**2. Orthogonal Polynomials**

Let V*n* be an *n*-dimensional vector space. A basis of this vector space is the minimal set of vectors that spans the vector space. An orthogonal basis is a subset of bases consisting of exactly *n* basis vectors such that the inner product between any two basis vectors, *βm* and *β<sup>n</sup>*, is proportional to the Kronecker delta (*δmn*). Given mathematically with angle brackets, this orthogonal inner product takes the following form:

$$
\langle \beta\_{m\nu} \beta\_n \rangle = \mathfrak{c} \delta\_{mn} \mathfrak{c}
$$

where *m* and *n* are part of the set of positive integers, including zero. In the event *c* = 1, the set is termed orthonormal. The *i*th standard basis vector of V*n* is generally the *i*th vector of the *n*-dimensional identity; however, there are infinitely many bases for each vector space. It should be noted that it is not a requirement that a basis be orthogonal, merely linearly independent; however, the use of non-orthogonal bases is practically unheard-of.

An element *α* ∈ V can be expressed in terms of an ordered basis B = {*β*1, *β*2, ... , *βn*}, as the linear combination

$$
\alpha = a\_1 \beta\_1 + a\_2 \beta\_2 + \dots + a\_n \beta\_n \, , \tag{1}
$$

where [*<sup>a</sup>*1, *a*2, ... , *an*] is the coordinate of *α*. While any set of independent vectors can be used as a basis, different bases can prove beneficial—possibly by making the system more intuitive or more mathematically straightforward. When expressing the state of a physical system, the selection of a coordinate frame is effectively choosing a basis for the inhabited vector space. Consider a satellite in orbit. If the satellite's ground track is of high importance (such as weather or telecommunications satellites), an Earth-fixed frame would be ideal. However, in cases where a satellite's actions are

dictated by other space-based objects (such as proximity operations), a body-fixed frame would be ideal.

It is common to constrict the term vector space to the spaces that are easiest to visualize, most notably a Cartesian space, where the bases are vectors radiating from the origin at right angles. The term vector space is much more broad than this though. A vector space need only contain the zero element and be closed under both scalar addition and multiplication, which applies to much more than vectors.

Most notable in this work is the idea of a polynomial vector space. Let P*n*+<sup>1</sup> be an (*n*+<sup>1</sup>)-dimensional vector space made up of all polynomials of positive degree *n* or less with standard basis B = {1, *x*, ... , *<sup>x</sup><sup>n</sup>*}. The inner product with respect to the function *ω* on the real-valued polynomial space is given by

$$\langle f(\mathbf{x}), \mathbf{g}(\mathbf{x}) \rangle\_{\omega(\mathbf{x})} = \int\_{\mathbb{B}} f(\mathbf{x}) \mathbf{g}(\mathbf{x}) d\omega(\mathbf{x}) \,\mathrm{d}\omega$$

where *ω*(*x*) is a non-decreasing function with support S and *f* and *g* are any two polynomials of degree *n* or less. A polynomial family <sup>Φ</sup>(*x*) is a set of polynomials with monotonically increasing order that are orthogonal. The orthogonality condition is given mathematically as

$$
\langle \phi\_m(\mathbf{x}), \phi\_n(\mathbf{x}) \rangle\_{\omega(\mathbf{x})} = \int\_{\mathbb{S}} \phi\_m(\mathbf{x}) \phi\_n(\mathbf{x}) d\omega(\mathbf{x}) = 0 \tag{2a}
$$

$$
\langle \phi\_m^2(\mathbf{x}) \rangle\_{\omega(\mathbf{x})} = \int\_{\mathbb{S}} \phi\_m(\mathbf{x}) \phi\_m(\mathbf{x}) d\omega(\mathbf{x}) = \mathbf{c},\tag{2b}
$$

where *φk*(*x*) is the polynomial of order *k*, *c* is a constant, and S is the support of the non-decreasing function *<sup>ω</sup>*(*x*). Note that while polynomials of negative orders (*k* < 0), referred to as Laurent polynomials, exist, they are not covered in this work.

The most commonly used polynomial families are categorized in the Askey scheme, which groups the polynomials based on the generalized hypergeometric function (*pFq*) from which they are generated [29–31]. Table 1 lists some of the polynomial families, their support, the non-decreasing function they are orthogonal with respect to (commonly referred to as a weight function), and the hypergeometric function they can be written in terms of. For completeness, Table 1 lists both continuous and discrete polynomial groups; however, the remainder of this work only considers continuous polynomials.


**Table 1.** Common Orthogonal Polynomials.

#### *2.1. Polynomials Orthogonal on the Unit Circle*

While the Askey polynomials are useful in many applications, their standard forms place them in the polynomial ring <sup>R</sup>[*x*], or all polynomials with real-valued coefficients that are closed under polynomial addition and multiplication. These polynomials are orthogonal with respect to measures on the real line. In the event that a set of polynomials orthogonal with respect a measure on a curved interval (e.g., the unit circle) is desired, the Askey polynomials would be insufficient.

