Fast Computation of Singular Oscillatory Fourier Transforms

Table 1

(a) Absolute errors in 𝑛-point approximations by the proposed method to the integral ∫¹₀𝑥^−(1/2)(𝑥−1)^−(1/2)cos𝑥𝑒^𝑖𝜔𝑥𝑑𝑥. (b) Absolute errors in 𝑛-point approximations by the Clenshaw-Curtis-Filon-type method [31] to the integral ∫¹₀𝑥^−(1/2)(𝑥−1)^−(1/2)cos𝑥𝑒^𝑖𝜔𝑥𝑑𝑥.

(a)  

𝜔⧵𝑛	2	3	4
10	2.7×10^−5	1.0×10^−6	3.9×10^−8
20	1.1×10^−6	9.9×10^−9	2.2×10^−10
40	6.3×10^−8	2.4×10^−10	1.6×10^−12
80	1.1×10^−9	1.3×10^−12	6.3×10^−15

(b)  

𝜔⧵𝑛	2	3	4
10	8.6×10^−1	7.8×10^−1	6.9×10^−1
20	7.6×10^−1	5.1×10^−1	3.4×10^−1
40	6.3×10^−1	4.6×10^−1	3.0×10^−1
80	5.9×10^−1	4.4×10^−1	2.2×10^−1