Pseudocontinuable Functions Research Articles

Interpolation and duality in spaces of pseudocontinuable functions

Given an inner function theta on the unit disk, let K^p_theta :=H^pcap theta {overline{z}}overline{H^p} be the associated star-invariant subspace of the Hardy space H^p. Also, we put K_{*theta }:=K^2_theta cap mathrm{BMO}. Assuming that B=B_{{mathcal {Z}}} is an interpolating Blaschke product with zeros {mathcal {Z}}={z_j}, we characterize, for a number of smoothness classes X, the sequences of values {mathcal {W}}={w_j} such that the interpolation problem fbig |_{{mathcal {Z}}}={mathcal {W}} has a solution f in K^2_Bcap X. Turning to the case of a general inner function theta , we further establish a non-duality relation between K^1_theta and K_{*theta }. Namely, we prove that the latter space is properly contained in the dual of the former, unless theta is a finite Blaschke product. From this we derive an amusing non-interpolation result for functions in K_{*B}, with B=B_{{mathcal {Z}}} as above.

An abstract approach to approximation in spaces of pseudocontinuable functions

We provide an abstract approach to approximation with a wide range of regularity classes X X in spaces of pseudocontinuable functions K ϑ p K^p_\vartheta , where ϑ \vartheta is an inner function and p > 0 p>0 . More precisely, we demonstrate a general principle, attributed to A. Aleksandrov, which asserts that if a certain linear manifold X X is dense in K ϑ q K^{q}_\vartheta for some q > 0 q>0 , then X X is in fact dense in K ϑ p K^p_{\vartheta } for all p > 0 p>0 . Moreover, for a rich class of Banach spaces of analytic functions X X , we describe the precise mechanism that determines when X X is dense in a certain space of pseudocontinuable functions. As a consequence, we obtain an extension of Aleksandrov’s density theorem to the class of analytic functions with uniformly convergent Taylor series.

Factorization and non-factorization theorems for pseudocontinuable functions

Let θ be an inner function on the unit disk, and let Kθp:=Hp∩θH0p‾ be the associated star-invariant subspace of the Hardy space Hp, with p≥1. While a nontrivial function f∈Kθp is never divisible by θ, it may have a factor h which is ‘‘not too different” from θ in the sense that the ratio h/θ (or just the anti-analytic part thereof) is smooth on the circle. In this case, f is shown to have additional integrability and/or smoothness properties, much in the spirit of the Hardy–Littlewood–Sobolev embedding theorem. The appropriate norm estimates are established, and their sharpness is discussed.

A pseudo-continuous neural network approach for developing water retention pedotransfer functions with limited data

Summary In this study, a new approach, which we called pseudo-continuous, to develop pedotransfer functions (PTFs) for predicting soil–water retention with an artificial neural network (ANN) was introduced and tested. It was compared with ANN PTFs developed using traditional point and parametric approaches. The pseudo-continuous approach has a continuous performance, i.e. it enables to predict water content at any desirable matric potential, but without the need to use a specific equation, such as the one by van Genuchten. Matric potential is considered as an input parameter, which enables to increase the number of samples in the training dataset with a factor equal to the number of matric potentials used to determine the water retention curve of the soil samples in the dataset. Generally, the pseudo-continuous functions performed slightly better than the point and parametric functions. The root mean square error (RMSE) of the pseudo-continuous functions when considering local data for training and testing, and with both bulk density and organic matter as extra input variables on top of sand, silt and clay content, was 0.027 m 3 m −3 compared to 0.029 m 3 m −3 for both the point and parametric PTF. The increased number of samples in the training phase and the selection of matric potential as an input variable enabling to predict water content at any desired matric potential are the most important reasons why pseudo-continuous functions would need more intention in the future. Uniformity in the training and test dataset was shown to be important in deriving PTFs. We finally recommend the use of pseudo-continuous PTFs for further improvement and development of PTFs, in particular when datasets are limited.

