The wedge-of-the-edge theorem

The wedge-of-the-edge theorem: edge-of-the-wedge type phenomenon within the common real boundary

J. E. Pascoe J. E. Pascoe
Department of Mathematics
Washington University in St. Louis
One Brookings Drive
St. Louis, MO 63130
[
July 20, 2019
Abstract.

The edge-of-the-wedge theorem in several complex variables gives the analytic continuation of functions defined on the poly upper half plane and the poly lower half plane, the set of points in with all coordinates in the upper and lower half planes respectively, through a set in real space, The geometry of the set in the real space can force the function to analytically continue within the boundary itself, which is qualified in our wedge-of-the-edge theorem. For example, if a function extends to the union of two cubes in which are positively oriented, with some small overlap, the functions must analytically continue to a neighborhood of that overlap of a fixed size not depending of the size of the overlap.

2010 Mathematics Subject Classification:
Primary 32A40
Partially supported by National Science Foundation Mathematical Science Postdoctoral Research Fellowship DMS 1606260

J. E. Pascoe]pascoej@math.wustl.edu

1. Introduction

Let denote the open upper half plane in That is,

Figure 1. A picture of .

Let be an open set. One can show using elementary complex analysis, for example Morera’s theorem, that any continuous map which is analytic on is indeed analytic on the whole of When is real-valued, this observation is the essential part of the proof of the Schwarz reflection principle.

The analogue in several variables was proven by Bogoliubov in 1956 and is known as the edge-of-the-wedge theorem. A quality yet concise and fully comprehensible discussion was given by Rudin in Lectures on the edge-of-the-wedge theorem[10]. The edge-of-the-wedge theorem is an important stem theorem in several complex variables.

Theorem 1.1 (The edge-of-the-wedge theorem).

Let be an open set. There is an open set in containing such that every function which is analytic on analytically continues to

Some degustation of the edge-of-the-wedge theorem is needed before we proceed. Two some what remarkable notes are:

  1. The set is not open. That is, the open set containing must intersect the exterior, somehow leaking out the sides in a big way. For example, when this means and must be nonempty.

    Figure 2. A projection of the edge-of-the-wedge theorem onto the imaginary axes. Note that all of appears as a point in this projection.

    Thus, we can imagine the result is somewhat deeper than in the one variable. Additionally, we are presented with a mystery: what is the geometry of the maximal

  2. The set is independent of the function On face, this fact is somewhat surprising, but in hindsight, and in the light of Montel type theorems, how could it be otherwise.

The metaphor evoked in the phrasing edge-of-the-wedge merits some additional discussion. The phrase thin edge of the wedge refers to something minor with major, often spurious, implications, evoking the classical use of a metal wedge to split logs for firewood. The continuous continuation through the superficially thin looking set between the wedges and gives way to the major analytic continuation throughout the fatter set

We note that various generalizations of the edge-of-the-wedge theorem to weaker distributional notions of agreement between the two wedges, for example [6]. Rudin’s book [10] gives some other generalizations and some more modern surveys include[12]. Our goals are somewhat orthogonal to these matters: during our current enterprise, we will consider the geometry of such a continuation once we already know that it exists. That is, given the edge-of-the-wedge theorem automatically manufactures a set and we would like to understand how shapes

Specifically, we will be particularly interested in In principle, we might be tempted to assume we are forced to have which is often the case– for example, when is a cube, that is, When one may simply take take a dense sequence in and define

and for products of such functions will suffice.

While a general qualitative description of remains somewhat elusive, when has some nontrivial geometry in several variables, the situation is more exciting. Under certain nice geometric conditions, specifically if is the union of two cubes with a small overlap which are positively oriented with respect to each other, we will see that our continuation is somewhat larger than within This is the content of our wedge-of-the-edge theorem.

Theorem 1.2 (The wedge-of-the-edge theorem for hypercubes).

There is an open set containing such that for any every continuous function which is analytic on analytically continues to

Theorem 1.2 follows from the more general Theorem 2.1. In fact, can be taken to contain a uniform ball around which we discuss in Subsection 3.1.

The geometric situation in the wedge-of-the-edge theorem has and pieces of the edge, emulating the roles of the wedges and in the classical edge-of-the-wedge theorem. Hence the name wedge-of-the-edge theorem.

Figure 3. A pictographic representation of the wedge-of-the-edge theorem. Note the visual analogy with the edge-of-the-wedge theorem.

Several examples showing the necessity of the orientation and small overlap. The function can fit two squares in the opposite orientation with a small overlap in the opposite orientation as in the wedge-of-the-edge theorem, but cannot be analytically continued to a uniform neighborhood of for large . The function cannot analytically continue through the point demonstrating the necessity of the small overlap. Our qualitative understanding is still incomplete. For example, we have not eliminated the possibility that cannot be taken to contain the whole

Figure 4. Figures of the domains of and respectively. In the figure of the curve represents the singular set of the function, which as goes to infinity approaches the coordinate axes.

