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Abstract

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,
$$ \begin{gathered} {W_c}^ + (y)f \cdot (x) = {c_s}f(x) + \int\limits_0^y {{c_a}(x - r)f(r)dr\quad 0 \leqq x \leqq y,} \hfill \\ {W_d}^ + (m)g \cdot (n) = \sum\limits_{k = 0}^m {d(n - k)g(k)\quad 0 \leqq n \leqq m.} \hfill \\ \end{gathered} $$
.

Keywords

Linear Order Toeplitz Operator Banach Algebra Compact Abelian Group Invertible Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Basel AG 1969

Authors and Affiliations

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

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