Orthogonal systems for time-dependent spectral methods
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This paper is concerned with orthonormal systems in real intervals, given with zero Dirichlet boundary conditions. More specifically, our interest is in systems with a skew-symmetric differentiation matrix (this excludes orthonormal polynomials). We consider a simple construction of such systems and pursue its ramifications. In general, given any C^1(a,b) weight function such that w(a)=w(b)=0, we can generate an orthonormal system with a skew-symmetric differentiation matrix. Except for the case a=-∞, b=+∞, only a limited number of powers of that matrix is bounded and we establish a connection between properties of the weight function and boundedness. In particular, we examine in detail two weight functions: the Laguerre weight function x^αe^-x for x>0 and α>0 and the ultraspherical weight function (1-x^2)^α, x∈(-1,1), α>0, and establish their properties. Both weights share a most welcome feature of separability, which allows for fast computation. The quality of approximation is highly sensitive to the choice of α and we discuss how to choose optimally this parameter, depending on the number of zero boundary conditions.
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