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Let $0\leq f\in\mathscr{D}(\mathbb{R}^n)$. As shown e.g. by J.-M. Bony, F. Broglia, F. Colombini and L. Pernazza, Nonnegative functions as squares or sums of squares, J. Funct. Anal. 232 (2006) 137-147 (see also this MO question and this math.SE question), not every such an $f$ is of the form $f=g^2$ for $g\in\mathscr{D}(\mathbb{R}^n)$ or even a finite sum of such. On the other hand, as mentioned in the paper above, C. Fefferman and D. H. Phong sketched a proof (On positivity of pseudo-differential operators, Proc. Natl. Acad. Sci. U.S.A. 75 (1978) 4673-4674) of the fact that any $0\leq f\in\mathscr{C}^\infty(\mathbb{R}^n)$ can be written as a sum of squares $$f=\sum^k_{j=1}g_j^2$$ with $g_j\in\mathscr{C}^{1,1}(\mathbb{R}^n)$ (i.e. $g_j$ is a differentiable function whose derivatives are locally Lipschitz) for all $1\leq j\leq k$ for some $k\in\mathbb{N}$. This fact was a key ingredient of the proof of the important inequality for scalar pseudodifferential operators with non-negative symbols that bears their name. For a modern, more detailed proof of the above formula, see N. Lerner, Some Facts About the Wick Calculus, in L. Rodino, M.W. Wong (eds.), Pseudodifferential Operators: Quantization and Signals, Lectures given at the C.I.M.E. Summer School held in Cetraro, Italy June 19–24, 2006 (Springer Lecture Notes in Mathematics 1949, 2008), pp. 135-174, particularly Theorem 5.2, pp. 167-172, and the discussion right after Theorem 1.1 in the paper by Bony et alii above on page 139. It immediately follows by multiplication by squares of smooth bump functions that the $g_j$'s may be chosen to be compactly supported if $f\in\mathscr{D}(\mathbb{R}^n)$. Bony et alii showed above that this regularity for $n\geq 4$ is sharp.

All this leads naturally to the following

Question: Is every $0\leq f\in\mathscr{D}(\mathbb{R}^n)$ the limit of a sequence of sums of squares in $\mathscr{D}(\mathbb{R}^n)$ in the latter's topology?

In other words, is the cone of non-negative elements of $\mathscr{D}(\mathbb{R}^n)$ the (sequential) closure of the cone of sums of squares in $\mathscr{D}(\mathbb{R}^n)$?

I am particularly interested in arguments that do not rely on the result by Fefferman and Phong.

EDIT - Follow-up subquestion: (suggested by André Henriques) Is every $0\leq f\in\mathscr{D}(\mathbb{R}^n)$ the limit of an increasing sequence of sums of squares in $\mathscr{D}(\mathbb{R}^n)$ in the latter's topology? Particularly, can $f$ be written as $$f=\sum^\infty_{j=1}g_j^2$$ with $g_j\in\mathscr{D}(\mathbb{R}^n)$ for all $j$ and convergence in $\mathscr{D}(\mathbb{R}^n)$?

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  • $\begingroup$ In dimension $n=1$, we can simply modify $f$ slightly near the edges of an interval where $f>0$ to make it a square there, and then we do this for finitely many of these intervals and set $g=0$ otherwise to obtain an approximation. In general, can't we do essentially the same thing with the components of the open set $\{f>0\}$ ? $\endgroup$ – Christian Remling Dec 23 '17 at 1:18
  • $\begingroup$ Hmm... The difficulty with your idea is that there may be points $x$ where $f(x)=0$ but the derivatives of $f$ at such $x$ may be large (such points are in the interior of $\mathrm{supp} f$, of course). It's not clear to me how to modify $f$ near $f^{-1}(0)$ so as to put it in square form (to do it away from there is easy if we use squares of smooth functions as elements of a suitable partition of unity since there simply taking the square root of $f$ is harmless) while keeping at a finite but arbitrarily large number of derivatives not too far off the derivatives of $f$ near $f^{-1}(0)$. $\endgroup$ – Pedro Lauridsen Ribeiro Dec 23 '17 at 3:39
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    $\begingroup$ Maybe it would have been more interesting to ask whether every $0\leq f\in\mathscr{D}(\mathbb{R}^n)$ is the increasing limit of a sequence of sums of squares in $\mathscr{D}(\mathbb{R}^n)$? Or, even better, whether it can be written as $f=\sum_{j=1}^\infty g_j^2$. $\endgroup$ – André Henriques Dec 23 '17 at 22:05
  • $\begingroup$ I think this, if true, is bound to be (much?) harder, specially if we want convergence to take place in $\mathscr{D}(\mathbb{R}^n)$, since the "cheapest" way to get the result I asked for is to keep the approximating sequence away from zero at the zeros of $f$ which are cluster points of $\{f>0\}$, as Pietro Majer proposed below. In your case, one does the opposite. A possibility would be to somehow "flatten" the approximating functions near these troublesome points, but this probably rules out convergence in $\mathscr{D}(\mathbb{R}^n)$ since, again, the derivatives of $f$ may be large there. $\endgroup$ – Pedro Lauridsen Ribeiro Dec 23 '17 at 23:43
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    $\begingroup$ Yes, it's going to be much harder... but only because Pietro Majer's answer was so easy. ;-) $\endgroup$ – André Henriques Dec 24 '17 at 0:14
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Given $f\in\mathcal{D}(\mathbb{R}^n)$, $f\ge0$ choose a $g\in\mathcal{D}(\mathbb{R}^n)$, $g\ge0$ such that $\mathrm{supp}(f)\subset \{g>0\}$, and let $\epsilon>0$. Then $\sqrt{f+\epsilon^2 g^2}$ is $C^\infty$: by composition, at any point where $g(x)>0$, and because it locally coincides with $\epsilon g$, at any point where $g(x)=0$.

