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For a continuous function $f:[a,b]\to R$ there is a natural and obvious procedure to approximate it with a sequence of continuous, piecewise linear functions: take $N$ equally spaced points in $[a,b]$ and join the corresponding points in the graph of $f$ with segments. If $f$ is $C^1$, this method has the additional benefit that it converges in the Lipschitz norm, without any tweaking.

Is there an equally simple procedure for functions of several variables? What is the most natural definition of a continuous, piecewise linear approximation for a continuous function $f:[a,b]^2\to R$, having the 'best possible' properties concerning $Df$ when $f$ is assumed to be $C^1$? The last part of the question is vague since I do not know exactly what to expect; ideally, I would like the approximating sequence to converge also in the Lipschitz norm. Note that piecewise linear functions, even in 1D, are not dense in Lip, but I want to approximate a $C^1$ function.

Background: I am writing some notes for an advanced calculus course, and a simple piecewise linear approximation procedure would greatly simplify some proofs. Since this must be a very familiar problem to people working in more applied fields, I thought it might be a good idea to ask them before rediscovering the wheel.

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  • $\begingroup$ In 2D a natural analogue of your procedure is to subdivide $[a, b]^2$ into triangles and join the corresponding points in the graph by the unique triangle passing through them. $\endgroup$
    – Qiaochu Yuan
    Commented Sep 12, 2016 at 7:49
  • $\begingroup$ But what about the convergence in Lip? $\endgroup$
    – Piero D'Ancona
    Commented Sep 12, 2016 at 7:50
  • $\begingroup$ Does'nt it do your job to approximate the derivative the way you describe and then integrate ? $\endgroup$
    – Duchamp Gérard H. E.
    Commented Sep 12, 2016 at 8:40
  • $\begingroup$ Yes but how to integrate? not obvious how to get a continuous piecewise linear function $\endgroup$
    – Piero D'Ancona
    Commented Sep 12, 2016 at 8:45
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    $\begingroup$ I think the "applied" solution to this is that instead of using piecewise linear approximations, they use piecewise $n$-linear approximations for most direct interpolation needs. (Or piecewise $n$-cubic if they want something that is $C^1$.) For 2D this would be bilinear approximations (4 data points and 4 unknowns) on the square. // I need to think a bit about whether Lip convergence holds; it should follow from the fact that $C^1$ implies strong differentiability. $\endgroup$
    – Willie Wong
    Commented Sep 12, 2016 at 15:54

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