In [32], Szeg˝o uses the connection between points on the unit circle and points on a finite real interval to develop polynomials that are orthogonal on the unit circle. Polynomials of this type are now known as Szeg˝o polynomials. Since the unit circle is defined to have unit radius, every point can be described on a real interval of length 2*π* and mapped to the complex variable *ϑ* = *<sup>e</sup>iθ*, where *i* is the imaginary unit. All use of the variable *ϑ* in the following corresponds to this definition. The orthogonality expression for the Szeg˝o polynomials, *φk*(*ϑ*), is

$$\langle \phi\_{\mathfrak{m}}(\theta), \phi\_{\mathfrak{n}}(\theta) \rangle\_{\omega(\theta)} = \frac{1}{2\pi} \int\_{-\pi}^{\pi} \phi\_{\mathfrak{m}}(\theta) \overline{\phi\_{\mathfrak{n}}(\theta)} \omega(\theta) d\theta = \delta\_{mn}\omega$$

where *φn*(*ϑ*) is the complex conjugate of *φn*(*ϑ*) and *ω*(*θ*) is the monotonically increasing weight function over the support. Note that, as opposed to Equation (2a), the Kronecker delta is not scaled, implying all polynomials using Szeg˝o's formulation are orthonormal.

While the general formulation outlined by Szeg˝o is cumbersome—requiring the calculation of Fourier coefficients corresponding to the weight function and large matrix determinants—it does provide a framework for developing a set of polynomials orthogonal with respect to any conceivable continuous weight function. In addition to the initial research done by Szeg˝o, further studies have investigated polynomials orthogonal on the unit circle [33–38].

Fortunately, there exist some polynomial families that are given explicitly, such as the Rogers-Szeg˝o polynomials. The Rogers-Szeg˝o polynomials have been well-studied [39–41] and were developed by Szeg˝o based on work done by Rogers over the *q*-Hermite polynomials. For a more detailed description of the relationship between the Askey scheme of polynomials and their *q*-analog, the reader is encouraged to reference [31,42].

The generating function for the Rogers-Szeg˝o polynomials is given as

$$\phi\_n(\vartheta; q) = \sum\_{k=0}^n \binom{n}{k}\_q \vartheta^k \, , \tag{3}$$

where (*nk*)*q*is the *q*-binomial

$$\binom{n}{k}\_q = \frac{(q;q)\_n}{(q;q)\_k (q;q)\_{n-k}} \qquad \text{and} \qquad \qquad (a;q)\_n = \prod\_{j=0}^{n-1} (1-aq^j) \cdot 1$$

The weight function that satisfies the orthogonality condition of these polynomials is

$$\omega(\theta) = \frac{1}{\sqrt{-2\pi \ln(q)}} \sum\_{j=-\infty}^{\infty} \exp\left\{ \frac{(\theta + 2\pi j)^2}{2\ln(q)} \right\} \qquad 0 < q < 1. \tag{4}$$

In addition to the generating function, a three-step recurrence [43] exists, which is given by

$$
\phi\_{n+1}(\theta;q) = (1+\theta)\phi\_n(\theta;q) - (1-q^n)\theta\phi\_{n-1}(\theta;q) \,. \tag{5}
$$

For convenience, the first five polynomials are:

$$\begin{aligned} \phi\_0 &= 1 \\ \phi\_1 &= \theta + 1 \\ \phi\_2 &= \theta^2 + (q+1)\theta + 1 \\ \phi\_3 &= \theta^3 + (q^2+q+1)\theta^2 + (q^2+q+1)\theta + 1 \\ \phi\_4 &= \theta^4 + (q+1)(q^2+1)\theta^3 + (q+1)(q^2+q+1)\theta^2 + (q+1)(q^2+1)\theta + 1 \end{aligned}$$

As is apparent, the *q*-binomial term causes the coefficients to be symmetric, which eases computation, and additionally, the polynomials are naturally monic.

#### *2.2. Distributions on the Unit Circle*

With the formulation of polynomials orthogonal on the unit circle, the weight function *ω*(*θ*) has been continuously mentioned but not specifically addressed. In the general case, the weight function can be any non-decreasing function; however, the most common polynomial families are those that are orthogonal with respect to well-known pdfs, such as the ones listed in Table 1. Because weight functions must exist over the same support as the corresponding polynomials, pdfs over the unit circle are required for polynomial orthogonal on the unit circle.

#### 2.2.1. Von Mises Distribution

One of the most common distributions used in directional statistics is the von Mises/von Mises-Fisher distribution [44–46]. The von Mises distribution lies on S1 (the subspace of R<sup>2</sup> containing all points that are unit distance from the origin), whereas the von Mises-Fisher distribution has extensions into higher dimensional spheres. The circular von Mises pdf is given as [24]

$$p\_{\mathfrak{m}}(\theta;\mu,\kappa) = \frac{e^{\mathfrak{x}\cos(\theta-\mu)}}{2\pi l\_0(\kappa)}\,'$$

where *μ* is the mean angular direction on a 2 *π* interval (usually [− *π*, *π*]), *κ* ≥ 0 is a concentration parameter (similar to the inverse of the standard deviation), and *I*0 is the zeroth order modified Bessel function of the first kind. The reason this distribution is so common is its close similarity to the normal distribution. This can be seen in Figure 1a, where von Mises distributions of various concentration parameters are plotted.

 **Figure 1.** Common Circular probability density functions (pdfs). (**a**) Circular von Mises distribution for multiple values of *κ*. (**b**) Wrapped normal distribution for multiple values of *σ*.
