Optimal Polynomial Admissible Meshes on Some Classes of Compact Subsets of
We show that any compact subset of which is the closure of a bounded star-shaped Lipschitz domain , such that has positive reach in the sense of Federer, admits an optimal AM (admissible mesh), that is a sequence of polynomial norming sets with optimal cardinality. This extends a recent result of A. Kroó on star-shaped domains.
Moreover, we prove constructively the existence of an optimal AM for any where is a bounded domain. This is done by a particular multivariate sharp version of the Bernstein Inequality via the distance function.
Key words and phrases:
1991 Mathematics Subject Classification:
- 1 Introduction
- 2 Optimal AMs for star-shaped sets having complement with positive reach
- 3 Optimal AM for domains by distance function
- 4 Acknowledgements
- A Sets of positive reach
- B (Oriented) distance function and domains
Let us denote by the space of polynomials of d real variables having degree at most . We recall that a compact set is said to be polynomial determining if any polynomial vanishing on is necessarily the null polynomial.
Let us consider a polynomial determining compact set and let be a subset of . If there exists a positive constant such that for any polynomial the following inequality holds
then is said to be a norming set of constant for Here and throughout the paper we use this notation: for any bounded function on
Let be a sequence of norming sets for with constants , and suppose that both and grow at most polynomially with (i.e., for a suitable ), then is said to be a weakly admissible mesh (WAM) for K; see111The original definition in  is actually a little weaker (sub-exponential growth instead of polynomial growth is allowed), here we prefer to use the present one which is the most common in the literature. . If , then is said an admissible mesh (AM) for ; in the sequel, with a little abuse of notation, we term (weakly) admissible mesh not only the whole sequence but also its -th element .
Observe that necessarily
since a (W)AM is -determining, i.e., any polynomial in vanishing on is the zero polynomial. When , following Kroó , we speak of an optimal admissible mesh.
We recall that AMs are preserved by affine transformations and can be constructed incrementally by finite union and product. Moreover they are stable under small perturbations and smooth mappings; see  and . For a survey on WAMs properties and applications we refer to .
The study of AMs has several computational motivations. Indeed, it has been proved by Calvi and Levenberg that discrete least squares polynomial approximations based on (W)AMs are nearly optimal in the uniform norm, see [14, Thm. 1]. Moreover, discrete extremal sets extracted from (W)AMs (see for instance ,), are known to be good interpolation sets and to behave asymptotically like Fekete points, namely the corresponding sequences of uniform probability measures converge weak to the pluripotential equilibrium measure of the underlying compact set; see   or the survey . We recall that it is possible to construct an admissible mesh with points on any real compact set satisfying a Markov Inequality  with exponent The mesh can be obtained by intersecting the compact set with a uniform grid having step size by [14, Thm. 5].
Indeed, the hypotheses of [14, Thm. 5] are not too restrictive. For instance one has a Markov Inequality with exponent for any compact set satisfying a uniform cone condition . Thus also for the closure of any bounded Lipschitz domain. However a Markov Inequality holds with an exponent possibly greater than 2 even for more general classes of sets; see  and  for details.
The cardinality growth order of AMs built by this procedure, however, causes severe computational drawbacks already for This gives a strong practical motivation to construct low-cardinality admissible meshes, in particular optimal ones.
It has been proved in  that for any compact polynomial determining there exists an admissible mesh with cardinality, unfortunately the method relies on the determination of Fekete points, which are not known in general and whose construction is an extremely hard task.
In order to build meshes with nearly optimal cardinality growth order one can restrict his attention to sets with simple geometry such as simplices, squares, balls and their images under any polynomial map (see for instance ) or can look at some specific geometric-analytic classes of sets; the present paper follows the latter idea.
In particular he notices that this implies the existence of optimal AMs for the closure of any star-shaped bounded domain.
While writing this paper we received a new preprint (now published) by A. Kroó where the author improves his estimate above by a fine use of Minkowski Functional smoothness; [23, Theorem 3].
In  he also conjectured that any real convex body has an optimal admissible mesh. In this work we build such optimal admissible meshes on two relevant classes of compact sets.
The paper is organized as follows.
In Section 2 we work on star-shaped compact sets in with nearly minimal boundary regularity assumptions. We prove in Theorem 2.3 that if is a bounded star-shaped Lipschitz domain such that has positive reach (see Definition A.1), then has an optimal admissible mesh.
In Section 3 we address the same problem but we drop the star-shape assumption on , it turns out that a little stronger boundary regularity is needed. In Theorem 3.6 we prove that if is a bounded domain of , then there exists an optimal admissible mesh for .
In the Appendices we provide, for the readerâs convenience, a quick review of some definitions and results from non-smooth and geometric analysis and geometric measure theory that are involved in the framework of this paper.
2. Optimal AMs for star-shaped sets having complement with positive reach
In approximation theory it is customary to consider as mesh parameter the fill distance of a given finite set of points with respect to a compact subset of .
In this definition it is not important whether the segment lies in or not. If one wants to control the minimum length of paths joining to and supported in then one may consider the following straightforward extension of the concept of fill distance given above.