An interesting matter to consider is the large rescaling limit of the wedge-of-the-edge theorem. That is, functions which are continuously defined on the entire positive and negative orthants plus some small overlap and the upper and lower multivariate half planes are entire.

Corollary 1.3 (The limiting wedge-of-the-edge theorem).

For any every continuous function which is analytic on analytically continues to all of

We now give a nice, somewhat amusing corollary of the main result along the lines of Hartog’s theorem. If we have a function defined on a upper and lower multivariate upper half plane and a region in missing a single point, we see that it must continue to that point.

Corollary 1.4 (A weak Hartog’s type theorem).

Let be an open set. Let If which is continuous and analytic on the interior of its domain, then, it has a continuous extension to

The following sketch gives a picture of the proof. The formal details are left to the reader.

Figure 5. Applying the wedge-of-the-edge theorem for hypercubes where the intersection is squeezed sufficiently near gives the proof of Corollary 1.4 since the function is forced to analytically continue to .

The origin of the wedge-of-the-edge theorem lies in the theory of Pick functions in several variables and their boundary values, which has received some amount of interest since the connection to multivariable operator monotonicity was established by Agler, McCarthy and Young[4]. Using strong aspects of their structure, a very detailed specialized version of the wedge-of-the-edge theorem was obtained by the author in [9] which can be used to relax the main result in [4] which was done in [8]. Some of the methods from [9] apply for our current endeavor, but, in general, the analysis is significantly more opaque here.

2. The wedge-of-the-edge theorem in general

We define a real wedge which is

  1. contained within the positive orthant, that is

  2. is a Borel set,

  3. has positive measure,

  4. is starlike with respect to the origin, that is, if then whenever

Figure 6. A real wedge.

The full version of the wedge-of-the-edge theorem is as follows.

Theorem 2.1 (The wedge-of-the-edge theorem).

Let be a real wedge. There is an open set containing such that for any real neighborhood of every continuous function which is analytic on analytically continues to

Figure 7. The general wedge-of-the-edge theorem.

The remainder of the section will consist of a proof the wedge-of-the-edge theorem, along with some remarks on the limitations of our techniques. The trick is to bound the polynomials occurring in the power series for at by interpolating their values after some superficially elaborate reduction.

Proof.

Without loss of generality our wedge will be contained inside

We first note that must be analytic at by the classical edge-of-the-wedge theorem. So, has a power series at Write

(2.1)

near where are the homogeneous terms of the power series for Our goal will be to show that the series for is absolutely convergent of some neighborhood of which has size which depends on and not on the size of That is it will be our goal to show that

Under our assumptions it is clear to see that the series in Equation (2.1) converges almost everywhere on Namely, the series converges absolutely on each For if we consider the function

we see that the rightmost series for converges on a and therefore absolutely when Here, we are using the assumption that is starlike to ensure the definition of on the whole of and since is contained in the positive orthant,

That is, it is now sufficient to prove the following lemma.

Lemma 2.2.

Fix a real wedge There is a constant such that, given a sequence of homogeneous polynomials in variables each of degree and converges almost everywhere on there is a such that

First note converges for since the function is defined and analytic for

Define We note that Alaoglu’s theorem implies that is relatively closed in That is a sequence of elements of which converge pointwise to a limit in must satisfy Here, given a sequence and a limit point we are taking and Namely, each is Borel.

Moreover, the measure of converges to the measure of as goes to infinity. So, there is some such that has measure greater than half the measure of

Now, note that each on So, it is now sufficient to prove the following lemma to establish Lemma 2.2.

Lemma 2.3.

Fix Fix There is a constant and a constant such that, for each a Borel set inside with positive measure greater than to , given a homogeneous polynomial of degree in variables such that on there is a such that

Now we need to introduce the norm on polynomials, which is given by the sum of the moduli of the coefficients. That is, given a polynomial

We define the via the formula

Notably, if is a polynomial in some set of variables and is a polynomial in we see that Moreover, we note that, for homogeneous polynomials

So, now it is sufficient to show the following to establish Lemma 2.3.

Lemma 2.4.

Fix Fix There is a constant and a constant such that, for each a Borel set inside with positive measure greater than given a polynomial of degree in variables such that on there is a such that

The proof will go by induction on the number of variables.

In no variables, that is, we see that Clearly, the maximum modulus that the constant value may have if our monomial is to be bounded on is So we get that

Figure 8. The idea of the proof is to approximate on one lower dimensional pieces which are relatively large and relatively far apart, where relatively large and far apart are constrained by the measure of

Now fix a general number of variables One may pick points such that if we define each has dimensional measure greater than and each is at least apart. The worst case is when all the measure is on one side of the box. We note that the Borel nature of ensures each slice will be Borel and hence measurable.

Let be a polynomial in variables which gives on each Now, by induction we see that there is and such that

Now apply the Lagrange interpolation theorem, noting that interpolation recovers the polynomial exactly since we have enough nodes, to get that

So we see that

Above the in the last line we needed to apply Stirling’s estimate which says that grows like This completes the proof.

