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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?

Gyu Eun Lee
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user157623
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    For $n=1$ you are right. For $n>1$ it becomes complicated... – user251257 Aug 11 '15 at 01:01
  • For $n\geq1$ can I think of a tensor product of monomial basis to form a basis and do something similar, I know of multivariate polynomial interpolation, but I am also looking for qualitative answers. Like a general result of existence even without an actual way of computing. – user157623 Aug 11 '15 at 01:04
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    try to interpolate $((-1,0), 0), ((0,0), 1), ((1,0), 0)$ with an affine function $\mathbb R^2\to\mathbb R$ (degree of freedom is $3$). In case of $n > 1$ the vandermonde matrix need not be non-singular for distinct sites. You could use some kind of bell function with compact support. But probably it is not what you have expected. – user251257 Aug 11 '15 at 01:12

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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.

  • Thanks for the answer! One question, your answer seems to work well for n=1, does it also work for n>1. – user157623 Aug 13 '15 at 22:15
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    Same thing. Bump functions are available in all dimensions; typically they are taken to be radially symmetric. (By the way: there is a much studied topic of interpolation with Radial Basis Functions that are not necessarily compactly supported -- Gaussian for example -- which makes the problem harder.) –  Aug 13 '15 at 22:18
  • Great answer thanks – user157623 Aug 13 '15 at 22:28