Gegenbauer function

Gegenbauer polynomials and coefficients

The Gegenbauer polynomials

c("$\n$", "{C_k^(\\lambda)(x)}_{k = 0}^\\infty")

form a family of orthogonal polynomials on the interval $[-1, 1]$

with respect to the weight function $(1 - x^2)^{\lambda - 1/2}$, for $\lambda > -1/2$, $\lambda \neq 0$. They usually appear when dealing with functions defined on c("$\n$", "$S^{p-1} := {x \\in R^p : ||x|| = 1}$") with index $\lambda = p / 2 - 1$.

The Gegenbauer polynomials are somehow simpler to evaluate for $x = \cos(\theta)$, with $\theta \in [0, \pi]$. This simplifies also the connection with the Chebyshev polynomials ${T_k(x)}_{k = 0}^\infty$, which admit the explicit expression

$T_k(\cos(\theta)) = \cos(k\theta)$. The Chebyshev polynomials appear as the limit of the Gegenbauer polynomials (divided by $\lambda$) when $\lambda$ goes to $0$, so they can be regarded as the extension by continuity of c("$\n$", "{C_k^(p/2 - 1)(x)}_{k = 0}^\\infty") to the case $p = 2$.

For a reasonably smooth function $\psi$ defined on $[0, \pi]$, [REMOVE_ME]$\psi(\theta) = \sum_{k = 0}^\infty b_{k, p}C_k^{(p/2 - 1)}(\cos(\theta)),\psi(\theta) = \sum_{k = 0}^\infty b_{k, p}C_k^(p/2 - 1)(\cos(\theta)), [REMOVE_ME_2]$

provided that the coefficients [REMOVE_ME]$b_{k, p} := \frac{1}{c_{k, p}} \int_0^\pi \psi(\theta)C_k^{(p/2 - 1)}(\cos(\theta)) (\sin(\theta))^{p - 2}\,\mathrm{d}\thetab_{k, p} := \frac{1}{c_{k, p}} \int_0^\pi \psi(\theta)C_k^(p/2 - 1)(\cos(\theta)) (\sin(\theta))^{p - 2} d\theta [REMOVE_ME_2]$

are finite, where the normalizing constants are [REMOVE_ME]$c_{k, p} := \int_0^\pi (C_k^{(p/2 - 1)}(\cos(\theta)))^2(\sin(\theta))^{p - 2} \,\mathrm{d}\theta.c_{k, p} := \int_0^\pi (C_k^(p/2 - 1)(\cos(\theta)))^2(\sin(\theta))^{p - 2} d\theta. [REMOVE_ME_2]$

The (squared) "Gegenbauer norm" of $\psi$ is [REMOVE_ME]$\|\psi\|_{G, p}^2 := \int_0^\pi \psi(\theta)^2C_k^{(p/2 - 1)}(\cos(\theta)) (\sin(\theta))^{p - 2}\,\mathrm{d}\theta.||\psi||_{G, p}^2 := \int_0^\pi \psi(\theta)^2C_k^(p/2 - 1)(\cos(\theta)) (\sin(\theta))^{p - 2} d\theta. [REMOVE_ME_2]$

The previous expansion can be generalized for a 2-dimensional function $\psi$ defined on $[0, \pi] \times [0, \pi]$: [REMOVE_ME] \psi(\theta_1, \theta_2) = \sum_{k = 0}^\infty \sum_{m = 0}^\inftyb_{k, m, p} C_k^{(p/2 - 1)}(\cos(\theta_1))C_k^{(p/2 - 1)}(\cos(\theta_2)),\psi(\theta_1, \theta_2) = \sum_{k = 0}^\infty \sum_{m = 0}^\inftyb_{k, m, p} C_k^(p/2 - 1)(\cos(\theta_1)) C_k^(p/2 - 1)(\cos(\theta_2)) [REMOVE_ME_2]

