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I have a list (list) in the form of {x,y} as

list = {{0., 16}, {0.00392157, 19}, {0.00784314, 6}, {0.0117647, 
  5}, {0.0156863, 8}, {0.0196078, 5}, {0.0235294, 6}, {0.027451, 
  7}, {0.0313725, 4}, {0.0352941, 10}, {0.0392157, 7}, {0.0431373, 
  9}, {0.0470588, 8}, {0.0509804, 11}, {0.054902, 9}, {0.0588235, 
  15}, {0.0627451, 10}, {0.0666667, 13}, {0.0705882, 16}, {0.0745098, 
  21}, {0.0784314, 24}, {0.0823529, 20}, {0.0862745, 25}, {0.0901961, 
  19}, {0.0941176, 24}, {0.0980392, 27}, {0.101961, 22}, {0.105882, 
  23}, {0.109804, 29}, {0.113725, 28}, {0.117647, 28}, {0.121569, 
  37}, {0.12549, 41}, {0.129412, 44}, {0.133333, 41}, {0.137255, 
  44}, {0.141176, 43}, {0.145098, 22}, {0.14902, 50}, {0.152941, 
  39}, {0.156863, 59}, {0.160784, 48}, {0.164706, 58}, {0.168627, 
  45}, {0.172549, 56}, {0.176471, 52}, {0.180392, 70}, {0.184314, 
  53}, {0.188235, 61}, {0.192157, 61}, {0.196078, 58}, {0.2, 
  41}, {0.203922, 47}, {0.207843, 53}, {0.211765, 59}, {0.215686, 
  54}, {0.219608, 58}, {0.223529, 63}, {0.227451, 55}, {0.231373, 59},
  {0.235294, 77}, {0.239216, 72}, {0.243137, 66}, {0.247059, 
  70}, {0.25098, 53}, {0.254902, 61}, {0.258824, 67}, {0.262745, 
  58}, {0.266667, 54}, {0.270588, 51}, {0.27451, 57}, {0.278431, 
  62}, {0.282353, 57}, {0.286275, 63}, {0.290196, 59}, {0.294118, 
  64}, {0.298039, 50}, {0.301961, 46}, {0.305882, 52}, {0.309804, 
  58}, {0.313725, 51}, {0.317647, 39}, {0.321569, 63}, {0.32549, 
  54}, {0.329412, 41}, {0.333333, 48}, {0.337255, 48}, {0.341176, 
  47}, {0.345098, 56}, {0.34902, 42}, {0.352941, 65}, {0.356863, 
  46}, {0.360784, 36}, {0.364706, 45}, {0.368627, 41}, {0.372549, 
  43}, {0.376471, 34}, {0.380392, 43}, {0.384314, 37}, {0.388235, 
  40}, {0.392157, 34}, {0.396078, 32}, {0.4, 32}, {0.403922, 
  50}, {0.407843, 40}, {0.411765, 31}, {0.415686, 34}, {0.419608, 
  45}, {0.423529, 37}, {0.427451, 28}, {0.431373, 44}, {0.435294, 
  28}, {0.439216, 32}, {0.443137, 24}, {0.447059, 24}, {0.45098, 
  28}, {0.454902, 34}, {0.458824, 38}, {0.462745, 36}, {0.466667, 
  25}, {0.470588, 31}, {0.47451, 15}, {0.478431, 24}, {0.482353, 
  27}, {0.486275, 20}, {0.490196, 24}, {0.494118, 31}, {0.498039, 
  19}, {0.501961, 11}, {0.505882, 25}, {0.509804, 25}, {0.513725, 
  32}, {0.517647, 17}, {0.521569, 17}, {0.52549, 21}, {0.529412, 
  25}, {0.533333, 16}, {0.537255, 16}, {0.541176, 20}, {0.545098, 
  17}, {0.54902, 17}, {0.552941, 16}, {0.556863, 19}, {0.560784, 
  22}, {0.564706, 14}, {0.568627, 18}, {0.572549, 16}, {0.576471, 
  20}, {0.580392, 12}, {0.584314, 15}, {0.588235, 17}, {0.592157, 
  5}, {0.596078, 17}, {0.6, 11}, {0.603922, 19}, {0.607843, 
  6}, {0.611765, 8}, {0.615686, 9}, {0.619608, 10}, {0.623529, 
  10}, {0.627451, 9}, {0.631373, 9}, {0.635294, 5}, {0.639216, 
  5}, {0.643137, 7}, {0.647059, 9}, {0.65098, 7}, {0.654902, 
  7}, {0.658824, 10}, {0.662745, 5}, {0.666667, 6}, {0.670588, 
  6}, {0.67451, 6}, {0.678431, 4}, {0.682353, 8}, {0.686275, 
  6}, {0.690196, 3}, {0.694118, 2}, {0.698039, 8}, {0.701961, 
  3}, {0.705882, 6}, {0.709804, 5}, {0.713725, 3}, {0.717647, 
  1}, {0.721569, 3}, {0.72549, 1}, {0.729412, 5}, {0.733333, 
  1}, {0.737255, 1}, {0.741176, 2}, {0.745098, 2}, {0.74902, 
  3}, {0.752941, 2}, {0.756863, 1}, {0.760784, 1}, {0.764706, 
  2}, {0.768627, 1}, {0.776471, 1}, {0.784314, 1}, {0.788235, 
  1}, {0.792157, 1}, {0.8, 1}, {0.807843, 1}, {0.827451, 
  1}, {0.835294, 1}, {0.870588, 1}};

Since the x values are not equally spaced, I am not sure whether I can extract the frequency components using Fourier.

How can I find the frequency components present in the list?

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  • $\begingroup$ Try to interpolate your data in an equally spaced grid, now you can calculate the FourierSeries. Or you can approximate(least square) your data by a trigonometric polynom... $\endgroup$ – Ulrich Neumann May 25 '18 at 13:50
  • $\begingroup$ @UlrichNeumann Yes, I thought about interpolation. Are there any other alternatives? $\endgroup$ – Majis May 25 '18 at 13:55
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    $\begingroup$ I think interpolation is the fastest way(see Hugh's answer). What's your purpose using FourierSeries for your data which doesn't look "periodic"? $\endgroup$ – Ulrich Neumann May 25 '18 at 14:02
  • $\begingroup$ @UlrichNeumann I have a set of such lists and I want to compare them by looking at their frequency contents. $\endgroup$ – Majis May 25 '18 at 14:06
  • $\begingroup$ See also: en.wikipedia.org/wiki/Non-uniform_discrete_Fourier_transform $\endgroup$ – Daniel Lichtblau May 26 '18 at 16:20
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Here is a way using Interpolation.

f = Interpolation[list];
{x1, x2} = list[[All, 1]][[{1, -1}]];
inc = (x2 - x1)/(Length[list] - 1);

Then your list with equal x increments is

newList = Table[{x, f[x]}, {x, x1, x2, inc}];
ListLinePlot[{list, newList}]

Mathematica graphics

Is the newList a good enough approximation for you? It looks very noisy. Also it does not have a definite periodic content so I don't know the purpose of using Fourier. Is this for smoothing? Anyway

ft = Fourier[newList[[All, 2]], FourierParameters -> {-1, -1}];
ListLinePlot[Abs[ft]]

Mathematica graphics

See here for notes on using Fourier and getting axes etc.

Hope that helps.

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