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  "body": "\n## Definition\n\nLet $A \u003e 0$ and let $\\Lambda = (\\lambda_n)_{n \\in \\Z}$ be a sequence of distinct complex numbers.\nFor $n \\in \\Z$ let $e_n(t) = e^{i \\lambda_n t}$, a function in the [[Lebesgue space|SASP14MA1Z]] $L^2(-A, A)$.\nA *nonharmonic Fourier series* with frequencies $\\Lambda$ is a series\n$$\\sum_{n \\in \\Z} c_n \\, e^{i \\lambda_n t}, \\quad t \\in (-A, A),$$\nwith complex coefficients $c_n$, considered with convergence in $L^2(-A, A)$.\n\n## Remarks\n\nWhen $A = \\pi$ and $\\lambda_n = n$, this is an ordinary [[Fourier series|G12KXDHPPS]], and the $e_n / \\sqrt{2\\pi}$ form an orthonormal basis.\nIn general the $e_n$ are not orthogonal, and the theory asks which sequences $\\Lambda$ make $(e_n)$ complete, a [[Riesz basis|ZYDYD7FB1Z]] or a frame of $L^2(-A, A)$.\nWhen $(e_n)$ is a Riesz basis, every $f \\in L^2(-A, A)$ has a unique expansion $f = \\sum c_n e_n$, and $\\sum \\abs{c_n}^2$ is comparable to $\\norm{f}^2_{L^2}$.\n\nExample: let $A = \\pi$ and $\\lambda_n = n + \\delta_n$ with real $\\delta_n$ and $\\sup_n \\abs{\\delta_n} \u003c 1/4$.\nBy Kadec's $1/4$ theorem, $(e^{i\\lambda_n t})$ is a Riesz basis of $L^2(-\\pi, \\pi)$, so every function in $L^2(-\\pi, \\pi)$ is the sum of a unique nonharmonic Fourier series with these frequencies.\nThe constant $1/4$ cannot be replaced by a larger number.\n\nNon-example: let $A = \\pi$ and $\\lambda_n = 2n$.\nThe functions $e^{2int}$ all have period $\\pi$, so they cannot represent $f = 1_{(0, \\pi)}$, which is not $\\pi$-periodic on $(-\\pi, \\pi)$.\nSo the closed span of $(e_n)$ is not all of $L^2(-\\pi, \\pi)$ and the system is not complete.\n",
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