## Linear Operators: Spectral operators |

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Page 976

By using the form of the characters on R* the

more concrete formulation in the present case also. It is easily seen that this

theorem asserts that if f is in L2(R"), the limit +N +N g(t1,..., t,) = (2:1)-"so lim - - s f(

x1, ...

By using the form of the characters on R* the

**Plancherel theorem**may be given amore concrete formulation in the present case also. It is easily seen that this

theorem asserts that if f is in L2(R"), the limit +N +N g(t1,..., t,) = (2:1)-"so lim - - s f(

x1, ...

Page 1155

Q.E.D. When discussing a group R with Haar measure A this theorem enables us

to refer to the Haar measure on the ...

characters forms a complete orthonormal set in L2(R), which fact was also ...

Q.E.D. When discussing a group R with Haar measure A this theorem enables us

to refer to the Haar measure on the ...

**Plancherel's theorem**asserts that the set ofcharacters forms a complete orthonormal set in L2(R), which fact was also ...

Page 1160

measure in R is regular, the set X of functions in Li o La(R) which vanish outside

of compact sets in R are dense in L2(R); by

in L.(R) and hence {rf - rgf, ge.Y) is dense in L(R). Now let f, g be in X and ...

measure in R is regular, the set X of functions in Li o La(R) which vanish outside

of compact sets in R are dense in L2(R); by

**Plancherel's theorem**(tffe 3 } is densein L.(R) and hence {rf - rgf, ge.Y) is dense in L(R). Now let f, g be in X and ...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 868 |

4 Exercises | 879 |

Copyright | |

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adjoint extension adjoint operator algebra analytic B-algebra B-space Borel set boundary conditions boundary values bounded operator closed closure Cº(I coefficients compact operator complex numbers constant continuous function converges Corollary deficiency indices Definition denote dense domain eigenfunctions eigenvalues element equation essential spectrum Exercise exists finite dimensional follows from Lemma follows from Theorem follows immediately formal differential operator formally self adjoint formula Fourier function defined function f Hence Hilbert space Hilbert-Schmidt operator identity inequality infinity integral interval kernel Lemma Let f Let Q linearly independent mapping matrix measure neighborhood non-zero norm open set operators in Hilbert orthogonal orthonormal basis Plancherel's theorem positive preceding lemma prove real numbers satisfies sequence singular ſº solution spectral spectral theorem square-integrable subspace Suppose symmetric operator theory To(r To(t topology transform uniformly unique unitary vanishes vector zero