Some Generalized Eigenfunction Expansions And Uniqueness Theorems
Some Generalized Eigenfunction Expansions And Uniqueness Theorems
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L8), as given by the authors noted above, depend upon explicit asymptotic estimates of the behavior of eigenfunctions and eigenvalues as A — »• 00 . For a proof of (2. 18) with respect to the second order system (2, 9)- (2. 11); and one which does not depend on specific asymptotic evalua- tions, see Peters [8].
The formula (2. L8) leads to the expansion of f(y) into an infinite sum of residues. If C is a circle with center at A^ containing no other eigenvalue we have 12 (2. 19) 00 -, r f(y) = -...) p^ r Q[G(t, y, z), f(t)] n=0 dz n 00 5 Q ^ 7^ G(t, y, z)clz, f(t) n If (X»(z) has a zero of order k at z = A, the corresponding term in the expansion (2. 19) is (2. 20) Q[?^(t, y, n), f(t)] where 9^(t, y, n)/(z ->^„) comes from the Laurent expansion of G(t, y, z) for the neighborhood of z = A . The function 9 can be obtained by substituting the expansion 00 G(t, y, z) = y— (z - A^)* 0_^(t, y, n) + T~ [z - A^)* 9^(t, y, n) in the equation LG(t, y, z) -A^r(t)G - (z -A^)rG = 5(t - y) and the boundary conditions G^(0, y, z) +PQG(0, y, z) = a^A^G(0, y, z) + a^(z - A^)G(0, y, z), G^(l, y, z) + p^G(l, y, z) = a^A^G(l, y, z) + a^(z -A^)G(l, y, z) which define G(t, y, z).
What to read after Some Generalized Eigenfunction Expansions And Uniqueness Theorems?
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To quickly assess the difficulty of the text, read a short excerpt:
L8), as given by the authors noted above, depend upon explicit asymptotic estimates of the behavior of eigenfunctions and eigenvalues as A — »• 00 . For a proof of (2. 18) with respect to the second order system (2, 9)- (2. 11); and one which does not depend on specific asymptotic evalua- tions, see Peters [8].
The formula (2. L8) leads to the expansion of f(y) into an infinite sum of residues. If C is a circle with center at A^ containing no other eigenvalue we have 12 (2. 19) 00 -, r f(y) = -...) p^ r Q[G(t, y, z), f(t)] n=0 dz n 00 5 Q ^ 7^ G(t, y, z)clz, f(t) n If (X»(z) has a zero of order k at z = A, the corresponding term in the expansion (2. 19) is (2. 20) Q[?^(t, y, n), f(t)] where 9^(t, y, n)/(z ->^„) comes from the Laurent expansion of G(t, y, z) for the neighborhood of z = A . The function 9 can be obtained by substituting the expansion 00 G(t, y, z) = y— (z - A^)* 0_^(t, y, n) + T~ [z - A^)* 9^(t, y, n) in the equation LG(t, y, z) -A^r(t)G - (z -A^)rG = 5(t - y) and the boundary conditions G^(0, y, z) +PQG(0, y, z) = a^A^G(0, y, z) + a^(z - A^)G(0, y, z), G^(l, y, z) + p^G(l, y, z) = a^A^G(l, y, z) + a^(z -A^)G(l, y, z) which define G(t, y, z).
What to read after Some Generalized Eigenfunction Expansions And Uniqueness Theorems?
You can find similar books in the "Read Also" column, or choose other free books by Arthur S Peters to read onlineMoreLess
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