Linear Operators: Spectral theory |
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Page 942
... continuous function . By replacing s by st and u by ut and using the fact that μ ( Et ) = μ ( E ) it is seen that 9 √q8 ( su ̄1 ) q ( ut ) μ ( du ) = 2q ( st ) , i.e. , every translate q ' of an eigenfunction corresponding to λ is also ...
... continuous function . By replacing s by st and u by ut and using the fact that μ ( Et ) = μ ( E ) it is seen that 9 √q8 ( su ̄1 ) q ( ut ) μ ( du ) = 2q ( st ) , i.e. , every translate q ' of an eigenfunction corresponding to λ is also ...
Page 966
... continuous , we conclude that h , agrees almost everywhere with a continuous function . By redefining h , 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 ) , m ...
... continuous , we conclude that h , agrees almost everywhere with a continuous function . By redefining h , 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 ) , m ...
Page 1002
... continuous function f of two real variables x = ( x1 , x2 ) is called almost periodic if for each ɛ > 0 there exists ... function may be approximated uniformly by linear combinations of functions of the form exp i ( t , x1 + t2x2 ) . 6 A ...
... continuous function f of two real variables x = ( x1 , x2 ) is called almost periodic if for each ɛ > 0 there exists ... function may be approximated uniformly by linear combinations of functions of the form exp i ( t , x1 + t2x2 ) . 6 A ...
Contents
BAlgebras | 859 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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A₁ 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 countably deficiency indices Definition denote dense eigenfunctions eigenvalues element equation essential spectrum Exercise exists f₁ finite dimensional follows from Lemma follows from Theorem formal differential operator formally self adjoint formula Fourier function f Haar measure Hence Hilbert space Hilbert-Schmidt operator homomorphism identity inequality infinity integral interval kernel L₁ L₁(R L₂ L₂(I L₂(R Lemma Let f linearly independent mapping matrix measure neighborhood non-zero norm operators in Hilbert orthogonal Plancherel's theorem positive preceding lemma prove real axis real numbers satisfies sequence shows solution spectral set spectral theorem square-integrable subset subspace Suppose T₁ T₂ theory To(t topology tr(T transform uniformly unique unitary vanishes vector zero