# Nehari extension problem

Let $\varphi _ {0} , \varphi _ {- 1 } , \varphi _ {- 2 } , \dots$ be a given sequence of complex numbers. The Nehari extension problem is the problem to find (if possible) all $f \in L _ \infty ( \mathbf T )$ satisfying the following conditions:

i) the $n$ th Fourier coefficient $c _ {n} ( f )$ of $f$ is equal to $\varphi _ {n}$ for each $n \leq 0$;

ii) the norm constraint $\| f \| _ \infty \leq 1$ holds true. Here, $\| f \| _ \infty$ is the norm of $f$ as an element of the Lebesgue function space $L _ \infty ( \mathbf T )$ and $\mathbf T$ is the unit circle. Instead of condition ii) one may require $\| f \| _ \infty < 1$, and in the latter case one calls the problem suboptimal.

The Nehari extension problem is not always solvable. In fact (see [a12]), the problem has a solution if and only if the infinite Hankel matrix

$$\left ( \begin{array}{cccc} \varphi _ {0} &\varphi _ {- 1 } &\varphi _ {- 1 } &\cdot \\ \varphi _ {- 1 } &\varphi _ {- 2 } &\varphi _ {- 3 } &\cdot \\ \varphi _ {- 2 } &\varphi _ {- 3 } &\varphi _ {- 4 } &\cdot \\ \cdot &\cdot &\cdot &\cdot \\ \end{array} \right )$$

induces a bounded linear operator $\Phi$ on ${\mathcal l} ^ {2}$, the Hilbert space of all square-summable sequences, such that its operator norm is at most one, i.e., $\| \Phi \| \leq 1$. The suboptimal version of the problem is solvable if and only if $\| \Phi \| < 1$. If $\| \Phi \| = 1$, either the solution of the Nehari extension problem is unique or there are infinitely many solutions. If $\| \Phi \| < 1$, then the problem and its suboptimal version always have infinitely many solutions, which can be parametrized by a fractional-linear mapping.

For the suboptimal case, the set of all solutions $f$ in the Wiener algebra ${\mathcal W}$, i.e., when one requires additionally that $\sum _ {n = - \infty } ^ \infty | {c _ {n} ( f ) } | < \infty$, can be described as follows. In this case, it is assumed that the given sequence $\varphi _ {0} , \varphi _ {- 1 } , \varphi _ {- 2 } , \dots$ is absolutely summable. Let $\| \Phi \| < 1$. Then the operators $I - \Phi ^ {*} \Phi$ and $I - \Phi \Phi ^ {*}$ are boundedly invertible on ${\mathcal l} ^ {2}$, and one can build the following infinite column vectors:

$$\left ( \begin{array}{c} a _ {0} \\ a _ {- 1 } \\ a _ {- 2 } \\ \vdots \\ \end{array} \right ) = ( I - \Phi \Phi ^ {*} ) ^ {- 1 } \left ( \begin{array}{c} 1 \\ 0 \\ 0 \\ \vdots \\ \end{array} \right ) ,$$

$$\left ( \begin{array}{c} c _ {0} \\ c _ {1} \\ c _ {2} \\ \vdots \\ \end{array} \right ) = \Phi ^ {*} \left ( \begin{array}{c} a _ {0} \\ a _ {- 1 } \\ a _ {- 2 } \\ \vdots \\ \end{array} \right ) , \left ( \begin{array}{c} d _ {0} \\ d _ {1} \\ d _ {2} \\ \vdots \\ \end{array} \right ) = ( I - \Phi ^ {*} \Phi ) ^ {- 1 } \left ( \begin{array}{c} 1 \\ 0 \\ 0 \\ \vdots \\ \end{array} \right ) ,$$

$$\left ( \begin{array}{c} b _ {0} \\ b _ {- 1 } \\ b _ {- 2 } \\ \vdots \\ \end{array} \right ) = \Phi \left ( \begin{array}{c} d _ {0} \\ d _ {1} \\ d _ {2} \\ \vdots \\ \end{array} \right ) .$$

