2009, NO 4
PP. 769- 779
Abstract . Any closed set is a set of zeros of an infinitely differentiable function. On
the other hand, for many important classes of functions f (polynomials, eigenvalues of
elliptic operators, analytic functions with a controlled growth, smooth functions with a
control of high-order derivatives, etc.) their zero sets Z(f) behave qualitatively like semialgebraic sets, and the "size" of Z (f) (in various senses) can be explicitly bounded. In
this paper we relate the geometry of the zero set of a given function with the rate of its
piecewise-polynomial approximation.
1. Introduction.
It is well known that any closed set F C Rn
is a set of zeros of an infinitely differentiable function f : Rn - t JR. On
the other hand, for many important classes of functions f their zero sets
Z(f) behave qualitatively like semi-algebraic sets, and the "size" of Z(f) (in
various senses) can be explicitly bounded. Such classes include polynomials,
eigenvalues of elliptic operators, analytic functions with a controlled growth,
smooth functions with a control of high-order derivatives, etc. In particular,
in [24] it was shown that assuming that f : BJ: - t JR (where BJ: is the unit
ball in JRn) is a Ck-function, max if(x)l = 1 while the k-th derivative of
f is sufficiently small, the zero set Z (!) of f behaves qualitatively like a
semi-algebraic set, and its size can be explicitly bounded. Under roughly the
same assumptions the bound on the volume of an €-neighborhood of Z(J)
· · · ·····was··-obtained··-in···[2)-::············································-··· -·······················································································-·································-········ ··· ················
In the present paper we relate the geometry of the zero set of a given
function with the rate of its piecewise-polynomial approximation. This relation may be considered as an additional manifestation of the following
phenomenon (see [25, 26]):
* To the memory of Professor Michael Drahlin.
t Department of Mathematics, The Weizmann Institute of Science, Rehovot 76100,
Many important properties of functions, traditionally associated with their
regularity, are in fact determined by usemi-algebraic complexity" of these
functions, which is the rate of their semi-algebraic approximation.
In the present paper we push the above observation to its extreme, assuming no regularity of the involved functions at all, besides their measurability. We show that the "size" of the zero set Z (f) of f : Bf -+ JR, in
particular, the volume of this set and its <:-entropy are bounded in terms of
the complexity of piecewise-polynomial <:-approximations of f.
2. Metric entropy of semi-algebraic sets.
In this section we
recall results of G. Vitushkin ([22, 23]) concerning metric entropy of semialgebraic sets. Since for our applications we need a rather accurate bounds
on the "semi-algebraic constants" involved, we provide a simple proof of
Vitushkin's theorem in one specific case.
Let us recall a general definition of metric entropy. Let A C X be a
relatively compact subset in a metric space X.
DEFINITION 1. Fore> 0 the covering number M(e, A) is the minimal number of closed £-balls in X covering A. The binary logarithm of the covering
number, H(<::,A) = logM(e,A) is called the €-entropy of A.
The notion of "covering number" is very popular in fractal geometry
and can be traced at least to Minkovski (see [17, 12]). See [15, 16] and many
other publications for computation of €-entropy of various functional classes.
Intuitively, <::-entropy of a set A is the minimal number of bits we need to
memorize a specific element of the set A with the accuracy t::.
In our setting X = lR11 • It will be convenient in this case to modify
slightly the definition of the covering number, and to consider coverings by
the e:-cubes Q~ which are translations of the standard £-cube Q~ = [0, e:] 11 • Q~
are the ~ - balls in the t=-norm. In particular, this choice implies that the
leading t erm in the expression forM(£, A) has the form M(e:, A) = vo;£A) + ....
A semi-algebraic set A in lR11 is the one obtained as a result of the standard set-theoretic operations over the sets defined by polynomial inequalities
P(x1, ... , x11 ) ~ 0 or P(x 1 , ... , Xn) < 0. The diagram D(A) of A is the collection of the discrete data in the definition of A: it consists of the dimension
n of the ambient space, of the degrees of the participating polynomials and
of all the set-theoretic formulas applied.
The following bound on the covering number of semi-algebraic sets was
obtained by Vitushkin ([22, 23]; see also [13, 27]):
PROPOSITION 1. Let A C Q~ be a semi-algebraic set with the diagram D.
Then for any e > 0 we have
M(e, A)
~ Co(D) + C1(D)(?::) + · · · + Cn-l(D)(?:: )n-l + J.£n(A)(! )n.
Here J.£n(A) denotes the n-dimensional Lebesgue measure (or the volume) of
the set A. The constants C0 , C1 , ... , Cn-1 , Cn depend only on D.
