I am wondering if the following problem has an affirmative answer: Does any finite dataset can be interpolated by a continuous and smooth function. Formally, let $E\subset \mathbb{R}^{n}$ be a finite set of cardinality N and let $f:E\mapsto\mathbb{R}$ be a dataset. Is there always a smooth function $F\in\mathbf{C}^\infty(\mathbb{R}^n,\mathbb{R})$ such that the restriction to the domain E of F interpolates the data $F|_{E}=f$.
I think the answer is Yes, and it is trivial because I can think of a basis of monomials and form a matrix with this basis and solve the weights linearly and obtain a polynomial interpolator of an arbitrary degree for $p\geq N$.Is my understanding correct? I am missing something?
Yes, but it's easier to use bump functions than polynomials. Pick $r>0$ small enough so that the $r$-neighborhoods of the elements of $ E$ are disjoint. For each element $e\in E$, take a $C^\infty$ bump that is equal to $1$ at $e$ and is supported in its $r$-neighborhood. Multiply by $f(e)$. Add these together.