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  "body": "\n## Definition\n\nLet $a \u003c b$ and let $M = (M_k)_{k \\ge 0}$ be a sequence of positive real numbers.\nThe class $C^M([a, b])$ is the set of [[smooth functions|0BE3QDH73A]] $f \\colon [a, b] \\to \\R$ for which there are constants $A, B \u003e 0$ with\n$$\\abs{f^{(k)}(x)} \\le A B^k M_k \\quad \\text{for all } k \\ge 0 \\text{ and all } x \\in [a, b].$$\nThe class $C^M([a, b])$ is *quasi-analytic* if the following holds.\nIf $f \\in C^M([a, b])$ and $f^{(k)}(x_0) = 0$ for some $x_0 \\in [a, b]$ and all $k \\ge 0$, then $f = 0$ on $[a, b]$.\nA *quasi-analytic function* is a function in a quasi-analytic class.\n\n## Remarks\n\nEquivalently, two functions in a quasi-analytic class that have the same derivatives of all orders at one point are equal.\nAt the endpoints, the [[derivatives|084NGSTWM7]] are one-sided.\n\nExample: $M_k = k!$.\nA function with $\\abs{f^{(k)}} \\le A B^k k!$ on $[a, b]$ is [[analytic|0JD5ZNW0F4]], with Taylor series of radius at least $1/B$ around each point.\nIf all its derivatives vanish at $x_0$, then it vanishes near $x_0$, and by moving along $[a, b]$ in steps it vanishes everywhere.\nSo $C^M([a, b])$ is quasi-analytic.\n\nNon-example: $M_k = (k!)^2$.\nHere $\\sum_{k \\ge 1} M_{k-1}/M_k = \\sum_{k \\ge 1} 1/k^2$ converges, so by the [[Denjoy–Carleman theorem|KV0GW9K7JS]] the class is not quasi-analytic.\nIt contains a nonzero function that vanishes on a subinterval, and all its derivatives vanish at the points of that subinterval.\n",
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