View source for Nehari extension problem

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n1100101.png" /> be a given sequence of complex numbers. The Nehari extension problem is the problem to find (if possible) all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n1100102.png" /> satisfying the following conditions:

i) the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n1100103.png" />th Fourier coefficient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n1100104.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n1100105.png" /> is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n1100106.png" /> for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n1100107.png" />;

ii) the norm constraint <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n1100108.png" /> holds true. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n1100109.png" /> is the norm of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001010.png" /> as an element of the Lebesgue function space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001012.png" /> is the unit circle. Instead of condition ii) one may require <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001013.png" />, and in the latter case one calls the problem suboptimal.

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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001014.png" /></td> </tr></table>

induces a bounded [[Linear operator|linear operator]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001015.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001016.png" />, the [[Hilbert space|Hilbert space]] of all square-summable sequences, such that its operator norm is at most one, i.e., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001017.png" />. The suboptimal version of the problem is solvable if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001018.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001019.png" />, either the solution of the Nehari extension problem is unique or there are infinitely many solutions. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001020.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001021.png" /> in the Wiener algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001022.png" />, i.e., when one requires additionally that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001023.png" />, can be described as follows. In this case, it is assumed that the given sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001024.png" /> is absolutely summable. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001025.png" />. Then the operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001026.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001027.png" /> are boundedly invertible on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001028.png" />, and one can build the following infinite column vectors:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001029.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001030.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001031.png" /></td> </tr></table>

Now, consider the functions

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001032.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001033.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001034.png" /></td> </tr></table>

Then, each solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001035.png" /> of the suboptimal Nehari extension problem for the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001036.png" /> is of the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001037.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001038.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001039.png" /> is an arbitrary element of the Wiener algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001040.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001041.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001042.png" /> and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001043.png" />th Fourier coefficient of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001044.png" /> is zero for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001045.png" />. Moreover, (a1) gives a one-to-one correspondence between all such <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001046.png" /> and all solutions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001047.png" />. The central solution, i.e., the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001048.png" />, which one obtains when the free parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001049.png" /> in (a1) is identically zero, has a maximum entropy characterization. In fact, it is the unique solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001050.png" /> of the suboptimal Nehari extension problem that maximizes the entropy integral

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001051.png" /></td> </tr></table>

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 [[#References|[a13]]]. There exist many different approaches to treat the Nehari problem and its various generalizations. For instance, the method of one-step extensions (see [[#References|[a1]]]), the commutant-lifting approach (see [[#References|[a6]]] and [[Commutant lifting theorem|Commutant lifting theorem]]), the [[Band method|band method]] (see [[#References|[a10]]]), reproducing-kernel Hilbert space techniques (see [[#References|[a5]]]), and Beurling–Lax methods in Krein spaces (see [[#References|[a4]]] and [[Krein space|Krein space]]). The results are used in [[H^infinity-control-theory|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001052.png" /> control theory]] (see [[#References|[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 [[#References|[a9]]] and [[#References|[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,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001053.png" /></td> </tr></table>

to a full infinite matrix such that the resulting operator on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001054.png" /> 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 [[#References|[a2]]]. The main results for the stationary case carry over to the non-stationary case [[#References|[a11]]], [[#References|[a7]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  V.M. Adamjan,   D.Z. Arov,   M.G. Krein,   "Infinite Hankel block matrices and related extension problems"  ''Transl. Amer. Math. Soc.'' , '''111'''  (1978)  pp. 133–156  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''6'''  (1971)  pp. 87–112</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W.B. Arveson,   "Interpolation in nest algebras"  ''J. Funct. Anal.'' , '''20'''  (1975)  pp. 208–233</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J. Ball,   I. Gohberg,   L. Rodman,   "Interpolation of rational matrix functions" , ''Operator Theory: Advances and Applications'' , '''45''' , Birkhäuser  (1990)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J. Ball,   J.W. Helton,   "A Beurling–Lax theorem for Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001055.png" /> which contains classical interpolation theory"  ''J. Operator Th.'' , '''9'''  (1983)  pp. 107–142</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  H. Dym,   "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001056.png" /> contractive matrix functions, reproducing kernel Hilbert spaces and interpolation" , ''CBMS'' , '''71''' , Amer. Math. Soc.  (1989)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  C. Foias,   A.E. Frazho,   "The commutant lifting approach to interpolation problems" , ''Operator Theory: Advances and Applications'' , '''44''' , Birkhäuser  (1990)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  B.A. Francis,   "A course in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001057.png" /> control theory" , Springer  (1987)</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  K. Glover,   "All optimal Hankel-norm approximations of linear multivariable systems and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001058.png" />-error bounds"  ''Int. J. Control'' , '''39'''  (1984)  pp. 1115–1193</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  I. Gohberg,   S. Goldberg,   M.A Kaashoek,   "Classes of linear operators II" , ''Operator Theory: Advances and Applications'' , '''63''' , Birkhäuser  (1993)</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  Z. Nehari,   "On bounded bilinear forms"  ''Ann. of Math.'' , '''65'''  (1957)  pp. 153–162</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  V.V. Peller,   N.J. Young,   "Superoptimal analytic approximations of matrix functions"  ''J. Funct. Anal.'' , '''120'''  (1994)  pp. 300–343</TD></TR></table>