By Ingold L.

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SmB CHAPTER 3. LINEAR OPERATORS IN HILBERT SPACES 42 Since DnC)") = 0 we find sin(~ + 1)0 smO The solutions to this equation are given by = o. O=~ n+1 with k = 1,2, ... ,n. Since A = -2 cos 0, we find the eigenvalues Ak with k = 1,2, ... , n. = -2 cos (~) n+1 Consequently, IAkl < 2. (i) and Ak =j:. Ak' if k =j:. k'. If n -t (ii) 00, then infinitely many Ak with IAkl ::; 2 and (iii) Therefore specA = Ak - Ak+1 -t 0 for n -t 00. e. we have a continuous spectrum. Another approach to find the spectrum is as follows.

Therefore V(A*A) may be smaller than V(A). Next we summarize the algebraic properties of the operator norm. It follows from the definitions of the norm and the adjoint of a bounded operator, together with the triangular inequality that if A, B are bounded operators and c E C, then IlcA11 IclllAIl IIAI12 IIA*AII IIA+BII < IIAII+IIBII IIABII < IIAIIIIBII· Definition. Suppose that K is a subspace of 1l. L in K and K 1-, respectively, we may define a linear operator II by the formula IIf = h. This is termed the projection operator from 1l to K, or simply the projection operator or projector for the subspace K.

00 00 -00 -00 Thus ft(w) depends on f(u) only for t - T ~ u ~ t and (if 9 is continuous) gives little weight to the values of f near the endpoints. e. 9 E L2(R). When g(u) == 1 (so 9 fj. L2(R)), the windowed Fourier transform reduces to the ordinary Fourier transform. In the following we merely assume that 9 E L2(R). If we define gw,t(u) := e27riwug(u - t) we obtain ligw,tli = Ilgll· Consequently gw,t also belongs to L2(R), and the windowed Fourier transform can be expressed as the innner product of f with gw,t which makes sense if both functions are in L2 (R) .