Now, consider the functions

$$\alpha ( \lambda ) = \sum _ {j = - \infty } ^ { 0 } a _ {j} a _ {0} ^ {- {1 / 2 } } \lambda ^ {j} ,$$

$$\gamma ( \lambda ) = \sum _ {j = 0 } ^ \infty c _ {j} a _ {0} ^ {- {1 / 2 } } \lambda ^ {j} , \delta ( \lambda ) = \sum _ {j = 0 } ^ \infty d _ {j} d _ {0} ^ {- {1 / 2 } } \lambda ^ {j} ,$$

$$\beta ( \lambda ) = \sum _ {j = - \infty } ^ { 0 } b _ {j} b _ {0} ^ {- {1 / 2 } } \lambda ^ {j} .$$

Then, each solution $f \in {\mathcal W}$ of the suboptimal Nehari extension problem for the sequence $\varphi _ {0} , \varphi _ {- 1 } , \varphi _ {- 2 } , \dots$ is of the form

$$\tag{a1 } f ( \lambda ) = ( \alpha ( \lambda ) h ( \lambda ) + \beta ( \lambda ) ) ( \gamma ( \lambda ) h ( \lambda ) + \delta ( \lambda ) ) ^ {- 1 } ,$$

where $\lambda \in \mathbf T$ and $h$ is an arbitrary element of the Wiener algebra ${\mathcal W}$ such that $| {h ( \lambda ) } | < 1$ for $\lambda \in \mathbf T$ and the $n$ th Fourier coefficient of $h$ is zero for each $n \leq 0$. Moreover, (a1) gives a one-to-one correspondence between all such $h$ and all solutions $f$. The central solution, i.e., the solution $f _ {\textrm{ cen } } ( \lambda ) = \beta ( \lambda ) \delta ( \lambda ) ^ {- 1 }$, which one obtains when the free parameter $h$ in (a1) is identically zero, has a maximum entropy characterization. In fact, it is the unique solution $f \in {\mathcal W}$ of the suboptimal Nehari extension problem that maximizes the entropy integral

$${ \frac{1}{2 \pi } } \int\limits _ {- \pi } ^ \pi { { \mathop{\rm log} } ( 1 - \left | {f ( e ^ {it } ) } \right | ^ {2} ) } {dt } .$$

The Nehari extension problem has natural generalizations for matrix-valued and operator-valued functions, and it has two-block and four-block analogues. In the matrix-valued case, a superoptimal Nehari extension problem is studied also. In the latter problem the constraint is made not only for the norm, but also for a number of first singular values [a13]. There exist many different approaches to treat the Nehari problem and its various generalizations. For instance, the method of one-step extensions (see [a1]), the commutant-lifting approach (see [a6] and Commutant lifting theorem), the band method (see [a10]), reproducing-kernel Hilbert space techniques (see [a5]), and Beurling–Lax methods in Krein spaces (see [a4] and Krein space). The results are used in $H ^ \infty$ control theory (see [a8]), and when the data are Fourier coefficients of a rational matrix function, the formulas for the coefficients in the linear fractional representation (a1) can be represented explicitly in state-space form (see [a9] and [a3]).

The Nehari extension problem also has non-stationary versions, in which the role of analytic functions is taken over by lower-triangular matrices. An example is the problem to complete a given lower-triangular array of numbers,

$$\left ( \begin{array}{cccc} \varphi _ {00 } &{} &{} &{} \\ \varphi _ {10 } &\varphi _ {11 } &{} &{} \\ \varphi _ {20 } &\varphi _ {21 } &\varphi _ {22 } &{} \\ \cdot &\cdot &\cdot &\cdot \\ \end{array} \right ) ,$$

to a full infinite matrix such that the resulting operator on ${\mathcal l} ^ {2}$ is bounded and has operator norm at most one. The non-stationary variants of the Nehari extension problem have been treated in terms of nest algebras [a2]. The main results for the stationary case carry over to the non-stationary case [a11], [a7].

How to Cite This Entry:
Nehari extension problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Nehari_extension_problem&oldid=49323
This article was adapted from an original article by I. GohbergM.A. Kaashoek (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article