Optimal AM

Optimal Polynomial Admissible Meshes on Some Classes of Compact Subsets of

Federico Piazzon room 712 Department of Mathematics, Universitá di Padova, Italy. Phone +39 0498271260 fpiazzon@math.unipd.it http://www.math.unipd.it/ fpiazzon/ (work in progress)
July 16, 2019
Abstract.

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:
Supported by INdAM GNCS and Doctoral School on Mathematical Sciences Univ. of Padua

1. Introduction

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

(1)

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 [14] 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. [14]. 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 [22], 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 [28] and [29]. For a survey on WAMs properties and applications we refer to [11].

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 [11],[12]), 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 [5] [6] or the survey [24]. We recall that it is possible to construct an admissible mesh with points on any real compact set satisfying a Markov Inequality [10] 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 [3]. 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 [26] and [27] 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 [7] 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 [8]) or can look at some specific geometric-analytic classes of sets; the present paper follows the latter idea.

In [23] the author proves that any compact star-shaped set with Minkowski Functional (see for instance [13, pg. 6]) having -Lipschitz gradient has an admissible mesh with

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 [22] 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 reader’s 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 .

(2)

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

and define

(3)

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.

Proposition 2.1.

Let be a bounded Lipschitz domain in , then there exists such that there exists and the following hold:

  1. as .

  2. .

Figure 1. The geodesic mesh in the proof of Proposition 2.1 is built by lifting the grid mesh by the local parametrization of the boundary . The curve connecting to is similarly produced by lifting the segment
Proof.

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

Now we use Area Formula [20] [18][Th. 1 pg. 96] to compute the length of the Lipschitz curve

Here is the Jacobian of a Lipschitz mapping, see [18][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 .

Theorem 2.2.

Let be a bounded star-shaped Lipschitz domain such that has positive reach (see Definition A.1), then has an optimal polynomial admissible mesh.

Proof.

We can suppose without loss of generality the center of the star to be by stability of AM under euclidean isometries [11].

Let us set for any . By a well known result ([17]) the set of all ’s (varying the index ) is an admissible mesh of degree and constant for the interval :

(5)

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.

Figure 2. The geometry of .

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

(6)

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

  1. and

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 [9] and the references therein), moreover the constant of such an inequality is the inverse of the radius of the ball:

(7)

Let us recall (see for instance [2][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 [18][Th.2 pg 81]) for a.e. we have

(8)
(9)

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

(10)

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 [23].

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 [1]) 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.

Corollary 2.2.1.

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 [15] and [16].

In the planar case a similar result holds under slightly weaker assumptions.

Theorem 2.3 ([30]).

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 [21]. 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 [19]. 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 [22] 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.

(11)

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 [14].

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

(12)

Let us introduce the following notation illustrated in Figure 3.

(13)
(14)
Remark 3.2.

In the case when is a domain one has the estimate where see Definition A.1 and thereafter.

Figure 3. Here and is the length of the shortest segment inside containing and having direction

The following consequence of Bernstein Inequality will play a central role in our construction.

Proposition 3.1.

Let be a bounded domain in and let us introduce the sequence of functions

(15)

For any let , then for any we have

(16)

If moreover we have , let us pick any and define the sequence of functions

(17)

Then the above polynomial estimate (16) still holds when is replaced by

Proof.

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

(18)

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

(19)

The last statement follows directly by the special choice of . The right hand side in (17) dominates (case by case) the r.h.s. in (15) when cases are chosen accordingly to (17). □∎

Actually the above proof proves also the following corollary, it suffices to take (17) and substitute by in the second case.

Corollary 3.2.1.

Let be an open bounded domain and a positive number such that . Then for any we have

(20)

We introduce the following in the spirit of [31]. Let us denote by the standard length measure in .

Proposition 3.2.

Let be a bounded domain in such that and let . Then

  1. for any the map

    is constant, let be its value.

  2. We have

    (21)

    In particular extends continuously to .

  3. 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 .

  4. We have

  5. Let then for any we have

    (22)
Figure 4. A plot of a section of along a segment of metric projection, where , . Abscissa here is the distance from the boundary.
Proof.

(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

(23)