat- ical Society, Providence, RI, 2002

work page 2002
[24]

Nikolski, Operators, functions, and systems: an eas y reading

N. Nikolski, Operators, functions, and systems: an eas y reading. Vol. 2. Model op- erators and systems. Mathematical Surveys and Monographs, 93. American Mathe- matical Society, Providence, RI, 2002. 19

work page 2002
[25]

Nitzan, A

S. Nitzan, A. Olevskii, A. Ulanovskii, Exponential fra mes on unbounded sets. Proc. Amer. Math. Soc. 144 (2016), no. 1, 109–118

work page 2016
[26]

A. G. Poltoratkii, The boundary behavior of pseudocont inuable functions, Algebra i Analiz 5 (1993) no. 2, 189-210 (in Russian). English transl ation in St.Petersburg Math. J. 5 (1994) 389–406

work page 1993
[27]

Rudin, Real and complex analysis, Third edition, McG raw-Hil, New York, 1987

W. Rudin, Real and complex analysis, Third edition, McG raw-Hil, New York, 1987

work page 1987
[28]

Saksman, An elementary introduction to Clark measur es

E. Saksman, An elementary introduction to Clark measur es. Topics in complex anal- ysis and operator theory, Univ. Málaga, Málaga (2007) 85–13 6

work page 2007
[29]

Sarason, Sub-Hardy Hilbert spaces in the unit disk, U niversity of Arkansas Lecture Notes in the Mathematical Sciences, 10

D. Sarason, Sub-Hardy Hilbert spaces in the unit disk, U niversity of Arkansas Lecture Notes in the Mathematical Sciences, 10. John Wiley & Sons, In c., New York, 1994

work page 1994
[30]

Alberto P. Calderón

K. Stephenson, Isometries in the Nevanlinna class, Ind iana Univ. Math. J. 26 (1977) 307-324. Departamento de Matemática, Universidad Nacional de La Pla ta and, Instituto Ar- gentino de Matemática "Alberto P. Calderón" (IAM-CONICET) , Buenos Aires, Ar- gentina E-mail address : antezana@mate.unlp.edu.ar Departamento de Matemática, Universidad Nacional de L...

work page 1977

[1] [1]

A. B. Aleksandrov, Isometric embeddings of coinvariant subspaces of the Shift oper- ator, J. Math. Sci. 92 (1998) 3543–3549

work page 1998

[2] [2]

A. B. Aleksandrov, On the existence of angular boundary v alues of pseudocontinuable functions, Zap. Nauchn. Semin. POMI 222 (1995) 5-17; transl ation in J. Math. Sci. 87 (1997) 3781–3787

work page 1995

[3] [3]

J. Cima, A. Matheson, W. Ross, The Cauchy transform, Math ematical Surveys and Monographs Volumen 125, American Mathematical Society, Pr ovidence, RI, 2006, pp. 179–208

work page 2006

[4] [4]

D. N. Clark, One dimensional perturbations of restricte d shifts, J. Anal. Math. 25 (1972) 169–191

work page 1972

[5] [5]

Dai, Spectra of Cantor measures, Math

X.-R. Dai, Spectra of Cantor measures, Math. Ann. 366 (20 16), no. 3-4, 1621–1647

work page

[6] [6]

Dai, X.-G

X.-R. Dai, X.-G. He, C.-K. Lai, Spectral property of Cant or measures with consecu- tive digits. Adv. Math. 242 (2013), 187–208

work page 2013

[7] [7]

D. E. Dutkay, C.-K. Lai, Uniformity of measures with Four ier frames. Adv. Math. 252 (2014), 684–707

work page 2014

[8] [8]

D. E. Dutkay, C.-K. Lai, Y.W ang, Fourier bases and Fourie r frames on self-aﬃne measures. Preprint (2016), arXiv:1602.04750

work page internal anchor Pith review Pith/arXiv arXiv 2016

[9] [9]

Dutkay, D

D.E. Dutkay, D. Han, Q. Sun, On the spectra of a Cantor meas ure, Adv. Math. 221 (2009) 251–276

work page 2009

[10] [10]

D. E. Dutkay, Palle E. T. Jorgensen, Aﬃne fractals as bou ndaries and their harmonic analysis, Proc. Amer. Math. Soc. 139 (2011) 3291–3305

work page 2011

[11] [11]

R.Duﬃn, A.Schaeﬀer, A class of non-harmonic Fourier se ries, Trans. Amer. Math. Soc. 72 (1952) 341–366

work page 1952

[12] [12]

Haller, R

R. Haller, R. Szwarc, Kaczmarz algorithm in Hilbert spa ce, Studia Math. 169 (2005), no. 2, 123–132

work page 2005

[13] [13]

He, C.-K

X.-G. He, C.-K. Lai, K.-S. Lau, Exponential spectra in L2(µ). Appl. Comput. Har- mon. Anal. 34 (2013), no. 3, 327–338

work page 2013

[14] [14]

