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- W1998360055 abstract "The solution of the integral equation f(x;k)−ν ∫ − 1 1 c ( x , y ) exp ( i k | x − y | ) f ( y ; k ) d y = g(x;k), −1 < x + 1, is studied when the real parameter k tends to ∞ while v remains bounded, |v| ⩽v0. It is shown that for large k > k0(v0) this equation is non-singular and that there is a constant A(v0), independent ofv and k, such that max|f(x; k)| ⩽ A(ν0) max|g(x; k)|. When g(x; k) = g(x) is independent of k it follows that f(x; k)−g(x) tends to 0 as k tends to ∞. The argument depends on the solution of two associated Volterra equations not involving k. When the factor exp(ik|x−y|) in the kernel is replaced by cos k(x−y) the situation is more complicated. The corresponding results hold for the modified equation if and only if v is not near a characteristic value of the equation F(x)−½ν ∫ − 1 1 c ( x , y ) F ( y ) d y = G(x)." @default.
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- W1998360055 date "1969-01-01" @default.
- W1998360055 modified "2023-09-27" @default.
- W1998360055 title "Integral Equations with a Rapidly Oscillating Kernel" @default.
- W1998360055 doi "https://doi.org/10.1112/jlms/s1-44.1.449" @default.
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