"Unique" band limited function (Digital Show and Tell) 2014-12-11 09:29:46 I recently watched monty's excellent video, "Digital Show and Tell", but found one of the claims confusing. He asserts that there is a unique band-limited function that fits the data points. Thus, higher sampling rates add no additional information in the lower frequencies.This doesn't match my understanding of the Fourier Transform. I learned about the DFT from a background in mathematics---I have never used it for signal analysis.To make my confusion concrete, run this program in octave:function f = interp_fft(y) n = size(y,2); F = 0:n-1; P = fft(y); f = @(x) sum(arrayfun(@(f,p) p*exp(2*pi*i*f*x/n), F, P))/n;endfunctionn=32x_dot=0:n-1;x_all=0:0.01:n;N = rand(1,n);M = N(1:n/2);f = interp_fft(N);g = interp_fft(M);y = arrayfun(f, x_all);z = arrayfun(g, x_all);plot(x_dot,N,'ro',x_all,y,'b-',x_all,z,'k-');In the resulting graph, we have random data points represented by the 'red dots'. I produced two fitting curves. One curve fits the first half only and the second curve fits all the dots. You can see, that despite agreeing on the first dots, between the dots the functions differ.Define "16" units on the x-access to be 1 second. Then, as far as I understand this, I turned the 16 data point function into frequency powers at 0Hz, 1Hz, 2Hz, ..., 15Hz. The second curve has power at frequencies at 0Hz, 0.5Hz, 1Hz, 1.5Hz, ...15.5Hz.Wouldn't both functions still be called "band-limited"? So, I have two functions that pass through the first 16 data points ... and they differ.What gives?