Symbols of truncated Toeplitz operators

We consider three topics connected with coinvariant subspaces of the backward shift operator in Hardy spaces H p : – properties of truncated Toeplitz operators; – Carleson-type embedding theorems for the coinvariant subspaces; – factorizations of pseudocontinuable functions from H 1 . These problems turn out to be closely connected and even, in a sense, equivalent. The new approach based on the factorizations allows us to answer a number of challenging questions about truncated Toeplitz operators posed by D. Sarason.

Existence of minmax points in discontinuous strategic games

In this note we prove the existence of minmax points for strategic form games where the sets of strategies are topological spaces and the payoff functions satisfy conditions weaker than continuity. The employed tools are the class of transfer weakly upper continuous functions and the class of weakly lower pseudocontinuous functions. An example shows that our result is of minimal character.

Pseudocontinuity is Necessary and Sufficient for Order-preserving Continuous Representations

Using some properties of pseudocontinuous functions - a recent generalization of continuous functions - and without invoking Debreu's Open Gap Theorem, we solve the following problem: given a pseudocontinuous function $v$, find a continuous function $u$ such that $u(x)>u(y)$ if and only if $v(x)>v(y)$. We show that this problem can be solved only for pseudocontinuous functions. Finally, we obtain a new proof on the existence of continuous numerical representations for continuous, transitive and total binary relations.

VARIATIONAL STABILITY OF SOCIAL NASH EQUILIBRIA

New results on the variational stability of social Nash equilibria are obtained using the class of sequentially pseudocontinuous functions. This class of functions strictly includes the class of sequentially continuous functions and finds a natural motivation in the framework of Choice and Economic Theory since it characterizes the continuity of the preference relations on first countable topological spaces. We investigate the connections with previous results and we show that it is not possible to improve our results further on.

Pseudocontinuous functions and existence of Nash equilibria

In topological spaces, we introduce a new class of functions (pseudocontinuous functions) and we present some characterizations and properties. In particular, we show that any preference relation endowed of utility functions is continuous if and only if any utility is pseudocontinuous. A maximum theorem is proved for such a class of functions and connections with similar results are investigated. Finally, the existence of Nash equilibria for games with pseudocontinuous payoffs is obtained.

Boundary convergence of vector-valued pseudocontinuable functions

In the first part of the paper we discuss a multi-dimensional analogue of the well-known construction by D. Clark that allows one to study families of spectral measures of perturbations of the model contraction. In the second part we present extensions of the relevant results on the boundary behavior of pseudocontinuable functions. We show that, although the most direct analogue of the scalar theorem on the existence of boundary values for pseudocontinuable functions with respect to Clark measures fails in the non-scalar situation, suitable vector-valued versions of such results can be found.

Discontinuous but Well‐Posed Optimization Problems

Using classes of sequentially pseudocontinuous functions, recently introduced by the authors, the aim of the paper is to investigate Tikhonov and parametric well-posedness for optimization problems when the objective functions are not necessarily sequentially lower semicontinuous. Sequential pseudocontinuity is a property more general than sequential semicontinuity and finds motivations in choice theory, since the continuity of preference relations on first countable topological spaces is characterized by the sequential pseudocontinuity of any utility function. Examples show that it is not possible to improve the results with other well-known classes of discontinuous functions.

Pseudocontinuity in Optimization and Nonzero-Sum Games

We enlarge the class of sufficient conditions which guarantee, on the one hand, the sequential closedness of set-valued functions defined by minimum points and by social Nash equilibria and, on the other hand, the existence of solutions for MinSup problems, using new classes of functions, called sequentially lower pseudocontinuous and sequentially upper pseudocontinuous functions. Properties and characterizations of such classes are investigated.