The original proof was obtained in a much more general situation, via
the ''Vitushkin variations" (see [22, 23, 13, 27]). We give here a detailed
proof only in dimensions 1 and 2 and only for sets A being the sub-level sets
of polynomials. Because of a natural scaling of the covering number we can
assume r = 1.
Let P(x) = P(x 1 , •.. , Xn) be a polynomial of degree d. We shall consider
heresemi-algebraicsetsoftheform Wp(P) = {x E B}, P(x) ~ p}. In particular, we denote by Vp(P) = Wp2(P2 ) the sub-level set {x E Bi, !P(x) ! ~ p}
of the polynomial P. The set Wp(P) is semi-algebraic, and it has a very special form: this set is defined by exactly one polynomial inequality P(x) ~ p.
So the diagram D (Wp(P)) is completely determined by the pair (n,d). In
this case the constants Ci (D) = Ci (n, d) can be given explicit ly.
Consider first the case n = 1. Here the set Wp(P) consists of at most
[~] + 1 subintervals ~i in [0, 1] since its boundary { P(x ) = p} consists of at
most d + 2 points (including the end points of [0, 1]) by the Bezout theorem.
Let us cover each of these subintervals ~i by the adjacent €-intervals, starting
with the left endpoint. Since all the adjacent £-intervals, except possibly one,
are inside ~i their number does not exceed l~il(:) + 1. Altogether we have
at most [~] + 1 + J.£1 (V,(P))(~) covering £-intervals.
In two dimensions we consider the sub-level set W = Wp(P) inside the
unit cube Qf = [0, 1]2 with P = P(x1 , x2 ) a polynomial of degree d. Let us
subdivide Qi into adjacent €-cubes Q!i by vertical and horizontal straight
lines. We have in total 2([~] + 1) straight segments. Mark for our covering
all those Q!i which intersect W.
Some of the marked Q!i are contained in W. The number of such boxes
1E ) 2 • Some other boxes Q!i cross the boundary 8W of
does not exceed ~-t2 (V)(
··~----· ..··- · ·w·:. ··To·eaclcsuclrQ?-we··crur·ru=rsuciate··(exclusively)--the·first--crossing-·point···---...................
of its boundary with 8W (assuming that an orientat ion of oW was fixed) .
Hence the number of the boxes of the second type does not exceed the total
number of the crossing points of 8W with the partition segments. By the
Bezout t heorem a straight line can meet 8W at not more than d points (since
8V = {P = p}). Altogether we have at most 2d( [ ~) + 1) ~ 4d(:) boxes of
the second type.
Finally, there may be covering boxes containing completely some of the
connected components of W . Their number does not exceed (d - 1) 2 since
inside each compact connected component of W there is a critical point of
P. The last are defined by the system of two polynomial equations
of degree d - 1 and the number of their solutions is bounded by (d - 1)2 via
the two-dimensional Bezout theorem. (We can assume that all the solutions
of this system are non-degenerate since in our proof the polynomial P can
be perturbed slightly).
Exactly the same approach can be extended to higher dimensions and
to more complicated semi-algebraic sets. The technical details become more
involved, but the general form of the bounds remains the same. In particular,
for the sub-level sets W = Wp(P) and V = Vp(P) the constants Ci(n,d) have
the form Ci(n, d) = Ci(n)d(n-i). Indeed, these constants appear, as in the
proof for the dimension 2 given above, as the bounds on the number of the
connected components of n - i - dimensional plane sections of our semialgebraic set, which have the required form (see, for example, [27]).
This completes the proof of the Vitushkin theorem in our special case.
Let us summarize the specific bounds obtained.
PROPOSITION 2. For a sub-level set W = Wp(P) or V = Vp(P) inside the
cube Q; in IR, or the cube Q; in IR 2 , we have, respectively,
M(c, W) ~ [2]
+ 1 + Pl(W)(; ),
M(t, V)::; d + 1 + Pl(V)( ~ ),
M(t, W) ~ (d- 1) 2 + 4d(-) + p 2 (W)(- ?,
M(t, V) ::; (2d- 1) 2 + 8d(-) + p 2 (V)(- ?.
For a sub-level set W (V) inside the cube
Q~ C
IR"' we have
M(t, W) ::;
L Ci(n)d<n-i}(~Y + f.ln(W)( ~t,
i= O
M(£, V) ::;
L Ci(n) (2d)(n- i) (:c )i + f.ln(V)( ~ t.