J. E. Herr, E. S. W eber, Fourier series for singular meas ures, Axioms 6(2) (2017) 7

work page 2017

[15] [15]

J. E. Herr, P. E. T. Jorgensen, E. S. W eber, Positive matr ices in the Hardy space with prescribed boundary representations via the Kaczmarz algothim, J. d’Analyse Mathematique, to appear

work page

[16] [16]

Hruschev, N.K

S.V. Hruschev, N.K. Nikolskii, B.S. Pavlov, Unconditi onal bases of exponentials and of reproducing kernels. Complex analysis and spectral theo ry (Leningrad, 1979/1980) pp. 214–335 (in Russian). English translation in Lecture No tes in Math., 864, Springer, Berlin-New York, 1981

work page 1979

[17] [17]

P. E. T. Jorgensen, S. Pedersen, Dense analytic subspac es in fractal L2-spaces. J. Anal.Math. 75 (1998), 185–228

work page 1998

[18] [18]

Kaczmarz, Approximate solution of systems of linear equations, Bulletin Inter- national de l’Académie Plonaise des Sciences et des Lettres

S. Kaczmarz, Approximate solution of systems of linear equations, Bulletin Inter- national de l’Académie Plonaise des Sciences et des Lettres . Classe des Sciences Mathématiques et Naturelles. Série A. Sciences Mathématiq ues 35 (1937) 355-357 (in German). English translate in Internat. J. Control 57 (1 993) 1269–1271

work page 1937

[19] [19]

Kwapie ´n, J Mycielski, On the Kaczmarz algorithm of approximation i n inﬁnitedi- mensional spaces, Studia Math

S. Kwapie ´n, J Mycielski, On the Kaczmarz algorithm of approximation i n inﬁnitedi- mensional spaces, Studia Math. 148 (2001) no. 1. 75–86

work page 2001

[20] [20]

I. Laba, Y. W ang, Some properties of spectral measures. Appl. Comput. Harmon. Anal.20 (2006), no. 1, 149–157

work page 2006

[21] [21]

Lai, On Fourier frame of absolutely continuous me asures

C.-K. Lai, On Fourier frame of absolutely continuous me asures. J. Funct. Anal. 261 (2011),no. 10, 2877–2889

work page 2011

[22] [22]

A. W. Marcus, D. A. Spielman, N. Srivastava, Interlacin g families II: Mixed charac- teristic polynomials and the Kadison-Singer problem. Ann. of Math. (2) 182 (2015), no. 1, 327–350

work page 2015

[23] [23]

Nikolski, Operators, functions, and systems: an eas y reading

N. Nikolski, Operators, functions, and systems: an eas y reading. Vol. 1. Hardy, Han- kel, and Toeplitz. Mathematical Surveys and Monographs, 92 . American Mathemat- ical Society, Providence, RI, 2002

work page 2002

[24] [24]

Nikolski, Operators, functions, and systems: an eas y reading

N. Nikolski, Operators, functions, and systems: an eas y reading. Vol. 2. Model op- erators and systems. Mathematical Surveys and Monographs, 93. American Mathe- matical Society, Providence, RI, 2002. 19

work page 2002

[25] [25]

Nitzan, A

S. Nitzan, A. Olevskii, A. Ulanovskii, Exponential fra mes on unbounded sets. Proc. Amer. Math. Soc. 144 (2016), no. 1, 109–118

work page 2016

[26] [26]

A. G. Poltoratkii, The boundary behavior of pseudocont inuable functions, Algebra i Analiz 5 (1993) no. 2, 189-210 (in Russian). English transl ation in St.Petersburg Math. J. 5 (1994) 389–406

work page 1993

[27] [27]

Rudin, Real and complex analysis, Third edition, McG raw-Hil, New York, 1987

W. Rudin, Real and complex analysis, Third edition, McG raw-Hil, New York, 1987

work page 1987

[28] [28]

Saksman, An elementary introduction to Clark measur es

E. Saksman, An elementary introduction to Clark measur es. Topics in complex anal- ysis and operator theory, Univ. Málaga, Málaga (2007) 85–13 6

work page 2007

[29] [29]

Sarason, Sub-Hardy Hilbert spaces in the unit disk, U niversity of Arkansas Lecture Notes in the Mathematical Sciences, 10

D. Sarason, Sub-Hardy Hilbert spaces in the unit disk, U niversity of Arkansas Lecture Notes in the Mathematical Sciences, 10. John Wiley & Sons, In c., New York, 1994

work page 1994

[30] [30]

Alberto P. Calderón

K. Stephenson, Isometries in the Nevanlinna class, Ind iana Univ. Math. J. 26 (1977) 307-324. Departamento de Matemática, Universidad Nacional de La Pla ta and, Instituto Ar- gentino de Matemática "Alberto P. Calderón" (IAM-CONICET) , Buenos Aires, Ar- gentina E-mail address : antezana@mate.unlp.edu.ar Departamento de Matemática, Universidad Nacional de L...

work page 1977