update PiecewisePolynomialKernel

Author: st--

docs/src/kernels.md

@@ -155,11 +155,19 @@ where $r$ has the same dimension as $x$ and $r_i > 0$.
 
 The [`PiecewisePolynomialKernel`](@ref) is defined for $x\in \mathbb{R}^D$ and $V \in \{0,1,2,3\}$ as
 ```math
-  k(x,x'; P, V) &= \max(1 - r, 0)^{j + V} f(r, j),
+  k(x,x'; P, V) = \max(1 - r, 0)^{j + V} f_V(r, j),
+```
+where $r = x^\top P x'$ (with $P$ a positive-definite matrix) and $j = \lfloor \frac{D}{2}\rfloor + V + 1$.
+
+The polynomials $f_V$ are defined as follows:
+````math
+\begin{aligned}
+    f_0(r, j) &= 1, \\
+    f_1(r, j) &= 1 + (j + 1) r, \\
+    f_2(r, j) &= 1 + (j + 2) r + ((j^2 + 4j + 3) / 3) r^2, \\
+    f_3(r, j) &= 1 + (j + 3) r + ((6 j^2 + 36j + 45) / 15) r^2 + ((j^3 + 9 j^2 + 23j + 15) / 15) r^3.
+\end{aligned}
 ```
-where $r = x^\top P x'$ (with $P$ a positive-definite matrix),
-$j = \lfloor \frac{D}{2}\rfloor + V + 1$, and
-$f$ is a piecewise polynomial (see source code).
 
 ## Polynomial Kernels
 

src/basekernels/piecewisepolynomial.jl

@@ -2,13 +2,12 @@
     PiecewisePolynomialKernel{V}(maha::AbstractMatrix)
 
 Piecewise Polynomial covariance function with compact support, V = 0,1,2,3.
-The kernel functions are 2v times continuously differentiable and the corresponding
-processes are hence v times  mean-square differentiable. The kernel function is:
+The kernel functions are 2V times continuously differentiable and the corresponding
+processes are hence V times mean-square differentiable. The kernel function is:
 ```math
     κ(x, y) = max(1 - r, 0)^(j + V) * f(r, j) with j = floor(D / 2) + V + 1
 ```
 where `r` is the Mahalanobis distance mahalanobis(x,y) with `maha` as the metric.
-
 """
 struct PiecewisePolynomialKernel{V, A<:AbstractMatrix{<:Real}} <: SimpleKernel
     maha::A