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    $\begingroup$ This is fine - if $0\leq g\in\mathscr{D}(\mathbb{R}^n)$ is such that $\mathrm{supp} f\subset\{g>0\}$, then $0\leq f_\epsilon=f+\epsilon^2g^2=:g_\epsilon^2\rightarrow f$ in $\mathscr{D}(\mathbb{R}^n)$ as $\epsilon\rightarrow 0$ with $g_\epsilon=\sqrt{f_\epsilon}\in\mathscr{D}(\mathbb{R}^n)$ for all $\epsilon>0$ by your argument. I'm (even more) OK with a sum with just one square ;-) actually, I'm dumbfounded at how the answer could be this simple and how I could have missed it... Thanks! $\endgroup$ – Pedro Lauridsen Ribeiro Dec 23 '17 at 21:33
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    $\begingroup$ I've just found that an argument similar to yours is sketched in L. Schwartz's proof of the Bochner-Schwartz theorem on the characterization of distributions of positive type, as done in his book "Théorie des Distributions". $\endgroup$ – Pedro Lauridsen Ribeiro Dec 28 '17 at 3:14
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It is possible I have misunderstood the topology on $\mathcal{D}(\mathbb{R}^n)$; I assumed it means that all the derivatives must converge. With that understanding, let $a \in (0,3]$. I claim that $x^4 y^2+y^4 z^2+z^4 x^2 - a x^2 y^2 z^2$ cannot be written as a sum $\sum f_i(x,y,z)^2$ of $C^3$ functions. Note that this function is nonnegative by the arithmetic-geometric mean inequality.

If we had $x^4 y^2+y^4 z^2+z^4 x^2 - a x^2 y^2 z^2 = \sum f_i(x,y,z)^2$ then the degree $k$ part of the Taylor series of the right hand side would converge to the degree $k$ part of the Taylor series of the left hand side. From this we see that the $f_i$ vanish up to third order and, writing $c_i$ for the cubic part of the Taylor series of $f_i$, we have $x^4 y^2+y^4 z^2+z^4 x^2 - a x^2 y^2 z^2 = \sum c_i^2(x,y,z)$. This is the standard example of a nonnegative polynomial which cannot be written as a sum of squares, so now we just hav to show that the same proof shows that it cannot be written as an infinite sum of squares.

There are no $x^6$, $y^6$ or $z^6$ terms on the right hand side, so the $c_i$ have no $x^3$, $y^3$ or $z^3$ terms. Similarly, looking at the coefficients of $x^2 y^4$, $y^2 z^4$ and $z^2 z^4$, we see that there are no $x y^2$, $y z^2$ or $z x^2$ terms. So $c_i(x,y,z) = p_i x^2 y + q_i y^2 z + r_i z^2 x + s_i x y z$. But then $-a = \sum s_i^2$, a contradiction.

