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Firstly,I just need to find u_k for the function 2.1.22 and then
ı need to plot u(x) and INu(x) in MATLAB for N=4,8,16 .
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For any integer N > 0, consider the set of points X; = 2пј N j = 0,...,N-1, (2.1.24) referred to as nodes or grid points or knots. The discrete Fourier coefficients of a complex-valued function u in (0,27] with respect to these points are N-1 ūk = 2 u(;)e-ikt;, k=-N/2, ...,N/2 -1. (2.1.25) N Consequently, the polynomial N/2-1 Inu(x) = ūkeiku x k=-N/2 = Σ (2.1.28) (2) The function u(x) = sin(x/2) is infinitely differentiable in (0,21], = (2.1.22)
n NI 1.5 1 0.5 -0.50 0.5 N = 4 N = 8 N = 16 1 1.5 (e) 2