with coefficients [REMOVE_ME]$b_{k, m, p} := \frac{1}{c_{k, p} c_{m, p}} \int_0^\pi\int_0^\pi\psi(\theta_1, \theta_2) C_k^{(p/2 - 1)}(\cos(\theta_1))C_k^{(p/2 - 1)}(\cos(\theta_2)) (\sin(\theta_1))^{p - 2}(\sin(\theta_2))^{p - 2}\,\mathrm{d}\theta_1\,\mathrm{d}\theta_2.b_{k, m, p} := \frac{1}{c_{k, p} c_{m, p}} \int_0^\pi \int_0^\pi\psi(\theta_1, \theta_2) C_k^(p/2 - 1)(\cos(\theta_1))C_k^(p/2 - 1)(\cos(\theta_2)) (\sin(\theta_1))^{p - 2}(\sin(\theta_2))^{p - 2} d\theta_1 d\theta_2. [REMOVE_ME_2]$

The (squared) "Gegenbauer norm" of $\psi$ is [REMOVE_ME]$\|\psi\|_{G, p}^2 := \int_0^\pi\int_0^\pi \psi(\theta_1, \theta_2)^2C_k^{(p/2 - 1)}(\cos(\theta_1)) C_k^{(p/2 - 1)}(\cos(\theta_2))(\sin(\theta_1))^{p - 2} (\sin(\theta_2))^{p - 2}\,\mathrm{d}\theta_1\,\mathrm{d}\theta_2.||\psi||_{G, p}^2 := \int_0^\pi\int_0^\pi \psi(\theta_1, \theta_2)^2C_k^(p/2 - 1)(\cos(\theta_1)) C_k^(p/2 - 1)(\cos(\theta_2))(\sin(\theta_1))^{p - 2} (\sin(\theta_2))^{p - 2} d\theta_1 d\theta_2. [REMOVE_ME_2]$

Gegen_polyn(theta, k, p)

Gegen_coefs(k, p, psi, Gauss = TRUE, N = 320, normalize = TRUE,
  only_const = FALSE, tol = 1e-06, ...)

Gegen_series(theta, coefs, k, p, normalize = TRUE)

Gegen_norm(coefs, k, p, normalize = TRUE, cumulative = FALSE)

Gegen_polyn_2d(theta_1, theta_2, k, m, p)

Gegen_coefs_2d(k, m, p, psi, Gauss = TRUE, N = 320, normalize = TRUE,
  only_const = FALSE, tol = 1e-06, ...)

Gegen_series_2d(theta_1, theta_2, coefs, k, m, p, normalize = TRUE)

Gegen_norm_2d(coefs, k, m, p, normalize = TRUE)

Arguments

• theta, theta_1, theta_2: vectors with values in $[0, \pi]$.
• k, m: vectors with the orders of the Gegenbauer polynomials. Must be integers larger or equal than 0.
• p: integer giving the dimension of the ambient space $R^p$ that contains $S^{p-1}$.
• psi: function defined in $[0, \pi]$ and whose Gegenbauer coefficients are to be computed. Must be vectorized. For Gegen_coefs_2d, it must return a matrix of size c(length(theta_1), length(theta_2)).
• Gauss: use a Gauss--Legendre quadrature rule of N nodes in the computation of the Gegenbauer coefficients? Otherwise, call integrate. Defaults to TRUE.
• N: number of points used in the Gauss-Legendre quadrature for computing the Gegenbauer coefficients. Defaults to 320.
• normalize: consider normalized coefficients (divided by $c_{k, p}$)? Defaults to TRUE.
• only_const: return only the normalizing constants $c_{k, p}$? Defaults to FALSE.
• tol: tolerance passed to integrate's rel.tol and abs.tol if Gauss = FALSE. Defaults to 1e-6.
• ...: further arguments to be passed to psi.
• coefs: for Gegen_series and Gegen_norm, a vector of coefficients $b_{k, p}$ with length length(k). For Gegen_series_2d and Gegen_norm_2d, a matrix of coefficients $b_{k, m, p}$ with size c(length(k), length(m)). The order of the coefficients is given by k and m.
• cumulative: return the cumulative norm for increasing truncation of the series? Defaults to FALSE.