3. Volume of sub-level sets of polynomials: Remez inequality.
Let P(x) = P( x 1 , .•. , Xn) be a polynomial of degree d. In this section, to
fit the traditional notations, we consider a part of the polynomial sub-level
set inside the usual Euclidean ball. So we denote by V = Vp(P ) the sub-level
set {x E Br, !P(x)! ~ p} of P. For n = 1 the volume of V can be estimated
via the following Remez inequality ([21]):
THEOREM 1. Let P(x) be a polynomial of degree d. Then
~ Td(
- It),
where Jt = ~t 1 (Vi(P) and Td(x) = cos(darccos(x)) is the d-th Chebyshev
For other related bounds as well as for various multidimensional generalizations of the classical Remez inequality see, in particular, [3, 4, 5, 6, 11),
[18]-.[20]. Specifically, we shall use in this paper the following Yu.BrudnyiGanzburg inequality ([6]), which is, probably, the most direct generalization
of the classical Remez's one:
THEOREM 2. Let l3 c !Rn be a convex body and let n C l3 be a measurable
set. Then for a real polynomial P(x) = P(x1 , ... , xn) of degree d we have
sup IPI ~ Td(
1 + (1- .\):n
1 )
1- (1- .\)n
sup !Pl.
Here .\ = ~:~~~. This inequality is sharp and fo r n = 1 coincides with the
classical Remez inequality.
As a corollary (which is not sharp but sufficient for our applications) one
PROPOSITION 3. In the above notations
~----------------- ----------~-
Let us now assume that on the cube Q~ of the size r the SUPQ;: !PI = M.
Then from Proposition 3 we get
PROPOSITION 4. The volume of Vp(P) satisfies
4. Tubular neighborhoods of sub-level sets.
A 6-tabular neighborhood T0 (S) of a set S consists of all the points at the distance at most 8
from S. Metric entropy provides a convenient tool to study the geometry of
T0 (S) because of the following simple lemma:
LEMMA 1. M(2E, T~(S)) :5 M(E, 8).
Proof: Indeed, if certain €-balls cover S, the 2E-balls centered at the same
points cover ~ (S).
Now combining Lemma 1 with Propositions 2 and 4 we get the following:
THEOREM 3. For a polynomial P of degree dover the cube Q~ we have
M(2E, TdYp(P)
n Q~])::; ~ Ci(n)d(n-i)(~)i + 4nrn(~)~(}t.
In particular, for the volume of the tabular neighborhoods we get
2nCi(n)d(n-i)riEn-i+2(n+2)nrn( ~)~
L Ci(n)d(n-i)ri€n-i+2(n+2)nrn( ~)~'
where Ci(n) = 2nCi(n). Assuming € :5 ~ and r :5 1 we get
J.Ln(Te[Vp(P) n Q~]) ::;
L Ci(n)ri(dc)n-i + 4nrn( ~)~ =
= d€
L Ci(n)ri(dE)"-i-l + 4nrn( ~)~ :=:; C(n)dE + 4nrn( ~)~ .
We summarize this computation in the following proposition:
PROPOSITION 5. For a polynomial P of degree d over the cube
E :5 ~ we have
and for
In particular, for p = 0, i. e. for the zero set Z(P) we get
J.Ln(TdZ(P) n Q~]) :5 C(n)d€.
This last inequality was obtained by different methods in [14, 1] (see also
[2),[7]-[10]) in the course of study of tabular neighborhoods of the nodal sets
of the Laplacian eigenfunctions.
5. Piecewise-polynomial approximations.
We consider measurable functions f : Q1 ~ R
DEFINITION 2. A piecewise-polynomial £-approximation A off consists of
1. A covering of Q1 by a collection of sub-cubes Qi = Q~f, j
= 1, ... , p,
2. A collection of polynomials Pi of degree di ,
such that for any j = 1, ... ,p we have supQ~~ If- Pi I : : :; c
5.1. Volume of the zero set of f.
Now we define an invariant of a
piecewise-polynomial <:-approximation A of f which will bound the volume
of the zero set of f:
DEFINITION 3. Let 6 2:: 0 be given. We define v;-:(6, A ) as
v~(o,A) =4n I :rj(- ) d;,
where Mi = supQ~i JPil·
Now applying the inequality of Proposition 4 .we get one of our main
THEOREM 4. The volume of the zero set Z(f) = {x E Q].\ f (x ) = 0} satisfies
where A runs over all the piecewise-polynomial <:-approximations off.
Proof: Over each sub-cube Q~! the zero set Z(f) off is contained in the sub'
level set ~(Pj), since supQ~i lf-Pil
:::::; e by definition of the£ approximations.
The result now follows from Proposition 4 applied to each of the polynomials
Pi of degree di on the sub-cube Q~f.
