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  "body": "\n## Definition\n\nLet $H$ be a [[Hilbert space|HM2HHE3J51]] with [[inner product|0NB0AWGJBW]] $\\ip{\\cdot}{\\cdot}$, and let $(f_i)_{i \\in I}$ be a family of vectors of $H$.\nThe family is a *frame* if there are constants $0 \u003c A \\le B \u003c \\infty$ with\n$$A \\norm{f}^2 \\le \\sum_{i \\in I} \\abs{\\ip{f}{f_i}}^2 \\le B \\norm{f}^2 \\quad \\text{for all } f \\in H.$$\nThe numbers $A$ and $B$ are *frame bounds*.\nThe frame is *tight* if one can take $A = B$, that is, if $\\sum_{i \\in I} \\abs{\\ip{f}{f_i}}^2 = A \\norm{f}^2$ for all $f \\in H$.\nA tight frame with $A = B = 1$ is a *Parseval frame*.\n\n## Remarks\n\nFor a tight frame with bound $A$, every $f \\in H$ satisfies $f = A^{-1} \\sum_{i \\in I} \\ip{f}{f_i} f_i$, with convergence in $H$.\nEvery orthonormal basis is a Parseval frame, and the union of two orthonormal bases is a tight frame with $A = 2$.\n\nExample: in $\\R^2$, let $u_1 = (0, 1)$, $u_2 = (-\\tfrac{\\sqrt3}{2}, -\\tfrac12)$, $u_3 = (\\tfrac{\\sqrt3}{2}, -\\tfrac12)$.\nFor $f = (x, y)$, $\\sum_k \\ip{f}{u_k}^2 = y^2 + \\tfrac14(\\sqrt3 x + y)^2 + \\tfrac14(\\sqrt3 x - y)^2 = \\tfrac32 (x^2 + y^2)$.\nSo $(u_1, u_2, u_3)$ is a tight frame with bound $\\tfrac32$.\n\nNon-example: in $\\R^2$, the family $(e_1, e_1, e_2)$ gives $\\sum \\ip{f}{f_i}^2 = 2x^2 + y^2$.\nIt is a frame with optimal bounds $A = 1$ and $B = 2$, but it is not tight.\n",
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