View source for Toeplitz form, indefinite

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A quadratic form, defined on the space  $  \Phi $
of infinite sequences  $  x = \{ \xi _ {p} \} _ {- \infty }   ^  \infty  $
of finite support by the expression

$$ 
( x, x)  =  \sum _ {- \infty } ^ { {+ }  \infty }
c _ {p - q }  \xi _ {p} \overline \xi \; _ {q} ,
$$

where the sequence  $  c = \{ c _ {p} \} _ {- \infty }   ^  \infty  $, 
$  c _ {0} = \overline{ {c _ {0} }}\; $, 
is such that, from some dimension  $  N $
onwards, the form  $  ( x, x) $
reduces to canonical form as a sum of  $  \kappa $
squares on each subspace

$$ 
\Phi  ^ {N}  \subset   \Phi ,\ \ 
\Phi  ^ {N}  =  \{ {\xi _ {p} } : {\xi _ {p} = 0, | p | > N } \}
.
$$

An indefinite scalar product may be introduced in  $  \Phi $
by means of the Toeplitz form; after factorization over the isotropic subspace and completion,  $  \Phi $
becomes a [[Pontryagin space|Pontryagin space]].