Approximation in AC (  )

For a nonempty compact subset 𝜎 in the plane, the space AC ( 𝜎 ) is the closure of the space of complex polynomials in two real variables under a particular variation norm. In the classical setting, AC [0, 1] contains several other useful dense subsets, such as continuous piecewise linear functions, C 1 functions and Lipschitz functions. In this paper, we examine analogues of these results in this more general setting.


Introduction
A standard method in analysis is to first prove some fact for particularly wellbehaved functions, and then extend the result to a more general class via a limiting argument. To do this, one must of course know that the functions that you are interested in can be approximated by the simpler well-behaved ones.
The class of AC( ) spaces, comprising 'absolutely continuous' functions on a nonempty compact subset of the plane, was introduced in [4,5] to provide a functional calculus model which unified the theories for the well-bounded operators of Smart and Ringrose [13,14,16] and the trigonometrically well-bounded operators of Berkson and Gillespie [9], while overcoming some of the problems inherent in the definition of AC-operators [7,8]. Much is now known about the algebraic structure of these spaces [1-3, 10, 11], but rather less has been recorded about the relationship between these spaces and other standard spaces of functions.
Throughout the paper, we will freely swap between considering the plane as ℂ or ℝ 2 . Subsets of ℝ may be considered as embedded in ℂ or as subsets of the first coordinate axis in ℝ 2 as appropriate. Given a nonempty subset ⊆ ℝ 2 , the space AC( ) is defined to be the closure of the complex polynomials p(x, y) in two real variables inside the space BV( ) of functions of bounded variation (in the sense of [5]).
In the classical situation of AC[0, 1], it is useful to know that this space contains several standard spaces as dense subsets; C 1 [0, 1], the Lipschitz functions, and the continuous piecewise linear functions. The aim of this paper is to examine to what extent the analogous facts are true in this more general setting.
In Sect. 2, we shall briefly give the main definitions and properties of the spaces involved. A more detailed account of the spaces BV( ) and AC( ) can be found in [12]. Note that the definitions given here involve a significant simplification of the original ones used in [5]. A discussion of the equivalence of the different definitions can be found in the appendix to [3].
In Sect. 3, we shall introduce the space CTPP( ) of continuous piecewise planar functions on , which forms a natural analogue of the continuous piecewise linear functions on the line. In Sect. 4, we show that in the case of real scalars, every CTPP( ) function is absolutely continuous. An important fact which we shall use in this section is that being absolutely continuous is a 'local property' in the sense that a function is absolutely continuous on if and only if it is absolutely continuous on some neighbourhood of each point in .
We then show in Sect. 5 that (suitably interpreted) we always have the inclusions of the real spaces C 1 ℝ ( ) ⊆ AC ℝ ( ) ⊆ C ℝ ( ). (These inclusions are of course in general proper, since that is the case when = [0, 1]. ) Indeed the spaces C 1 ℝ ( ) and CTPP ℝ ( ) are always dense subsets of AC ℝ ( ). In the final section we show that the earlier results for real functions can be extended to cover complex-valued functions.

Crossing segments and variation factors
Suppose that is a nonempty compact subset of the plane and that f ∶ → ℂ. The main concept of interest to us will be the variation of f over . The definition looks reminiscent of the usual definition of variation for a function defined on an interval, but must now take account of the fact that there is no longer a natural ordering of the points of the set.
Suppose that S = x 0 , x 1 , … , x n = [x j ] n j=0 is a finite ordered list of points in the plane. Note that the elements of such a list do not need to be distinct. For the moment assume that n > 0. Let S denote the piecewise linear curve joining the points of S in order.

Definition 2.1
Suppose that is a line in the plane. We say that x j x j+1 , the line segment joining x j to x j+1 , is a crossing segment of S on if any one of the following hold: 1. x j and x j+1 lie on (strictly) opposite sides of ; 2. j = 0 and x 0 ∈ ; 3. j > 0, x j ∉ and x j+1 ∈ . 9 ] and the line shown in Fig. 1.
Let vf (S, ) denote the number of crossing segments of S on and define the variation factor of S to be To cover the case where S contains a single point we set vf ([x 0 ], ) = 1 if x 0 ∈ and zero otherwise, and set vf ([x 0 ]) = 1. Note that vf (S) is well defined since 1 ≤ vf (S) ≤ n. Informally, vf (S) may be thought of as the maximum number of times any line crosses S ; the above definitions make the notion of a crossing precise.
It is clear that the variation factor is invariant under rotations or translations of the set S, and indeed under any affine transformation of S.

Variation for functions f ∶ → ℂ
We can now define the variation of f ∶ → ℂ. Suppose that S = [x j ] n j=0 is an ordered list of points in . We define the curve variation of f on the list S to be vf (S) = max vf (S, ). We include the case S = x 0 by setting cvar ( f , x 0 ) = 0. The two-dimensional variation of f on is defined to be where the supremum is taken over all finite ordered lists of elements of . The variation norm is and this is used to define the set of functions of bounded variation on ,

Properties of BV() and AC()
We record for later the most important properties of these spaces. Proofs can be found in [12].

