## Linear Operators: Spectral theory |

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Page 951

To begin the study , some basic properties of convolutions will be obtained . 1

LEMMA . ( a ) If f is a - measurable , then the

measurable function . ( b ) For f , geL ( R ) the

integrable in y for ...

To begin the study , some basic properties of convolutions will be obtained . 1

LEMMA . ( a ) If f is a - measurable , then the

**function f**( x − y ) is a 2 x 2 -measurable function . ( b ) For f , geL ( R ) the

**function f**( x − y ) g ( y ) isintegrable in y for ...

Page 986

This shows that I 2 Land completes the proof of the lemma . Q . E . D . → 7

THEOREM . ( Wiener L , - closure theorem ) . Linear combinations of the

translates of a

does not ...

This shows that I 2 Land completes the proof of the lemma . Q . E . D . → 7

THEOREM . ( Wiener L , - closure theorem ) . Linear combinations of the

translates of a

**function f**in Ly ( R ) are dense in Ly ( R ) if and only if its transform fdoes not ...

Page 1075

if f is of bounded variation in the neighborhood of x . ... 15 Show that there exists

a

t ) e - itx dt , 21 J - A F denoting the Fourier transform

if f is of bounded variation in the neighborhood of x . ... 15 Show that there exists

a

**function f**in L ( - 00 , + 00 ) for which the family of functions 1 p + A f ( x ) = - T F (t ) e - itx dt , 21 J - A F denoting the Fourier transform

**of f**, fails to satisfy the ...### What people are saying - Write a review

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### Contents

IX | 859 |

Eigenvalues and Eigenvectors | 903 |

Spectral Representation | 911 |

Copyright | |

15 other sections not shown

### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

### Common terms and phrases

additive adjoint operator Akad algebra Amer analytic assume Banach spaces basis belongs Borel boundary conditions boundary values bounded called clear closed closure coefficients compact complex Consequently constant contains continuous converges Corollary corresponding defined Definition denote dense determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension finite follows formal differential operator formula function function f given Hence Hilbert space identity independent indices inequality integral interval Lemma limit linear mapping Math matrix measure multiplicity neighborhood norm obtained partial positive preceding present problem projection proof properties prove range regular remark representation respectively restriction result Russian satisfies seen sequence shown singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique vanishes vector zero