Definition 2.1 (Geodesic Fill-Distance).
Let be a finite subset of the set , then we set
the geodesic fill distance of over .
Here and throughout the paper we denote by the total variation of the curve ,
Notice that if we make the further assumption of the local completeness of then it ensures the existence of a length minimizer in provided it is not empty, that is if there exists a rectifiable curve connecting any and in such that . Thus if has finite geodesic diameter, which will be the case of all instances considered in this paper, then we can replace by in (3).
Now we want to build a mesh on the boundary of a bounded Lipschitz domain having a given geodesic fill distance but keeping as small as possible the cardinality of the mesh. Then we use such a “geodesic” mesh to build an optimal AM for the closure of the domain.
For the reader’s convenience we recall here that a domain is termed a (uniformly) Lipschitz domain if there exist and an open neighbourhood of in such that for any there exists and a rotation such that and
The following result, despite its rather easy proof, is a key element in our construction. For a bounded Lipschitz domain the euclidean and geodesic (on the boundary) distances restricted to the boundary are equivalent.
Let be a bounded Lipschitz domain in , then there exists such that there exists and the following hold:
Here we denote by the dimensional ball of radius centered at with respect to the norm i.e. the coordinate cube centered at and having sides of length
Since is a Lipschitz domain using the above notation we can write
Let us denote the graph function of by that is
By compactness we can pick such that
Let , take any and let us consider the grid of step-size in the dimensional cube
where is the ceil operator. Set
Now notice that
In order to verify the (ii) for any we explicitly find and build a curve connecting to whose variation gives an upper bound for the geodesic distance of from . For the following construction we refer to the Figure 1.
Take any then there exist (at least one) such that Let us pick such an
Let us denote by the canonical projection on the first coordinates acting from onto
Let , by the very construction we can find such that moreover the whole segment lies in
We consider the curve and we set the curve that joins x to obtained by mapping the segment under
Here is the Jacobian of a Lipschitz mapping, see [pg. 101].
We take the maximum over using (3), notice that our by the construction is an element of
Now we are ready to state and prove the main result of this section. We build an optimal mesh for a star shaped Lipschitz bounded domain having complement of positive reach by the following technique. First, we consider the hypersurfaces given by the images of the boundary of the domain under a one parameter family of homotheties, being the parameter chosen as Chebyshev points scaled to the suitable interval. We prove that this family of hypersurfaces is a norming set for the given compact. The second key element is that on each such hypersurface we can use a Markov Tangential Inequality with minimal (with respect to the degree of the considered polynomial) growth rate .
Let be a bounded star-shaped Lipschitz domain such that has positive reach (see Definition A.1), then has an optimal polynomial admissible mesh.
We can suppose without loss of generality the center of the star to be by stability of AM under euclidean isometries .
Let us set for any . By a well known result () the set of all ’s (varying the index ) is an admissible mesh of degree and constant for the interval :
Let us take any and consider the set , notice that because is star-shaped.
One can set , i.e., is the union of the images of under the homotheties having parameters See Figure 2.
Notice that the restriction of any polynomial of degree at most in variables to any segment is a univariate polynomial of degree at most , then due to (5) are norming sets for , that is
Therefore we are reduced to finding an admissible polynomial mesh of degree for .
Let us consider any222Notice that is compact connected, nonempty and consists of an infinite number of points, obviously it contains an infinite number of Lipschitz curves. Lipschitz curve , by Proposition A.1 for a.e. there exists such that
Hereafter is, as customary, the tangent space to at
Since the boundary of the ball is a compact algebraic manifold, it admits Markov Tangential Inequality of degree (see  and the references therein), moreover the constant of such an inequality is the inverse of the radius of the ball:
Let us recall (see for instance [Lemma 1.1.4]) that any Lipschitz curve can be re-parametrized by arclength by the inversion of , obtaining a Lipschitz curve
Therefore (using Rademacher Theorem, see for instance [Th.2 pg 81]) for a.e. we have
By Proposition 2.1 we can pick subsets on such that and . For notational convenience we write in place of
Let us now pick any and consider , an arc connecting a closest point of to and itself such that , parametrized in the arclength.
By the Lebesgue Fundamental Theorem of Calculus for any one has
where in the last line we used (9). Thus we have
By the properties of rescaling, setting we have also
for, consider the homothety , where and write the inequality (10) for each
Therefore, taking the union over and using and , we have
Hence, setting we can write
Now we can use (6) to get and hence
Thus is an admissible polynomial mesh for . The set is the disjoint union of sets ,thus
therefore is an optimal admissible mesh of constant . □∎
This result should be compared to the recent article [23, Theorem 3]. The results achieved by the author are set in a little more general context, still they do not cover the case of a Lipschitz domain with complement having Positive Reach but not being globally smooth. The key element here is that inward pointing corners and cusps are allowed in our setting, while they are not in .
From an algorithmic point of view an AM built by a straightforward application of Theorem 2.2 may be refined. Informally speaking such a collocation technique creates AMs that are clustered near the center of the star, while this seem to have no geometrical nor analytical meaning.
