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].

References

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[a5]	H. Dym, " contractive matrix functions, reproducing kernel Hilbert spaces and interpolation" , CBMS , 71 , Amer. Math. Soc. (1989)
[a6]	C. Foias, A.E. Frazho, "The commutant lifting approach to interpolation problems" , Operator Theory: Advances and Applications , 44 , Birkhäuser (1990)
[a7]	C. Foias, A.E. Frazho, I. Gohberg, M.A. Kaashoek, "Discrete time-variant interpolation as classical interpolation with an operator argument" Integral Eq. Operator Th. , 26 (1996) pp. 371–403
[a8]	B.A. Francis, "A course in control theory" , Springer (1987)
[a9]	K. Glover, "All optimal Hankel-norm approximations of linear multivariable systems and the -error bounds" Int. J. Control , 39 (1984) pp. 1115–1193
[a10]	I. Gohberg, S. Goldberg, M.A Kaashoek, "Classes of linear operators II" , Operator Theory: Advances and Applications , 63 , Birkhäuser (1993)
[a11]	I. Gohberg, M.A. Kaashoek, H.J. Woerdeman, "The band method for positive and contractive extension problems: An alternative version and new applications" Integral Eq. Operator Th. , 12 (1989) pp. 343–382
[a12]	Z. Nehari, "On bounded bilinear forms" Ann. of Math. , 65 (1957) pp. 153–162
[a13]	V.V. Peller, N.J. Young, "Superoptimal analytic approximations of matrix functions" J. Funct. Anal. , 120 (1994) pp. 300–343