Pseudocontinuations and the backward shift

Beurling's theorem characterizes the forward shift invariant subspaces in the Hardy space $H^2$ on the open unit disk $\bold D$. The description is in terms of an inner function, that is, a function in $H^2$ whose nontangential boundary values have modulus $1$ almost everywhere. If $S$ stands for the forward shift $Sf(z)=zf(z)$, then the adjoint $L=S^*$ is the backward shift, $Lf(z)=\break (f(z)-f(0))/z$. The annihilator of a forward shift invariant subspace is then backward shift invariant, and Beurling's theorem leads to a description also of the backward shift invariant subspaces, as noted by R. G. Douglas, H. S. Shapiro and A. L. Shields [Ann. Inst. Fourier (Grenoble) 20 (1970), fasc. 1, 37--76; MR0270196 (42 #5088)]. Whereas the forward invariant subspaces are described primarily in terms of zeros, the backward invariant subspaces are characterized in terms of pseudocontinuations. To be concrete, take $I$ to be the forward invariant subspace of all functions in $H^2$ that vanish along a given finite sequence $A$ of distinct points in $\bold D$. Its annihilator $I^\perp$ is finite-dimensional, and consists of all rational functions with simple poles along the sequence $A^*$ obtained by reflecting $A$ in the unit circle. Then, if we let the finite sequence $A$ ``grow'' to become in the limit a Blaschke sequence plus a negative singular mass on the circle, the annihilator will increase as well, but there will remain a ``connection'' between the behavior inside $\bold D$ and the behavior outside in the exterior disk ${\bold D}_{\rm e}$, the complement of the closed unit disk on the Riemann sphere. The connection is furnished by the pseudocontinuation across the circle: we have a holomorphic Nevanlinna class function on the inside, and a meromorphic Nevanlinna class function on the outside, and they have the same nontangential boundary values almost everywhere on the unit circle. The issue at hand is whether the Hardy space situation is typical of backward invariant subspaces in Banach spaces $\scr B$ of analytic functions on the disk. A dichotomy appears: if $\scr B$ is bigger than the corresponding Hardy space, then the backward invariant subspaces possess pseudocontinuations across the unit circle, whereas if $\scr B$ is smaller, this is no longer generally the case. What happens is best understood in terms of forward invariant subspaces. With the standard Cauchy duality (the extension of the $H^2$-self-duality), we can think of the dual ${\scr B}^*$ of $\scr B$ as a space of holomorphic functions on $\bold D$, and study the forward shift invariant subspaces on ${\scr B}^*$. Let us concentrate on the case when $\scr B$ is a Hilbert space, of Dirichlet or Bergman type; then ${\scr B}^*$ falls into the same category, too. Every forward invariant subspace $\scr M$ of Dirichlet type has index $1$, which means that $S\scr M$ has codimension $1$ in $\scr M$; this is analogous to the $H^2$ case. Apparently, this means that the annihilator $\scr M^\perp$ (which is a backward invariant subspace of a Bergman space) consists of pseudocontinuable functions. However, there are plenty of forward invariant subspaces of a Bergman space which have index bigger than $1$ [see, e.g., H. Hedenmalm, J. Reine Angew. Math. 443 (1993), 1--9; MR1241125 (94k:30092)]. The annihilator of such a forward invariant subspace is a backward invariant subspace of a Dirichlet space, and some playing around with the formulas for pseudocontinuations suggests that in this case, it should not be unique (and hence not exist as a pseudocontinuation). This is then worked out rigorously in the paper.

Generalized Hardy inequalities and pseudocontinuable functions

Given positive integers n1< n2<... we show that the Hardy-type inequality $\sum\limits_{k = 1}^\infty {\frac{{\left| {\hat f(n_k )} \right|}}{k}} \leqslant const\left\| f \right\|1$ holds true for all f∈H1, provided that the nk's, satisfy an appropriate (and indispensable) regularity condition. On the other hand, we exhibit inifinite-dimensional subspaces of H1 for whose elements the above inequality is always valid, no additional hypotheses being imposed. In conclusion, we extend a result of Douglas, Shapiro and Shields on the cyclicity of lacunary series for the backward shift operator.

Weithed Spaces of Pseudocontinuable Functions and Approximations by Rational Functions with Prescribed Poles

The main result of this paper provides an essential intermediate step for the proof of the authors theorem on left and right Blaschke-Potapov products which yields more insight into the structure of J -inner functions. The goal of this paper is to prepare an appropriate backward Schur-Potapov algorithm by solving a weighted approximation problem.