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    $\begingroup$ In principle, the (counter)example you proposed should be compactly supported, but (as argued in my question) this can be easily achieved by multiplying everything by a squared bump function, so that's OK. That being said, recall that $f_j\rightarrow f$ in $\mathscr{D}(\mathbb{R}^n)$ if there is $K\subset\mathbb{R}^n$ compact such that $\mathrm{supp}f,\mathrm{supp}f_j\subset K$ for all $j$ and all derivatives of $f_j$ converge uniformly in $K$ to those of $f$, so one can start from a (counter)example in $\mathscr{C}^\infty$ with uniform convergence of all derivatives in compacta. $\endgroup$ – Pedro Lauridsen Ribeiro Dec 28 '17 at 15:19
  • $\begingroup$ Moreover, one should probably add to your answer the (obvious but clarifying) fact that $x^4y^2+y^4z^2+z^4x^2−ax^2y^2z^2$, $a\in(0,3]$ only vanishes at the union of three straight lines $\{(t,0,0)\ |\ t\in\mathbb{R}\}\cup\{(0,t,0)\ |\ t\in\mathbb{R}\}\cup\{(0,0,t)\ |\ t\in\mathbb{R}\}$ due to the sharpness of the AM-GM inequality, and that one is considering the Taylor(-MacLaurin) expansion (with remainder) around $(0,0,0)$. $\endgroup$ – Pedro Lauridsen Ribeiro Dec 28 '17 at 18:12
  • $\begingroup$ Why the $f_i$ vanish up the third degree? $\endgroup$ – Pietro Majer Dec 28 '17 at 21:24
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    $\begingroup$ Compare the constant terms on each side to get that the constant terms of all the $f_i$ are $0$. Then compare the quadratic terms to get that the linear parts of each $f_i$ are $0$. Then compare the degree $4$ terms to get that the degree $2$ parts of each $f_i$ are zero. $\endgroup$ – David E Speyer Dec 28 '17 at 21:59
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    $\begingroup$ Motzkin polynomial strikes again (the function given by David is the famous Motzkin's example of a non-SOS non-negative polynomial). $\endgroup$ – Dima Pasechnik Jan 19 '18 at 10:57
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The last counterexample posted here led me to think a bit on the theme "What can be done then?". I came out with the following simple crank machine, that crumbles a positive $C^1_c$ function $f$ into a $C^1$ convergent series of squares, and that could also work in other ordered algebras (in fact I suspect it may be a known procedure). We may assume with no loss of generality $0\le f\le1/2$.

Consider the sequence of functions: $$\begin{cases} g_1=f\\ g_{n+1}=g_n-g_n^2 \end{cases}$$

It is immediate by the definition that $f=\sum_{k=1}^n g_k^2 +g_{n+1}$, so that we have $f=\sum_{k=1}^\infty g_k^2$ in various senses, provided that the remainder $g_{n}$ converges to zero in the same sense.

Uniform convergence of $g_n$: We have $0\le g_n\le1$ and ${1\over g_{n+1}}={1\over g_{n}}+{1\over 1-g_{n}}\ge {1\over g_{n}}+1,$ hence ${1\over g_{n}}\ge n$ and $g_n\le {1\over {n}}$.

Uniform convergence of $\partial_i g_n$: We have $\partial_i g_{n+1}=(1-2g_n)\partial_ig_n$. Since $0\le 1-2g_n\le1-g_n={g_{n+1}\over g_n}\le 1$ we also have $|\partial_i g_{n+1}|\le {g_{n+1}\over g_n}|\partial_ig_n|\le |\partial_ig_n|$. Iterating the latter inequalities we get

$$|\partial_i g_{n}|\le {g_{n}\over f}|\partial_if|\le|\partial_i f|$$ which immediately implies the uniform convergence of $\partial_i g_{n}$ to $0$ (indeed $\partial_i g_{n}$ converges uniformly to zero on any set $\{f\ge\epsilon\}$ and it is dominated by $|\partial_i f|$).

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Rmk if for instance $f(x)=x^2+o(x^2)$ for $x\to0$ then also $g_n(x)=x^2+o(x^2)$ so that $g_n''(0)$ can't converge to zero, as illustrated in the last post.

On the other hand, by David Speyer's example, we know that, in general, for a smooth $f$, no series can't be convergent in $C^3$.

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    $\begingroup$ It could be the case that $C^2$ convergence fails in certain cases. Bony et alii (hehe) showed in the paper cited in my question that if $n\geq 4$ there is a non-negative smooth function in $\mathbb{R}^n$ which is not a finite sum of squares of $C^2$ functions. Perhaps by mimicking their argument in a way similar to David Speyer's answer one could show that $C^2$ convergence of a would-be series of squares fails for the same counterexample. $\endgroup$ – Pedro Lauridsen Ribeiro Jan 18 '18 at 13:55
  • $\begingroup$ I made a cleaner version. As to $C^2$ convergence, I'm not sure. I think $\partial_{ij}g_n$ converges uniformly to zero on any $\{f\ge\epsilon\}$ at least. $\endgroup$ – Pietro Majer Jan 19 '18 at 10:22
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    $\begingroup$ In one variable, if $f(0)=f'(0)=0$ and $f''(0) =c>0$, then I get that $g_n''(0)=c$ for all $n$, so $g''_n$ does not approach $0$. $\endgroup$ – David E Speyer Jan 19 '18 at 15:01
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    $\begingroup$ I uploaded some figures in a separate answer. $\endgroup$ – David E Speyer Jan 19 '18 at 15:19
  • $\begingroup$ This is a neat counterexample - thanks David! $\endgroup$ – Pedro Lauridsen Ribeiro Jan 19 '18 at 15:47
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This is a comment on Pietro Majer's answer, with graphics. The first figure shows $x^2$ (red) and the first five functions of Pietro's method approaching it (blue). The second figure shows the double derivatives thereof. As you can see, the functions are converging but their second derivatives are not.

enter image description here

enter image description here

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  • $\begingroup$ Nice graph --In fact I think second derivatives only converge to zero point-wise in $\{f>0\}$ $\endgroup$ – Pietro Majer Jan 19 '18 at 16:28

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