Returns

• Gegen_polyn: a matrix of size c(length(theta), length(k)) containing the evaluation of the length(k) Gegenbauer polynomials at theta.

• Gegen_coefs: a vector of size length(k) containing the coefficients $b_{k, p}$.

• Gegen_series: the evaluation of the truncated series expansion, a vector of size length(theta).

• Gegen_norm: the Gegenbauer norm of the truncated series, a scalar if cumulative = FALSE, otherwise a vector of size length(k).

• Gegen_polyn_2d: a 4-dimensional array of size c(length(theta_1), length(theta_2), length(k), length(m))

  containing the evaluation of the length(k) * length(m)

  2-dimensional Gegenbauer polynomials at the bivariate grid spanned by theta_1 and theta_2.

• Gegen_coefs_2d: a matrix of size c(length(k), length(m)) containing the coefficients $b_{k, m, p}$.

• Gegen_series_2d: the evaluation of the truncated series expansion, a matrix of size c(length(theta_1), length(theta_2)).

• Gegen_norm_2d: the 2-dimensional Gegenbauer norm of the truncated series, a scalar.

Description

The Gegenbauer polynomials

c("$\n$", "{C_k^(\\lambda)(x)}_{k = 0}^\\infty")

form a family of orthogonal polynomials on the interval $[-1, 1]$

with respect to the weight function $(1 - x^2)^{\lambda - 1/2}$, for $\lambda > -1/2$, $\lambda \neq 0$. They usually appear when dealing with functions defined on c("$\n$", "$S^{p-1} := {x \\in R^p : ||x|| = 1}$") with index $\lambda = p / 2 - 1$.

The Gegenbauer polynomials are somehow simpler to evaluate for $x = \cos(\theta)$, with $\theta \in [0, \pi]$. This simplifies also the connection with the Chebyshev polynomials ${T_k(x)}_{k = 0}^\infty$, which admit the explicit expression

$T_k(\cos(\theta)) = \cos(k\theta)$. The Chebyshev polynomials appear as the limit of the Gegenbauer polynomials (divided by $\lambda$) when $\lambda$ goes to $0$, so they can be regarded as the extension by continuity of c("$\n$", "{C_k^(p/2 - 1)(x)}_{k = 0}^\\infty") to the case $p = 2$.

For a reasonably smooth function $\psi$ defined on $[0, \pi]$,

$$\psi(\theta) = \sum_{k = 0}^\infty b_{k, p}C_k^{(p/2 - 1)}(\cos(\theta)),\psi(\theta) = \sum_{k = 0}^\infty b_{k, p}C_k^(p/2 - 1)(\cos(\theta)),$$

provided that the coefficients

$$b_{k, p} := \frac{1}{c_{k, p}} \int_0^\pi \psi(\theta)C_k^{(p/2 - 1)}(\cos(\theta)) (\sin(\theta))^{p - 2}\,\mathrm{d}\thetab_{k, p} := \frac{1}{c_{k, p}} \int_0^\pi \psi(\theta)C_k^(p/2 - 1)(\cos(\theta)) (\sin(\theta))^{p - 2} d\theta$$

are finite, where the normalizing constants are

$$c_{k, p} := \int_0^\pi (C_k^{(p/2 - 1)}(\cos(\theta)))^2(\sin(\theta))^{p - 2} \,\mathrm{d}\theta.c_{k, p} := \int_0^\pi (C_k^(p/2 - 1)(\cos(\theta)))^2(\sin(\theta))^{p - 2} d\theta.$$

The (squared) "Gegenbauer norm" of $\psi$ is

$$\|\psi\|_{G, p}^2 := \int_0^\pi \psi(\theta)^2C_k^{(p/2 - 1)}(\cos(\theta)) (\sin(\theta))^{p - 2}\,\mathrm{d}\theta.||\psi||_{G, p}^2 := \int_0^\pi \psi(\theta)^2C_k^(p/2 - 1)(\cos(\theta)) (\sin(\theta))^{p - 2} d\theta.$$

