The study of model spaces, the closed invariant subspaces of the backward shift operator, is a vast area of research with connections to complex analysis, operator theory and functional analysis. This self-contained text is the ideal introduction for newcomers to the field. It sets out the basic ideas and quickly takes the reader through the history of the subject before ending up at the frontier of mathematical analysis. Open questions point to potential areas of future research, offering plenty of inspiration to graduate students wishing to advance further.

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Introduction to Model Spaces and their Operators

An H(b) space is defined as a collection of analytic functions which are in the image of an operator. The theory of H(b) spaces bridges two classical subjects: complex analysis and operator theory, which makes it both appealing and demanding. The first volume of this comprehensive treatment is devoted to the preliminary subjects required to understand the foundation of H(b) spaces, such as Hardy spaces, Fourier analysis, integral representation theorems, Carleson measures, Toeplitz and Hankel operators, various types of shift operators, and Clark measures. The second volume focuses on the central theory. Both books are accessible to graduate students as well as researchers: each volume contains numerous exercises and hints, and figures are included throughout to illustrate the theory. Together, these two volumes provide everything the reader needs to understand and appreciate this beautiful branch of mathematics.

The Theory of H(b) Spaces

The (Norlund) logarithmic means of the Fourier series of the integrable function f is:

$$ \begin{array}{*{20}c} {{\frac{1} {{l_{n} }}{\sum\limits_{k = 1}^{n - 1} {\frac{{S_{k} {\left( f \right)}}} {{n - k}}} },}} & {{{\text{where}}\;l_{n} : = {\sum\limits_{k = 1}^{n - 1} {\frac{1} {k}} }.}} \\ \end{array} $$

In this paper we discuss some convergence and divergence properties of this logarithmic means of the Walsh–Fourier series of functions in the uniform, and in the L1 Lebesgue norm. Among others, as an application of our divergence results we give a negative answer to a question of Moricz concerning the convergence of logarithmic means in norm.

/pdf/uniform-and-l-convergence-of-logarithmic-means-of-walsh-2lai8pntt2.pdf

Uniform and L–convergence of Logarithmic Means of Walsh–Fourier Series

In this paper we prove and discuss some new $\left( H_{p},weak-L_{p}\right) $ type inequalities of maximal operators of Vilenkin–Norlund means with monotone coefficients. We also apply these results to prove a.e. convergence of such Vilenkin–Norlund means. It is also proved that these results are the best possible in a special sense. As applications, both some well-known and new results are pointed out.

Maximal Operators of Vilenkin–Nörlund Means

Exposed points and extremal problems in H1

/pdf/quasi-tauberian-theorems-applied-to-the-summability-of-104q6hiirw.pdf

Quasi-Tauberian theorems, applied to the summability of Fourier series by Riesz's logarithmic means

Some uniqueness theorems for $H^{p}(U^{n})$ functions

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EXTREME POINTS AND OUTER FUNCTIONS IN {H^1}({U^n})

/pdf/remarks-on-extremum-problems-in-h-1-17eqp2b49u.pdf

Kôzô Yabuta

Papers

Quasi-Tauberian theorems, applied to the summability of Fourier series by Riesz's logarithmic means

Some uniqueness theorems for $H^{p}(U^{n})$ functions

EXTREME POINTS AND OUTER FUNCTIONS IN {H^1}({U^n})

Remarks on extremum problems in {h^1}

Funktionen mit nichtnegativem Realteil in abstrakten Hardyalgebren