2.1. Caveat emptor

There are several reasons the reader should beware.

Firstly, our methods generate some kind of abstract estimates on but we do not really understand the polynomial estimates obtained. Secondly, even if we understood an optimal phrasing of something like Lemma 2.2, there are severe limits to our technique. We, apparently did not fully use the continuation to the full poly upper half plane– in fact, not even the whole polydisk. (A modest abstract improvement can be obtained therefore by conformally mapping into the polydisk first.) However, more importantly, there are limits to the polynomial interpolation method itself which do not seem to be optimal. For example taking a sum over homogenized Chebychev polynomials (which are all bounded by one on a certain wedge) would give some slight transformation of their generating function, namely , see [11, page 69], which satisfies the hypotheses of Lemma 2.2, but also apparently has singularities in the poly upper half plane.

That is, the polynomial value interpolation appears to be a rather naive method to approach the optimal wedge-of-the-edge theorem.

3. Some concluding remarks and conjecture

3.1. On cones and some uniformity

Some formulations of the edge-of-the-wedge theorem state the theorem with respect to cones[6, 10]. The goal now will be to discuss the context of cones and show that in that context we obtain some uniform amount of analytic continuation.

Let be an open cone in with a distinguished element We define a for by

For example if is the positive orthant and we recover the norm. Another example would be to view as by Hermitian matrices and to be the identity matrix which recovers the maximum modulus eigenvalue norm on Hermitian matrices. We extend the norm to via the formula We define

Corollary 3.1.

Define There is an open set containing a ball of some radius in the norm independent of such that for any neighborhood of any continuous function which is analytic on analytically continues to

Proof.

The novel part here is the uniformity. We will show that along any ray pointing out of the origin, we get some absolute length of analytic continuation. The key is to reduce to a dimensional problem.

First we note that any such that can be written where each of the components are in have norm less than , namely etc. Now, consider which defines a continuous function on which is analytic on So applying the wedge of the edge theorem we get some small analytic continuation along the ray spanned by of independent size and we are done. ∎

3.2. Conjecture

We note that the wedge-of-the-edge theorem is much stronger in the case of rational functions where it is the case that any analytic function on which extends continuously through a neighborhood of a line segment from to must analytically continue to the whole square

Roughly speaking, fix a rational function is variables which is reduced. The singular set is then the set where If is analytic on we know that cannot intersect so by an argument involveing the inverse function theorem we get that cannot be orthogonal to any positive directions at points , and therefore must have all positive coordinates. A geometric argument then gives the original claim. For a more detailed understanding of such varieties, which has been rapidly developed in recent years, see [1, 3, 2, 7, 5].

We are led to conjecture the following.

Conjecture 3.2.

Let contain the line segment from to Any continuous function which is analytic on analytically continues to

We note that in later parts of [4], they give a stratagem to show that local matrix monotonicity implies global matrix monotonicity by rational approximation schemes which, in principle, would need the conjecture above to be true.

Index

  • real wedge §2

References

  • [1] J. Agler and J.E. McCarthy. Distinguished varieties. Acta Math., 194:133–153, 2005.
  • [2] J. Agler and J.E. McCarthy. Hyperbolic algebraic and analytic curves. Indiana Math. J., 56(6):2899–2933, 2007.
  • [3] J. Agler, J.E. McCarthy, and M. Stankus. Toral algebraic sets and function theory on polydisks. J. Geom. Anal., 16(4):551–562, 2006.
  • [4] J. Agler, J.E. McCarthy, and N.J. Young. Operator monotone functions and Löwner functions of several variables. Ann. of Math., 176:1783–1826, 2012.
  • [5] Kelly Bickel, J. E. Pascoe, and Alan Sola. Derivatives of rational inner functions: geometry of singularities and integrability at the boundary. Proc. Lond. Math. Soc., 2017. to appear.
  • [6] H. Epstein. Generalization of the “edge-of-the-wedge” theorem. J. Mathematical Phys, 1:524–531, 1960.
  • [7] G. Knese. Polynomials defining distinguished varieties. Trans. Amer. Math. Soc., 362(11):5635–5655, 2010.
  • [8] J. E. Pascoe. Note on Löwner’s theorem in several commuting variables of Agler, McCarthy and Young. submitted, 2014.
  • [9] J. E. Pascoe. A wedge-of-the-edge theorem: analytic continuation of multivariable Pick functions in and around the boundary. Bull. Lond. Math. Soc., 2017. to appear.
  • [10] W. Rudin. Lectures on the Edge-of-the-Wedge Theorem. AMS, Providence, 1971.
  • [11] G. Szegő. Orthogonal Polynomials. American Mathematical Society Colloquium Publications, Providence, Rhode Island, 1939.
  • [12] V. S. Vladimirov, V. V. Zharinov, and A. G. Sergeev. Bogolyubov’s edge-of-the-wedge theorem, its development and applications. Russian Math. Surveys, 5:51–65, 1994.
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