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 √。8 ( 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 ) = μ ( E ) it is seen that √。8 ( 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 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 ) ...
... 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 ) ...
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 ( t1x1 + t2x1⁄2 ) . 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 ( t1x1 + t2x1⁄2 ) . 6 A ...
Contents
BAlgebras | 859 |
Bounded Normal Operators in Hilbert Space | 887 |
Miscellaneous Applications | 937 |
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adjoint extension adjoint operator algebra analytic B-algebra B*-algebra Borel set boundary conditions boundary values bounded operator C₁ closed closure Co(I coefficients complex numbers converges Corollary deficiency indices Definition denote dense 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 defined function f Haar measure Hence Hilbert space Hilbert-Schmidt operator identity inequality integral interval kernel L₁(R L₂(I L₂(R Lemma Let f linearly independent mapping Math matrix measure neighborhood norm open set operators in Hilbert orthogonal orthonormal partial differential operator Plancherel's theorem positive preceding lemma prove real axis real numbers representation satisfies Section sequence solution spectral spectral theory square-integrable subspace Suppose symmetric operator T₁ T₂ theory To(t topology unique unitary vanishes vector zero