Linear Operators: Spectral operators |
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Page 942
... continuous function . By replacing s by st and u by ut and using the fact that μ ( Et ) = u ( E ) it is seen that √。g ( su ̄1 ) q ( ut ) μ ( du ) = ¿ q ( st ) , i.e. , every translate qt of an eigenfunction corresponding to 2 is also ...
... continuous function . By replacing s by st and u by ut and using the fact that μ ( Et ) = u ( E ) it is seen that √。g ( su ̄1 ) q ( ut ) μ ( du ) = ¿ q ( st ) , i.e. , every translate qt of an eigenfunction corresponding to 2 is also ...
Page 966
... continuous , we conclude that h , agrees almost everywhere with a continuous function . By redefining hm on a set of measure zero , we may take it to be continuous . A change of variables in [ * ] shows that for every f in L1 ( R ) ...
... continuous , we conclude that h , agrees almost everywhere with a continuous function . By redefining hm on a set of measure zero , we may take it to be continuous . A change of variables in [ * ] shows that for every f in L1 ( R ) ...
Page 1002
... continuous function f of two real variables x = ( X1 , X2 ) is called almost periodic if for each e > 0 there exists ... function may be approximated uniformly by linear combinations of functions of the form exp i ( t1x1 + t2x2 ) . 6 A ...
... continuous function f of two real variables x = ( X1 , X2 ) is called almost periodic if for each e > 0 there exists ... function may be approximated uniformly by linear combinations of functions of the form exp i ( t1x1 + t2x2 ) . 6 A ...
Contents
BAlgebras | 859 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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adjoint extension adjoint operator algebra analytic B-algebra Borel set boundary conditions boundary values bounded operator C₁ closed closure coefficients compact operator complex numbers continuous function converges Corollary deficiency indices Definition denote dense 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 Haar measure Hence Hilbert space Hilbert-Schmidt operator identity inequality integral interval isometric isomorphism kernel L₁ L₁(R L₂(I L₂(R Lemma Let f linear linearly independent mapping matrix measure neighborhood non-zero norm operators in Hilbert orthogonal orthonormal basis Plancherel's theorem positive preceding lemma PROOF prove real axis satisfies sequence solution spectral spectral theorem square-integrable subset subspace Suppose T₁ T₁(t T₂ theory To(t topology unique unitary vanishes vector zero