## Linear Operators, Part 1 |

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Page 603

Thus, by induction, a e Q(T"). This completes the proof. Q.E.D. 8 THEOREM. Let P

be a polynomial of degree n, and let the function f in 37 (T) have a zero of order m

, 0 < m soo, at

Thus, by induction, a e Q(T"). This completes the proof. Q.E.D. 8 THEOREM. Let P

be a polynomial of degree n, and let the function f in 37 (T) have a zero of order m

, 0 < m soo, at

**infinity**. (a) If a is in Q(T"), then f(T)t is in Q(T"+"), where m+n = co ...Page 604

If P is a polynomial, P(a(T)) = a(P(T)). PRoof. Let P be of degree n, and suppose

A & P(g(T)). If g($) = [Å–P($)]-1, g is in 37 (T), with no zeros on g(T), and a zero of

order n at

If P is a polynomial, P(a(T)) = a(P(T)). PRoof. Let P be of degree n, and suppose

A & P(g(T)). If g($) = [Å–P($)]-1, g is in 37 (T), with no zeros on g(T), and a zero of

order n at

**infinity**. Then, by Theorems 8 and 9, [g(T)]−1 = AI — P(T) with domain ...Page 641

This class will include functions which are analytic on G(A), but not necessarily

analytic at

transforms. We also discuss the inversion of these operators by limits of ...

This class will include functions which are analytic on G(A), but not necessarily

analytic at

**infinity**. The functions to be considered are bilateral Laplace-Stieltjestransforms. We also discuss the inversion of these operators by limits of ...

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### Contents

A Settheoretic Preliminaries | 1 |

Convergence and Uniform Convergence of Generalized | 26 |

Algebraic Preliminaries | 34 |

Copyright | |

27 other sections not shown

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