The previous expansion can be generalized for a 2-dimensional function $\psi$ defined on $[0, \pi] \times [0, \pi]$:

\psi(\theta_1, \theta_2) = \sum_{k = 0}^\infty \sum_{m = 0}^\inftyb_{k, m, p} C_k^{(p/2 - 1)}(\cos(\theta_1))C_k^{(p/2 - 1)}(\cos(\theta_2)),\psi(\theta_1, \theta_2) = \sum_{k = 0}^\infty \sum_{m = 0}^\inftyb_{k, m, p} C_k^(p/2 - 1)(\cos(\theta_1)) C_k^(p/2 - 1)(\cos(\theta_2))

with coefficients

$$b_{k, m, p} := \frac{1}{c_{k, p} c_{m, p}} \int_0^\pi\int_0^\pi\psi(\theta_1, \theta_2) C_k^{(p/2 - 1)}(\cos(\theta_1))C_k^{(p/2 - 1)}(\cos(\theta_2)) (\sin(\theta_1))^{p - 2}(\sin(\theta_2))^{p - 2}\,\mathrm{d}\theta_1\,\mathrm{d}\theta_2.b_{k, m, p} := \frac{1}{c_{k, p} c_{m, p}} \int_0^\pi \int_0^\pi\psi(\theta_1, \theta_2) C_k^(p/2 - 1)(\cos(\theta_1))C_k^(p/2 - 1)(\cos(\theta_2)) (\sin(\theta_1))^{p - 2}(\sin(\theta_2))^{p - 2} d\theta_1 d\theta_2.$$

The (squared) "Gegenbauer norm" of $\psi$ is

$$\|\psi\|_{G, p}^2 := \int_0^\pi\int_0^\pi \psi(\theta_1, \theta_2)^2C_k^{(p/2 - 1)}(\cos(\theta_1)) C_k^{(p/2 - 1)}(\cos(\theta_2))(\sin(\theta_1))^{p - 2} (\sin(\theta_2))^{p - 2}\,\mathrm{d}\theta_1\,\mathrm{d}\theta_2.||\psi||_{G, p}^2 := \int_0^\pi\int_0^\pi \psi(\theta_1, \theta_2)^2C_k^(p/2 - 1)(\cos(\theta_1)) C_k^(p/2 - 1)(\cos(\theta_2))(\sin(\theta_1))^{p - 2} (\sin(\theta_2))^{p - 2} d\theta_1 d\theta_2.$$

Details

The Gegen_polyn function is a wrapper to the functions gegenpoly_n and gegenpoly_array in the gsl-package, which they interface the functions defined in the header file gsl_sf_gegenbauer.h (documented c("\n", "here")) of the c("\n", "GNU Scientific Library").

Note that the function Gegen_polyn computes the regular unnormalized Gegenbauer polynomials.

For the case $p = 2$, the Chebyshev polynomials are considered.

Examples

## Representation of Gegenbauer polynomials (Chebyshev polynomials for p = 2)

th <- seq(0, pi, l = 500)
k <- 0:3
old_par <- par(mfrow = c(2, 2))
for (p in 2:5) {
  matplot(th, t(Gegen_polyn(theta = th, k = k, p = p)), lty = 1,
          type = "l", main = substitute(p == d, list(d = p)),
          axes = FALSE, xlab = expression(theta), ylab = "")
  axis(1, at = c(0, pi / 4, pi / 2, 3 * pi / 4, pi),
       labels = expression(0, pi / 4, pi / 2, 3 * pi / 4, pi))
  axis(2); box()
  mtext(text = expression({C[k]^{p/2 - 1}}(cos(theta))), side = 2,
        line = 2, cex = 0.75)
  legend("bottomleft", legend = paste("k =", k), lwd = 2, col = seq_along(k))
}
par(old_par)

