The scattering of electromagnetic waves by resonant systems is determined by the excitation of the quasinormal modes (QNMs), i.e. the eigenmodes, of the system. This Review addresses three fundamental concepts in relation to the representation of the scattered field as a superposition of the excited QNMs: normalization, orthogonality, and completeness. Orthogonality and normalization enable a straightforward assessment of the QNM excitation strength for any incident wave. Completeness guarantees that the scattered field can be faithfully expanded into the complete QNM basis. These concepts are not trivial for non-conservative (non-Hermitian) systems and have driven many theoretical developments since initial studies in the 70’s. Yet, they are not easy to grasp from the extensive and scattered literature, especially for newcomers in the field. After recalling fundamental results obtained in initial studies on the completeness of the QNM basis for simple resonant systems, we review recent achievements and the debate on the normalization, clarify under which circumstances the QNM basis is complete, and highlight the concept of QNM regularization with complex coordinate transforms.

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We determine the asymptotic behavior of the l p -norms of the sequence of Taylor coefficients of b n , where b = z−λ 1−λz is an automorphism of the unit disk, p ∈ [1, ∞], and n is large. It is known that in the parameter range p ∈ [1, 2] a sharp upper boundholds. In this article we find that this estimate is valid even when p ∈ [1, 4). We prove thatWe prove that our upper bounds are sharp as n tends to ∞ i.e. they have the correct asymptotic n dependence.the elementary Blaschke factor corresponding to λ. Clearly |b λ (z)| = 1 is equivalent to z ∈ ∂D. For any n we have that B = b n is a bounded, holomorphic on D and as such posses a natural identification with its boundary behavior on ∂D [NN]. It is well known that the Taylor-and Fourier-coefficients of such functions can be identified [NN] and we will use these terms interchangeably in what follows. Let B = k≥0 B(k)z k denote the Taylor expansion of B = b n . We writefor the usual l p -norm of the sequence of Taylor coefficients of B. In the limit of large p we set ||B|| l∞ := sup k | B(k)|. We observe that our l p -norms only depend on the absolute