## Linear Operators: Spectral theory |

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Page 974

J —oo J —oo Now the expression C+0° e-<*+">'l*dx J —OO defines an entire

function of z which is equal to (+a°e-*'/2dx = (271)1 J— oo for all real z and

therefore for all z. Hence we conclude that (r/)(m) = (2ji)4 >J/2. By

J —oo J —oo Now the expression C+0° e-<*+">'l*dx J —OO defines an entire

function of z which is equal to (+a°e-*'/2dx = (271)1 J— oo for all real z and

therefore for all z. Hence we conclude that (r/)(m) = (2ji)4 >J/2. By

**Plancherel's****theorem**...Page 1155

orthonormal set in L2(R), which fact was also proved in Theorem 1.6. We leave it

to the reader to show that if R is the compact Abelian group of the real numbers

modulo ...

**Plancherel's theorem**asserts that the set of characters forms a completeorthonormal set in L2(R), which fact was also proved in Theorem 1.6. We leave it

to the reader to show that if R is the compact Abelian group of the real numbers

modulo ...

Page 1666

In this case, we have also 1*15 = (*»)• 2 L by

that k 2? 0. Suppose that ||F||(M < oo. Then, by what has been proved and by

Lemma 40 (ii), F is in H^](C) and ;S where A is a finite constant depending only

on ...

In this case, we have also 1*15 = (*»)• 2 L by

**Plancherel's theorem**. Next supposethat k 2? 0. Suppose that ||F||(M < oo. Then, by what has been proved and by

Lemma 40 (ii), F is in H^](C) and ;S where A is a finite constant depending only

on ...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 860 |

Commutative BAlgebras | 874 |

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Acad adjoint extension adjoint operator algebra Amer analytic B-algebra B*-algebra Banach spaces Borel set boundary conditions boundary values bounded operator closed closure coefficients complex numbers constant continuous function converges Corollary deficiency indices Definition denote dense differential equations Doklady Akad domain eigenfunctions eigenvalues element essential spectrum exists finite dimensional follows from Lemma follows from Theorem follows immediately formal differential operator formally self adjoint formula Fourier function g Haar measure Hence Hilbert space Hilbert-Schmidt operator hypothesis identity inequality integral interval kernel Lemma linear operator linearly independent mapping Math matrix measure Nauk SSSR N. S. neighborhood norm open set operators in Hilbert orthogonal orthonormal partial differential operator Plancherel's theorem positive Proc prove real axis real numbers representation satisfies second order Section sequence singular solution spectral set spectral theory square-integrable subspace Suppose symmetric operator topology transform unique unitary vanishes vector zero