How can I construct a C^{\infty} approximation to a tent function?

[Hurkyl]

Today, I had the desire to construct a C^{\infty} approximation to a tent function. Specifically, for any positive real number e I want a C^{\infty} function f such that:

f(x) = 0 if |x| > 1 + e
|f(x) - g(x)| < e for all x

where g(x) is the tent function given by:

<br /> g(x) =<br /> \begin{cases}<br /> 0 &amp; |x| \geq 1 \\<br /> 1 - |x| &amp; |x| \leq 1<br /> \end{cases}<br />

I'm willing to accept on faith that such things exist, but it struck me today that I don't know how to go about constructing such a thing, or at least proving its existence.

Given time I could probably figure it out, but I'm interested in a different problem (for which I want to use this), and I imagine this is a well-known thing.

So I guess what I'm looking for is at least a "yes" or "no" answer to the existence of such a function, but a hint as to the proof would be nice too.

 

[AKG]

I have a theorem which states:

Let g\,:\,\mathbb{R}^m\to\mathbb{R} be a bounded, uniformly continuous function. Then

(\forall\epsilon &gt; 0)(\exists f\in C^{\infty}(\mathbb{R}^m,\,\mathbb{R}))(\forall x \in \mathbb{R}^m)(|g(x) - f(x)| &lt; \epsilon)

Define a function \sigma _0\,:\,x\mapsto\exp\left(\frac{-1}{x+1}\right) for x > -1 and x\mapsto 0 otherwise. Define a function \sigma\,:\,x\mapsto\sigma_0(x)\sigma_0(-x). Finally, define \beta\,:\,\mathbb{R}^m\to\mathbb{R} by:

\beta (x) = \frac{\sigma (|x|)}{\int _{|y|&lt;1}\sigma (|y|)dy}

By uniform continuity of g, given \epsilon &gt; 0, you can pick \delta &gt; 0 such that:

(\forall x,z \in \mathbb{R}^m)(|x-z|&lt;\delta \Rightarrow |g(x)-g(z)|&lt;\epsilon)

The function f that you want is:

f(x) = \int _{\mathbb{R}^m}g(x+\delta y)\beta (y)dy

 

[Hurkyl]

I'm not sure if that's good enough for the problem of interest. And there has to be a typo in that, because g is essentially never used! (was it supposed to be in the final integral?)


The problem I want to solve (which I'm also sure is well-known by those who know it well -- but I actually want to enjoy this one!) is the following:

Suppose I have a "nice" space of functions R²->R, with some topology. I want to prove that the subspace spanned by the functions of the form f(x)g(y) for "nice" f and g is dense in the original space.

So, I take a "nice" function on two variables, and I want to find a sequence of functions in that subspace that converge to it.

My plan of attack was to form a polyhedral approximation to the function, where each polyhedron is the product of two line segments. I can then decompose the polyhedra into (a product of) tent functions.

Or, equivalently, I pick a (sufficiently fine) lattice of points in R², and use (sums of) (products of) tent functions to interpolate between points.

I then want to smooth my tent functions into "nice" functions, so that I get actual elements of the subspace of interest, and then try to produce a sequence that converges to the target function in the "nice" way specified by the topology.


One example of the class of "nice" functions of interest are C-infinity functions with compact support. Another would be the "rapidly decreasing" functions: if you take any derivative of your function, and multiply by any polynomial, the result is bounded.

The approximation in your post would seem to be inadequate for these tasks.

 

[AKG]

I'm not sure if that's good enough for the problem of interest. And there has to be a typo in that, because g is essentially never used! (was it supposed to be in the final integral?)

Yes, there was a typo; I've fixed it. The reason was that my theorem had f given, and g was the desired function, so in rewriting it to match your notation I missed a g.

Anyways, would using a Taylor approximation help with your problem?

 

[Hurkyl]

I had thought about it, but had initially dismissed it because C-infinity functions are generally not equal to their Taylor series. The partial series are polynomials, which are distinctly not nice, due to their behavior "at infinity". I think, maybe, they'd be useful if I was only working over a subset of R².


Hrm, that does give me an idea. (Yes, the chain of reasoning did start with Taylor series. )

Maybe I can construct a sequence of "nice" functions that approximate a delta function. I could make a slight variation to everyone's favorite non-analytic smooth function, to produce:

<br /> f(x) :=<br /> \begin{cases}<br /> 0 &amp; x \leq 0 \\<br /> e^{-1/x^2} e^{-1 / (x - 1)^2} &amp; 0 \leq x \leq 1 \\ <br /> 0 &amp; 1 \leq x<br /> \end{cases}<br />

this would give me a C-infinity function that is zero outside of (0, 1). By playing with constants, I ought to be able to produce a sequence that would approximate a delta function, and if I convolve one with a tent function, maybe I'll get what I seek?

 

[Hurkyl]

Never start an analysis problem after bedtime.

Yes, this idea works. If I define:

<br /> h_d(x) :=<br /> \begin{cases}<br /> K_d e^{-1/(x-d)^2} e^{-1/(x+d)^2} &amp; |x| &lt; d \\<br /> 0 &amp; |x| \geq d<br /> \end{cases}<br />

then I have a C-infinity function that is nonzero outside the compact interval [-d, d]. Furthermore, it is strictly positive in (-d, d), and the constant K_d is chosen so that the integral over [-d, d] is equal to 1.

So as d goes to zero, this approaches a delta function, so I think it can be used for smoothing. (And yes, I think this is a similar idea to what you posted)

If I want to smooth the uniformly continuous function f(x), I can define:

<br /> g_d(x) := \int_{-d}^{d} f(x - y) h_d(y) \, dy<br />

Then, if I pick \delta such that |x - y| &lt; \delta \Rightarrow |f(x) - f(y)| &lt; \epsilon, we can upper and lower bound f(x - y), giving:

<br /> f(x) - \epsilon &lt; g_{\delta}(x) &lt; f(x) + \epsilon<br />

So is this C-infinity? Well, we can apply a change of variable:

<br /> g_d(x) = \int_{x-d}^{x+d} f(z) h_d(x - z) \, dz<br />

Bleh, it's been a while... oh bleh, I had the derivative slightly wrong. (But I was just missing multipliers of "1") Found it at Wikipedia

<br /> g_d&#039;(x) = f(x+d) h_d(-d) - f(x-d) h_d(d) + \int_{x-d}^{x+d} f(z) \frac{\partial}{\partial x}h_d(x - z) \, dz<br /> = \int_{x-d}^{x+d} f(z) \frac{\partial}{\partial x}h_d(x - z) \, dz<br />

and repeating,

<br /> \left( \frac{d}{dx} \right)^n g_d(x)<br /> = \int_{x-d}^{x+d} f(z) \left( \frac{\partial}{\partial x} \right)^n h_d(x - z) \, dz<br />


and therefore, I can produce a smooth approximation of a tent function!

That was actually more fun than I thought it would be!