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  "body": "\n## Summary\n\nA nonharmonic Fourier series is a series $\\sum_n c_n e^{i\\lambda_n t}$ on an interval, where the frequencies $\\lambda_n$ need not be the integers.\nOrdinary [[Fourier series|G12KXDHPPS]] on $[-\\pi, \\pi]$ are the case $\\lambda_n = n$.\nThe main questions ask when the exponentials $e^{i\\lambda_n t}$ are complete in $L^2$ of the interval.\nThey also ask when they form a [[Riesz basis|ZYDYD7FB1Z]] or a frame.\nThe answers involve the density of the frequencies and entire functions of exponential type.\nThey matter in sampling theory and control theory.\n\n## Explanations\n\n### L1 human\n\nAn ordinary Fourier series builds a function on $[-\\pi, \\pi]$ from the waves $e^{int}$, whose frequencies are the whole numbers.\nWhat if the frequencies are slightly moved, say $\\lambda_n = n + \\delta_n$ with small $\\delta_n$?\nThe waves are then no longer orthogonal, but they may still be able to represent every function in a stable way.\nA classical answer is Kadec's theorem: if all real $\\delta_n$ satisfy $\\abs{\\delta_n} \\le L \u003c 1/4$, the waves still form a Riesz basis.\n",
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