Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Fast Computation of Orthogonal Systems with a Skew-symmetric
  Differentiation Matrix   Orthogonal systems in $\mathrm{L}_2(\mathbb{R})$, once implemented in
spectral methods, enjoy a number of important advantages if their
differentiation matrix is skew-symmetric and highly structured. Such systems,
where the differentiation matrix is skew-symmetric, tridiagonal and
irreducible, have been recently fully characterised. In this paper we go a step
further, imposing the extra requirement of fast computation: specifically, that
the first $N$ coefficients {of the expansion} can be computed to high accuracy
in $\mathcal{O}(N\log_2N)$ operations. We consider two settings, one
approximating a function $f$ directly in $(-\infty,\infty)$ and the other
approximating $[f(x)+f(-x)]/2$ and $[f(x)-f(-x)]/2$ separately in $[0,\infty)$.
In each setting we prove that there is a single family, parametrised by
$\alpha,\beta > -1$, of orthogonal systems with a skew-symmetric, tridiagonal,
irreducible differentiation matrix and whose coefficients can be computed as
Jacobi polynomial coefficients of a modified function. The four special cases
where $\alpha, \beta= \pm 1/2$ are of particular interest, since coefficients
can be computed using fast sine and cosine transforms. Banded,
Toeplitz-plus-Hankel multiplication operators are also possible for
representing variable coefficients in a spectral method. In Fourier space these
orthogonal systems are related to an apparently new generalisation of the
Carlitz polynomials.

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The categories of this paper are: hep-th
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The categories of this paper are: cmp-lg
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The categories of this paper are: econ
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The categories of this paper are: comp-gas
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The categories of this paper are: alg-geom
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The categories of this paper are: plasm-ph
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The categories of this paper are: dg-ga
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The categories of this paper are: q-fin
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The categories of this paper are: stat
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The categories of this paper are: chao-dyn
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The categories of this paper are: bayes-an
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The categories of this paper are: eess
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The categories of this paper are: hep-ph
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The categories of this paper are: math-ph
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The categories of this paper are: funct-an
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The categories of this paper are: nucl-ex