Problem: Let $f: \mathbb{R}^2 \to \mathbb{R}$ and $g: \mathbb{R}^2 \to \mathbb{R}$ satisfies $$ f(y(x)) = g(x) \ \forall x \in \mathbb{R}^2 $$ where $y : \mathbb{R}^2 \to \mathbb{R}^2$ is a map that requires attention in this question.
Suppose $f$ is parameterizable in terms of coefficients $c_i, i=1,\dots,n$ with a set of basis functions $b_i$ $$ f(x) = \sum_{i=1}^{n} c_i b_i(x) \ \forall x \in \mathbb{R}^2. $$ Following the above definitions, we can say $$ g(x) = f(y(x)) = \sum_{i=1}^{n} c_i b_i(y(x)) \ \forall x \in \mathbb{R}^2. $$ Using the basis function representation for $g$ and considering $n$ distinct data points $(x_i, g(x_i))$, we form a linear system $$ \mathrm{g} = \mathrm{B_y c} $$ where $\mathrm{g} \in \mathbb{R}^n$ stores data points $g(x_i)$, $\mathrm{B_y} \in \mathbb{R}^{n \times n}$ is a matrix stores the values of basis functions at $y(x_i)$, and $c \in \mathbb{R}^n$ stores the coefficients.
I would like to find out the conditions on function $y$ that ensures the invertibility of matrix $ \mathrm{B_y}$.
My attempt: Suppose $y$ is an identity map. Assuming $b$ is a set of monomial basis functions of degree at most $(n - 1)$, $B_y$ is known as a Vandermonde matrix. We know from the uniqueness theorem for interpolating polynomial that we obtain a nonsingular Vandermonde matrix if all data points $x_i$ are distinct.
This implies that, for a general function $y$, $B_y$ be invertible if $y$ has a unique map for each distinct point $x_i$. This means that $y$ should be an injective function to preserve the distinctness.
I am so far unable to comment on if the function needs to be a bijective, invertible, or even a differentiable function for $B_y$ be a nonsingular matrix.
Can you please comment on this problem?