## Coefficients and series in p = 2

# Function in [0, pi] to be projected in Chebyshev polynomials
psi <- function(th) -sin(th / 2)

# Coefficients
p <- 2
k <- 0:4
(coefs <- Gegen_coefs(k = k, p = p, psi = psi))

# Series
plot(th, psi(th), type = "l", axes = FALSE, xlab = expression(theta),
      ylab = "", ylim = c(-1.25, 0))
axis(1, at = c(0, pi / 4, pi / 2, 3 * pi / 4, pi),
     labels = expression(0, pi / 4, pi / 2, 3 * pi / 4, pi))
axis(2); box()
col <- viridisLite::viridis(length(coefs))
for (i in seq_along(coefs)) {
  lines(th, Gegen_series(theta = th, coefs = coefs[1:(i + 1)], k = 0:i,
                         p = p), col = col[i])
}
lines(th, psi(th), lwd = 2)

## Coefficients and series in p = 3

# Function in [0, pi] to be projected in Gegenbauer polynomials
psi <- function(th) tan(th / 3)

# Coefficients
p <- 3
k <- 0:10
(coefs <- Gegen_coefs(k = k, p = p, psi = psi))

# Series
plot(th, psi(th), type = "l", axes = FALSE, xlab = expression(theta),
      ylab = "", ylim = c(0, 2))
axis(1, at = c(0, pi / 4, pi / 2, 3 * pi / 4, pi),
     labels = expression(0, pi / 4, pi / 2, 3 * pi / 4, pi))
axis(2); box()
col <- viridisLite::viridis(length(coefs))
for (i in seq_along(coefs)) {
  lines(th, Gegen_series(theta = th, coefs = coefs[1:(i + 1)], k = 0:i,
                         p = p), col = col[i])
}
lines(th, psi(th), lwd = 2)

## Surface representation

# Surface in [0, pi]^2 to be projected in Gegenbauer polynomials
p <- 3
psi <- function(th_1, th_2) A_theta_x(theta = th_1, x = cos(th_2),
                                      p = p, as_matrix = TRUE)

# Coefficients
k <- 0:20
m <- 0:10
coefs <- Gegen_coefs_2d(k = k, m = m, p = p, psi = psi)

# Series
th <- seq(0, pi, l = 100)
col <- viridisLite::viridis(20)
old_par <- par(mfrow = c(2, 2))
image(th, th, A_theta_x(theta = th, x = cos(th), p = p), axes = FALSE,
      col = col, zlim = c(0, 1), xlab = expression(theta[1]),
      ylab = expression(theta[2]), main = "Original")
axis(1, at = c(0, pi / 4, pi / 2, 3 * pi / 4, pi),
     labels = expression(0, pi / 4, pi / 2, 3 * pi / 4, pi))
axis(2, at = c(0, pi / 4, pi / 2, 3 * pi / 4, pi),
     labels = expression(0, pi / 4, pi / 2, 3 * pi / 4, pi))
box()
for(K in c(5, 10, 20)) {
  A <- Gegen_series_2d(theta_1 = th, theta_2 = th,
                       coefs = coefs[1:(K + 1), ], k = 0:K, m = m, p = p)
  image(th, th, A, axes = FALSE, col = col, zlim = c(0, 1),
        xlab = expression(theta[1]), ylab = expression(theta[2]),
        main = paste(K, "x", m[length(m)], "coefficients"))
  axis(1, at = c(0, pi / 4, pi / 2, 3 * pi / 4, pi),
       labels = expression(0, pi / 4, pi / 2, 3 * pi / 4, pi))
  axis(2, at = c(0, pi / 4, pi / 2, 3 * pi / 4, pi),
       labels = expression(0, pi / 4, pi / 2, 3 * pi / 4, pi))
  box()
}
par(old_par)

References

Galassi, M., Davies, J., Theiler, J., Gough, B., Jungman, G., Alken, P., Booth, M., and Rossi, F. (2009) GNU Scientific Library Reference Manual. Network Theory Ltd. http://www.gnu.org/software/gsl/

NIST Digital Library of Mathematical Functions. Release 1.0.20 of 2018-09-15. F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, and B. V. Saunders, eds. https://dlmf.nist.gov/