*3.5. Szeg˝o-Chaos*

For the complex angular case, the chaos expansion is transformed slightly, such that

$$\varepsilon(\mathbf{x}, \boldsymbol{\theta}) = \sum\_{k=0}^{\infty} \varepsilon\_k(\mathbf{x}) \overline{\Psi\_k(\boldsymbol{\theta})},\tag{21}$$

where, once again, *ϑ* = *<sup>e</sup>iθ*. The complex conjugate is not required in Equation (21), but it must be remembered that the expansion must be projected onto the *conjugate* of the expansion basis in Equation (20b). While ultimately a matter of choice, it is more convenient to express the expansion in terms of the conjugate basis, rather than the original basis.

Unfortunately, while the first moment is calculated the same for real and complex valued polynomials, the real valued process does not extend to complex valued polynomials. This is because of the slightly different orthogonality condition between real and complex valued polynomials. While the inner product given in Equation (2a) is not incorrect, it is only valid for real valued polynomials. The true inner product of two functions contains a complex conjugate, that is

$$\langle \Psi\_{\mathfrak{m}\prime} \Psi\_{\mathfrak{n}} \rangle\_{p(\mathfrak{x})} = \int\_{\mathfrak{X}} \Psi\_{\mathfrak{m}}(\mathfrak{x}) \overline{\Psi\_{\mathfrak{n}}(\mathfrak{x})} p(\mathfrak{x}) d\mathfrak{x} = c \,\, \delta\_{\mathfrak{m}\mathfrak{n}} \dots$$

The difference between <sup>R</sup>[*x*] and C[*x*] is that the complex conjugate has no effect on <sup>R</sup>[*x*]. Fortunately, the zeroth polynomial of the Szeg˝o polynomials is unitary just like the Askey polynomials. The complex conjugate has no effect; therefore the zeroth polynomial has no imaginary component and is calculated the same for complex and purely real valued random variables.

The complex conjugate of a real valued function has no effect; therefore, the first moment takes the form;

$$\mu\_1 = \sum\_{k=0}^{\infty} \varepsilon\_k(\mathbf{x}) \int\_{\mathcal{X}} \Psi\_k(\xi) \Psi\_0(\xi) p(\xi) d\xi = \sum\_{k=0}^{\infty} \varepsilon\_k(\mathbf{x}) \int\_{\mathcal{X}} \Psi\_k(\xi) \overline{\Psi\_0(\xi)} p(\xi) d\xi. \tag{22}$$

In general, calculation of the second raw moment and the covariance cannot be simplified beyond

$$\begin{split} \mu\_{2} &= \sum\_{j=0}^{\infty} \sum\_{k=0}^{\infty} \varepsilon\_{j} \varepsilon\_{k} \int\_{\mathcal{X}} \overline{\overline{\Psi\_{j}(\boldsymbol{\xi})} \Psi\_{k}(\boldsymbol{\xi})} p(\boldsymbol{\xi}) d\boldsymbol{\xi} \\ &= \sum\_{j=0}^{\infty} \sum\_{k=0}^{\infty} \varepsilon\_{j} \varepsilon\_{k} \left< \overline{\Psi\_{j}(\boldsymbol{\xi})}, \Psi\_{k}(\boldsymbol{\xi}) \right>\_{p(\boldsymbol{\xi})} \end{split} \tag{23}$$

$$\text{cov}(\mathbf{x}, z) = \sum\_{j=1}^{\infty} \sum\_{k=1}^{\infty} a\_j \beta\_k \int\_{\mathcal{X}} \int\_{\mathcal{Z}} \overline{\Psi\_j(\boldsymbol{\xi}) \Phi\_k(\boldsymbol{\xi})} p(\boldsymbol{\zeta}, \boldsymbol{\xi}) d\boldsymbol{\zeta} d\boldsymbol{\xi} \,\tag{24}$$

The simplification from Equation (14) to Equation (15) as a result of shared bases can similarly be applied to Equation (24). This simplifies Equation (24) to a double summation but only a singular inner product (i.e., integral), i.e.,

$$\begin{split} \text{cov}(\boldsymbol{x}, \boldsymbol{z}) &= \sum\_{j=1}^{\infty} \sum\_{k=1}^{\infty} \alpha\_{j} \beta\_{k} \int\_{\mathcal{X}} \overline{\Psi\_{j}(\boldsymbol{\xi})} \Psi\_{k}(\boldsymbol{\xi}) \, d\boldsymbol{\xi} \\ &= \sum\_{j=1}^{\infty} \sum\_{k=1}^{\infty} \alpha\_{j} \beta\_{k} \langle \overline{\Psi\_{j}(\boldsymbol{\xi})}, \Psi\_{k}(\boldsymbol{\xi}) \rangle\_{p(\boldsymbol{\xi})} . \end{split}$$

The familiar expressions for the second raw moment given in Equation (10b) and the covariance given in Equation (16) are special cases for <sup>R</sup>[*x*] rather than general expressions.
