Throughout this article, we use the following notation and conventions. For a complex vector , we denote by its transposed, complex conjugate, such that is the standard scalar product for two vectors . The corresponding norm is defined by . The norm of a complex, matrix is
The transposed, complex conjugate of is denoted by , such that for all and . For two Hermitian matrices and , the inequality means for all . The identity matrix is denoted by .
The spectrum of a complex, matrix is the set of all eigenvalues of ,
which is real for Hermitian matrices. The spectral radius of is defined as
Then, the matrix norm of a complex matrix can also be computed as .
Next, we denote by the class of measurable functions on the open subset of , which are square-integrable. Two functions , which differ from each other only by a set of measure zero, are identified. The scalar product on is defined as
and the corresponding norm is . According to the Cauchy–Schwarz inequality we have
The Fourier transform of a function , belonging to the class of infinitely-differentiable functions with compact support, is defined as
According to Parseval’s identities, for all , and the map , can be extended to a linear, unitary map called the Fourier–Plancharel operator; see, for example, . Its inverse is given by for and .
For a differentiable function , we denote by , , , its partial derivatives with respect to , , , .
Indices labeling gridpoints and number of basis functions range from to . Superscripts and subscripts are used to denote the numerical solution at some discrete timestep and gridpoint, as in
Living Rev. Relativity 15, (2012), 9
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