Wiener—Hopf operators such as
$$ {W_c}^ + f \cdot (x) = {x_s}f(x) + \int\limits_0^\infty {{c_a}(x - r)dr\quad 0 \leqq x < \infty ,} $$
and their discrete analogues
$$ {W_d}^ + g \cdot (n) = \sum\limits_{k = 0}^\infty {d(n - k)g(k)\quad n = 0,1, \ldots ,} $$
arise in many important contexts and have been extensively studied. These operators are obtained by truncating convolutions over the additive group of real numbers R and over the additive group of integers I, respectively, where the truncations are given in terms of the linear order. It is therefore quite natural to truncate a second time to obtain the finite section operators,
$$ \begin{gathered} {W_c}^ + (y)f \cdot (x) = {c_s}f(x) + \int\limits_0^y {{c_a}(x - r)f(r)dr\quad 0 \leqq x \leqq y,} \hfill \\ {W_d}^ + (m)g \cdot (n) = \sum\limits_{k = 0}^m {d(n - k)g(k)\quad 0 \leqq n \leqq m.} \hfill \\ \end{gathered} $$


Linear Order Toeplitz Operator Banach Algebra Compact Abelian Group Invertible Operator 
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© Springer Basel AG 1969

Authors and Affiliations

  • I. I. HirschmanJr.
    • 1
  1. 1.Dept. of Math.Washington UniversitySt. LouisUSA

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