4 Fourier expansions

 In order better to understand the poor convergence properties of the Gram-Charlier series, let us first discuss how it is related to the Fourier expansion. A Fourier expansion (Szegö 1978; Suetin 1979; Nikiforov & Uvarov 1988) for any function f(x) in the set of orthogonal polynomials P_n is given by  

$$f(x) \sim \sum_{n=0}^{\infty} a_nP_n(x) \; ,$$ (16)

with  

$$a_n = {1\over h_n}\int_{-\infty}^\infty w(t)f(t)P_n(t) {\rm d}t \; .$$ (17)

Here h_n is the squared norm  

$$h_n = \int_{-\infty}^\infty w(t)P_n^2(t) {\rm d}t \; ,$$ (18)

and  

$$h_n=\sqrt{2\pi}n! \qquad {\rm for} \qquad H\!e_n(x) \; ,$$ (19)

 

$$h_n=\sqrt{\pi}2^n n! \qquad {\rm for } \qquad H_n(x) \; .$$ (20)

Now we can see that the Gram-Charlier series (11) is just the Fourier expansion (16) of f(x)=p(x)/Z(x) in the set of Chebyshev-Hermite polynomials with c_n=(-1)ⁿ a_n.

The properties of the Gram-Charlier approximations of p(x) in Figs. 1 and 2 are to be considered in the general context of the convergence of Fourier expansions. The source of the divergence lies in the sensitivity of the Gram-Charlier series to the behavior of p(x) at infinity - the latter must fall to zero faster than $\exp(-x^2/4)$ for the series to converge (Cramér 1957; Kendall 1952). This is often too restrictive for practical applications. Our example of the $\chi^2$ distribution in (15), with its exponential behavior at infinity, clearly demonstrates this.

The Fourier expansion of p(x)/Z(x) in another set of Hermite polynomials H_n(x) (8) (not in Chebyshev-Hermite polynomials $H\!e_n$(7), as for the Gram-Charlier series) is sometimes used:  

$$p(x) \sim \sum_{n=0}^{\infty} a_nH_n(x)Z(x) \; ,$$ (21)

with  

$$a_n = {\sqrt{\pi}\over 2^{n-1} n!} \int_{-\infty}^\infty Z(t) p(t)H_n(t) {\rm d}t \; .$$ (22)

This series is often called the Gauss-Hermite expansion (see e.g. its application to spectral lines of galaxies in van der Marel & Franx 1993). Examples in Figs. 3 and 4 show its better convergence.

  

Figure 3: The normalized $\chi^2$ PDF (15) for $\nu=5$ (dashed line), and its Gauss-Hermite approximations with 2 and 6 terms in the expansion (solid line)

  

Figure 4: The same as in Fig. 3 but for 12 and 36 terms in the Gauss-Hermite expansion

Van der Marel & Franx (1993) use a theorem due to Myller-Lebedeff on the convergence of the Gauss-Hermite expansion: it converges when $x^3p(x) \rightarrow 0$ for any p(x) with finite and continuous second derivative. Actually, the conditions sufficient for the convergence are better: if p(t) obeys the Lipschitz condition  

$$\vert p(t)-p(x)\vert \leq M\vert t-x\vert^\alpha, \quad M=\mbox{const}, \quad 0<\alpha\leq 1 \; ,$$ (23)

in a vicinity of x and  

$$\int_{-\infty}^\infty\vert p(t)\vert(1+\vert t\vert^{3/2}){\rm d}t < \infty \; ,$$ (24)

then the Gauss-Hermite series converges to p(x) at x (see e.g. Suetin 1979). These weaker conditions imply that the class of PDFs with convergent Gauss-Hermite expansions is much wider than suggested by the Myller-Lebedeff theorem cited in van der Marel & Franx (1993). But the simple relation between the coefficients in the expansion and the moments of the PDF typical for the Gram-Charlier series is now lost (cf. Eqs. (12), (13) and (22)). It might be not important in many practical applications when the moments cannot be accurately determined from observations, but it is very important for a theoretical work based on the analysis of the moments.

Our Figs. 3 and 4 show that the $\chi^2$ PDF is well-suited for approximation by a Gauss-Hermite series. However, one should be cautious about the accuracy of the computation of the Fourier coefficients for higher order terms. We have not encountered the problem of numerical errors in our Gram-Charlier example, since there all coefficients can be calculated analytically. Yet in general care must be taken of the computational accuracy to avoid spurious numerical divergence of a series which converges theoretically.

  

Figure 5: The normalized $\chi^2$ PDF (15) for $\nu=2$ (dashed line), and its Edgeworth-Petrov approximations with 4 and 12 terms in the expansion (solid line)

  

Figure 6: The normalized $\chi^2$ PDF (15) for $\nu=5$ (dashed line), and its Edgeworth-Petrov approximations with 2 and 4 terms in the expansion (solid)