From Linear Algebra to the Foundations of Gabor Analysis11systems but also for designing fast numerical inversion schemes. In the last part of this section we give a small insight to fundamental results of the Gabor frame matrix. We start with the special case of so-called separable lattices of the form 螞 = 伪Zm 尾Zm where 伪 and 尾 are divisors of m. We dee 伪 = m/伪 and 尾 = m/尾. (The case that 伪尾 divides m corresponds to integer oversampling.) Due to the fact that 尾� e2ijm尾/m = 0 if j does not divide m=0 尾, the jl-th element of the frame matrix S is simply given by � �尾 伪� g(j �伪n)g(l �伪n) if |j �l| is divided by 尾 n=0 (8) (S)jl = � otherwise ,which is called the Walnut representation of S for the discrete case [37]. The discrete Walnut representation implies the following properties of S: (1) Only every 尾-th subdiagonal of S is non-zero. (2) Entries along a subdiagonal are 伪-periodic. (3) S is a block circulant matrix of the form ��A0 A1 . . . A伪� �A伪� A0 . . . A伪� � �� S=�. . �. .. . �. �. . . . . A1 A2 . . . A0 where As are non-circulant 伪 伪 matrices, with (As )j,l = (S)j+s伪,l+s伪 for s = 0, 1, . . . , 伪 �1 and j, l = 0, 1, . . . , 伪 �1, [42]. (9)This special case applies merely for separable lattices. In general, however, we have another powerful representation of the frame matrix, the so-called Janssen representation. For a better understanding of the Janssen representation in ite dimension we introduce the Frobenius norm for m n matricesm nAFro=i=1 j=1|aij |21/2=tr(A A)where the trace tr(B) of B is the sum of its diagonal entries. (The Frobenius norm corresponds to the Hilbert-Schmidt norm in inite dimension.) The
