By Hille E.

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**Example text**

M are called frame coeﬃcients. 3; it is called the canonical dual frame of {fk }m k=1 . 6 Let {ek }2k=1 be an orthonormal basis for a two-dimensional vector space V with inner product. Let f1 = e1 , f2 = e1 − e2 , f3 = e1 + e2 . Then {fk }3k=1 is a frame for V . Using the deﬁnition of the frame operator, 3 f, fk fk , Sf = k=1 we obtain that Se1 = e1 + e1 − e2 + e1 + e2 = 3e1 Se2 = and −(e1 − e2 ) + e1 + e2 = 2e2 . Thus, 1 1 e1 , S −1 e2 = e2 . 3 2 By linearity, the canonical dual frame is S −1 e1 = {S −1 fk }3k=1 = = {S −1 e1 , S −1 e1 − S −1 e2 , S −1 e1 + S −1 e2 } 1 1 1 1 1 { e1 , e1 − e2 , e1 + e2 }.

Using the deﬁnition of the frame operator, 3 f, fk fk , Sf = k=1 we obtain that Se1 = e1 + e1 − e2 + e1 + e2 = 3e1 Se2 = and −(e1 − e2 ) + e1 + e2 = 2e2 . Thus, 1 1 e1 , S −1 e2 = e2 . 3 2 By linearity, the canonical dual frame is S −1 e1 = {S −1 fk }3k=1 = = {S −1 e1 , S −1 e1 − S −1 e2 , S −1 e1 + S −1 e2 } 1 1 1 1 1 { e1 , e1 − e2 , e1 + e2 }. 5, the representation of f ∈ V in terms of the frame is given by 3 f f, S −1 fk fk = k=1 = 1 1 1 1 1 f, e1 e1 + f, e1 − e2 (e1 − e2 ) + f, e1 + e2 (e1 + e2 ).

Note that we consequently identify operators U : Cn → Cm with their matrix representations with respect to the canonical bases in Cn and Cm . Letting {ek }nk=1 denote the canonical orthonormal basis in Cn and {ek }m k=1 the canonical orthonormal basis in Cm , the matrix representation of U is the m × n matrix, where the kth column consists of the coordinates of the image under U of the kth basis vector in U , in terms of the given basis in W . The jkth entry in the matrix representation is U ek , ej .