Remark Using in the definition of v:(&, A) instead of Proposition 4 a sharp
...,-~----- --beund-fer·-J:Ln·(-~f.P))RQ?-j--whiGh-Gan--be--obtained..from..Theorem..3.2..we ..obtain........................... ..
a more accurate invariant v;-:(o, A). The corresponding version of Theorem
5.1 is given by the inequality
where A runs over all the piecewise-polynomial e-appro:ximations of f . As
we shall see below, inequality (3) is sharp, while Theorem 5.1 is sharp only
up to a constant C(n).
5.2. Metric entropy of the zero set of f.
Our next result is the
bound on the metric entropy of Z(f) similar to the bound on the volume
provided by Theorem 5.1. We need the following definition:
DEFINITION 4. For a piecewise-polynomial f.-approximation A off the "i-th
variation" vf(A) fori = 0, 1, ... , n- 1 is defined as
vin(A) =
C ( )d(n-i) rJ,
~ i n J
where Ci(n) are the constants defined in Proposition 2.2.
The following theorem is the second main result of this paper:
THEOREM 5. The covering number M(E, Z(f)) of the zero set Z(f) = {x
Q1, f(x) = 0} satisfies
where A runs over all the piecewise-polynomial f.-approximations off.
Proof: Over each sub-cube Q~J the zero set Z(f) off is contained in the sub'
level set ~(PJ), since supQ~j If -.Pjl
:::; e by definition of thee approximations.
Applying Proposition 2 to each of the polynomials PJ of degree d5 on the subcubes Q~J1 we have
n- 1
M(t,Z(f)nQ~f) :5 M(e, Ve(Pj)nQ;f)::; LCi(n)dJn-i)(:)i+JLn(~(PJ))(~ti=O
Now by Proposition 4 we have Jln(~(PJ))) :::; 4nrj(~) dj. Summing up for
j = 1, ... ,p and using Definitions 2 and 4 we get the required result.
6. Some examples.
In this section we construct examples of functions f for which the rate of piecewise-polynomial approximation (versus the
complexity of the approximant) may be quite arbitrary while the bounds
of Theorems 5.1 and 5.2 are essentially sharp. Let Q1 = Ufr= 1 Q~j be a
non-overlapping partition of the cube Qf. For each j = 1, 2, ... we fix a
polynomial PJ(x) = PJ(xb . .. ,xn) of degree dj· Now we define a piecewisepolynomial function g(x) by
g(x) =
L XQ;:jPJ(x),
where XQ~? is the characteristic function of the cube Q~j. Now let
given. We define 9e(x) by
9e(x) = g(x) , !g(x)l 2
> 0 be
9e(x) = 0, !g(x)l < €.
Clearly, lge(x) - g(x) l ::; € and hence according to Definition 2 the partition Qf = U~ 1 Q~j and the function g define a piecewise-polynomial €approximation A of gE.
For the zero set Z(ge) we have Z(gE) = U~ 1 Ve(Pj) n Q~j and this union
is disjoint. Hence J.Ln(Z(ge)) =
1 J.Ln(Ve(Pi) n Q~j).
Now since the Remez inequality in Theorem 3.1 is sharp (and its version
given by Propositions 3, 4 is sharp up to constants C(n)) we can choose the
polynomials Pi in such a way that for each j = 1, 2, ... the volume of the
set "Ve(Pj) on Qr:j is given, up to a constant C(n), by the right hand side of
the expression (4) of Proposition 4. Therefore we have
00 = LJ.Ln(Ve(Pj) n Q;f) 2
j =l
where v;: (€, A) is given by Definition· 3. This inequality shows that for the
function 9e the upper bound of Theorem 5.1 is sharp, up to a constant C(n ).
Notice that the complexity of the piecewise-polynomial function g depends
on the choice of the covering Q";;i and of the degrees dj of the polynomials
Exactly in the same way we can show that the metric entropy bound
provided by Theorem 5.2 is sharp up to a constant C(n) . The only difference
is that we have to take into account that for each specific € > 0 some of the
cubes Q~!3 can be entirely covered by one €-ball. However, if we consider only
-,-----finite_padition!Lan<Ltake~umfficie.ntly_s_mJill, the §h~gpness of the bound m
Theorem 5.2 can be shown exactly as above.
Remark. It would be important to investigate the asymptotic behavior of
the bounds in Theorems 5.1 and 5.2 as € tends to zero, at least for some
natural classes of the approximated functions f. However, while the control
of the approximation accuracy is provided by classical results of Approximation Theory, the control of the constants Mi turns to be a rather subtle
problem. It is somewhat similar to the problems treated in [2, 24j and we
plan to present some results in this direction separately.
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-······~····¥••··· ···· ·--------- ----- -· ·· ······························ ·· -- - ---- · · -------·········-············------ ----- ------ ------ ··············~·-····----------- -- - -- - -- -····························· --- ---- -------- -----· · ······ ·· ········-······¥······· · ·· ·· · --- -