By Daveau C., Zaghdani A.

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A. Kaashoek, Classes of Linear Operators I, Operator Theory: Advances and Applications, 49, Birkh¨ auser Verlag, Basel, 1990. [13] I. Gohberg, S. A. Kaashoek, Classes of Linear Operators II, Operator Theory: Advances and Applications, 63, Birkh¨ auser Verlag, Basel, 1993. [14] H. Helson and D. , 99 (1958), pp. 165–202. [15] H. Helson and D. , 106 (1961), pp. 175–213. A. G. Zeinstra, The band method and generalized Carath´eodoryToeplitz interpolation at operator points, Integral Equations and Operator Theory, 33 (1999) pp.

Then the following equality holds ΦΩ∆1/2 − dα Φ(α) dα ϕα y 2 = (∆y, y) − d2α Φ(α)y 2 . 1) In particular, if α = 0, then we have ΦΩ∆1/2 y − Φ(0)y 2 = (∆y, y) − Φ(0)y 2 . 2) Proof. Notice that C(I − λA)−1 xopt = dα ϕα Ω∆1/2 y. 1) holds. This completes the proof. 2 . I. T. Georgiou, and A. Lindquist, A generalized entropy criterion for Nevanlinna-Pick interpolation: A convex optimization approach to certain problems in systems and control, IEEE Transactions on Automatic Control, 42 (2001) pp. 822–839.

21) is a lower triangular Toeplitz matrix with the identity on the main diagonal. In particular, this matrix is invertible. For the moment assume that X is ﬁnite dimensional. 21) implies that {C, J} is observable. Notice that D is invertible, and thus, {DC, J} is also observable. 18) and {DC, J} is observable, it follows that J is stable. The stability of A and J imply that Ω and Ω−1 are both function in H ∞ (L(U)). In other words, Θ = Ω−1 is an invertible outer function. Now assume that X is inﬁnite dimensional.

### A hp-discontinuous Galerkin method for the time-dependent Maxwell’s equation: a priori error estimate by Daveau C., Zaghdani A.

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