This issue can be partially removed by some minor modifications of the construction which turn the proof of Theorem 2.2 in a more efficient algorithm.
Theorem 2.2 is formulated in a rather general way, here we provide two corollaries that specialize such result.
It has been shown (see ) that domains (see B.1) of are characterized by the so called uniform double sided ball condition, that is, is a domain iff there exists such that for any there exist such that we have and , this property in particular says that (and itself) has positive reach A.1. Therefore the following is a straightforward corollary of our main result.
Let be a bounded star-shaped domain, then its closure has an optimal AM.
It is worth recalling that such domains can also be characterized by the behavior of the oriented distance function of the boundary (i.e. ). For any such domain there exists a (double sided) tubular neighborhood of the boundary where the oriented distance function has the same regularity of the boundary, this condition characterizes domains too. This framework is widely studied in  and .
In the planar case a similar result holds under slightly weaker assumptions.
Theorem 2.3 ().
Let be a bounded star-shaped domain in satisfying a Uniform Interior Ball Condition UIBC (see Definition A.4), then has an optimal polynomial admissible mesh.
A comparison of the statements of Theorem 2.2 and Theorem 2.3 reveals that actually in the second one we are dropping two assumptions, first the domain is no longer required to be Lipschitz, second we ask the weaker condition UIBC instead of complement of positive reach.
The first property is assumed to hold in the proof of the general case to make possible the construction of the geodesic mesh with a control on the asymptotics of the cardinality. In the boundary of a bounded domain satisfying the UIBC is rectifiable; see . Therefore, the geodesic mesh can be created by equally spaced (with respect to arc-length) points.
On the other hand the role of the second missing property is recovered by a deep fact in measure theory. If a set has the UIBC then then the set of points where the normal space (see Definition A.2) has dimension greater or equal to has locally finite Hausdorff measure; [19, 25]. In our bi-dimensional (i.e., ) case this result reads as follow: the normal space has dimension greater or equal to on a subset having Hausdorff measure equal to that is a finite set . Moreover it can be proved that, apart from this small set, the single valued normal space is Lipschitz.
3. Optimal AM for domains by distance function
As we mentioned above, in  the author conjectures that any real compact set admits an optimal AM, in this section we prove (in Theorem 3.3) that this holds at least for any real compact set which is the the closure of a bounded domain see B.1
We denote by the distance function w.r.t. the complement of , i.e.
and by the metric projection onto i.e., is the set of all minimizer of (11). We continue to use the same notation as in the previous section for the closure and the boundary of , namely and
Let us give a sketch of the overall geometric construction before giving details.
First for a given domain we take where is the maximum radius of the ball of the uniform interior ball condition satisfied by
We can split as follows
To construct an AM of degree on we work separately on and to obtain inequalities of the type
for where and are suitably chosen finite sets.
In the case of this is achieved by the trivial observation implies and therefore one can bound any directional derivative of a given polynomial using the univariate Bernstein Inequality (see Theorem 3.1 below). The resulting inequality is a variant of a Markov Inequality with exponent which is convenient and allow us to build a low cardinality mesh by a modification of the reasoning in .
The construction of an AM on is more complicated. The resulting mesh is given by points lining on some properly chosen level surfaces of The result is proved using the regularity property of the function in a small tubular neighborhood of and the Markov Tangential Inequality for the sphere.
3.1. Bernstein-like Inequalities and polynomial estimates via the distance function.
For the reader’s convenience we recall here the Bernstein Inequality.
Theorem 3.1 (Bernstein Inequality).
Let , then for any we have
Let us introduce the following notation illustrated in Figure 3.
In the case when is a domain one has the estimate where see Definition A.1 and thereafter.
The following consequence of Bernstein Inequality will play a central role in our construction.
Let be a bounded domain in and let us introduce the sequence of functions
For any let , then for any we have
If moreover we have , let us pick any and define the sequence of functions
Then the above polynomial estimate (16) still holds when is replaced by
Pick . Let us take such that We denoted by the segment where is as above and due to The restriction of to this segment is an univariate polynomial of degree not exceeding , then we can use the Bernstein Inequality 3.1 to get
evaluating at we get
thus establishing the first case of (17).
Let be such that . Notice that and hence (the standard unit dimensional sphere) we can pick a segment in the direction of having length lying in and having as midpoint. The Bernstein Inequality gives
Actually the above proof proves also the following corollary, it suffices to take (17) and substitute by in the second case.
Let be an open bounded domain and a positive number such that . Then for any we have
We introduce the following in the spirit of . Let us denote by the standard length measure in .
Let be a bounded domain in such that and let . Then
for any the map
is constant, let be its value.
In particular extends continuously to .
is constant on any level set of and .
Let us set where and is any positive integer greater than , we denote by the -level set of .
Let then for any we have
(i) The function depends on its argument only by the distance function, . The length of the segment is clearly constant when varies in the set .
Moreover for any let us denote by an euclidean isometry that maps onto , one trivially has for any . This is because for any by the Triangle Inequality and thus .
Thus we have
(ii) Let us parametrize the segment as , then we have