(Affine invariance)
If is an invertible affine transformation of the plane and � = ( ), then Φ(f ) = f • −1 is an isometric isomorphism from BV( ) to BV( � ). 5. Let ℝ = {Re z ∶ z ∈ } and suppose that g ∈ BV( ℝ ). Then, the function f (z) = g(Re z) is in BV( ) and ‖f ‖ BV( ) ≤ ‖g‖ BV( ℝ ) . By affine invariance, this result can be applied to any function which is constant on a family of parallel lines. 6. BV( ) always contains the polynomials in two real variables (considering as a subset of ℝ 2 ).
It follows from (6) that the closure of the polynomials in two real variables is always a closed subalgebra of BV( ). This subalgebra is called the algebra of absolutely continuous functions on and is denoted AC(  (3), (4) and (5) also hold for AC( ) spaces.

Joining and patching
It is easy to construct examples where a function is of bounded variation on each of two compact sets, but not of bounded variation on their union. The following result from [6] gives a sufficient condition on subsets 1 and 2 of which ensures that one can 'join' BV functions on those subsets to form a function of bounded variation on .
An important fact is the following result from [12] that says that being absolutely continuous is a local property. We shall say that a set U is a compact neighbourhood

Lipschitz, piecewise planar and differentiable functions
We shall denote the Lipschitz functions on by Lip( ). This is a Banach algebra under the norm ‖f ‖ Lip( ) = ‖f ‖ ∞ + L (f ), where (For the degenerate case of a singleton set, we set L (f ) to be zero for all f.) In the following proposition [12, Theorem 3.11], C stands for the variation constant of , which is the variation of the identity function (z) = z over (considered as a subset of ℂ ). If lies inside a rectangle, then C is at most the sum of the width plus the height of the rectangle.
Unfortunately, depending on , there may be Lipschitz functions which are not absolutely continuous. Examples are given in [5,12].
Continuous piecewise linear functions on an interval are of course Lipschitz and hence absolutely continuous. The analogue for subsets of the plane are continuous piecewise planar functions. We shall denote the space of continuous piecewise .
planar functions on by CTPP( ). The main part of this paper is to show that this space is always a dense subset of AC( ).
For k = 1, 2, 3, … , let C k ( ) denote the algebra of functions f for which there is an open neighbourhood U of and a k-times continuously differentiable function F on U such that f = F| .
In the classical case, C 1 [0, 1] ⊆ AC( ). We shall use the fact that all C 1 ( ) functions can be approximated by functions in CTPP( ) to show that this result extends to arbitrary .

CTPP functions
As before, we shall assume that is a nonempty compact subset of the plane considered here as ℝ 2 . We will say that a function This is no longer valid if f is complex valued. For complex-valued planar functions, one can split f into its real and imaginary components, and find that Applying (3.1) to the real and imaginary parts shows that f is of bounded variation. Note that f is planar if and only if Re(f ) and Im(f ) are both planar.
We shall say that P ⊆ ℝ 2 is a polygon if it is a compact simply connected set whose boundary consists of a finite number of line segments. By the Two Ears Theorem, all polygons can be triangulated.
Suppose that P is a polygon in ℝ 2 and let A = {A i } n i=1 be a triangulation of P. That is, the triangles A i are proper, closed, and have pairwise disjoint interiors, and ∪ i A i = P. We say that F ∶ P → ℂ is triangularly piecewise planar over A if F|A i is planar for each A i . The set of all such functions will be denoted by CTPP(P, A), and the set of continuous and triangularly piecewise planar functions on P is defined to be For a given triangulation A of a polygon P, note that f ∈ CTPP(P, A) is necessarily continuous since if A i and A j are two triangles in A, then f |A i and f |A j must agree on A i ∩ A j .
If A and A ′ are two triangulations of P, we shall say that A is a refinement of A ′ if every triangle in A ′ is the union of triangles in A. Clearly in this case, if f ∈ CTPP(P, A � ) then f ∈ CTPP(P, A). Any two triangulations will admit a common refinement.
Definition 3.1 A function f ∶ → ℂ is continuous and triangularly piecewise planar if there exists a polygon P containing , and F ∈ CTPP(P) such that F| = f . The set of all such functions will be denoted by CTPP( ).
If a suitable polygon P exists, one can in fact choose any polygon which contains as the basis for the triangulation.
Proof Suppose that P 0 is a polygon that contains . By definition, there exists a polygon P, a triangulation A = {A i } n i=1 of P, and F ∈ CTPP(P, A) such that F| = f . If P contains P 0 , then we can take F 0 = F|P 0 . Otherwise, let R be a rectangle whose interior contains both P and P 0 . Then R�int(P) can be triangulated, resulting in a triangulation of R given by The 'ear-clipping' triangulation algorithm allows us to do this in a way that for n + 1 ≤ k ≤ m, the triangle A k has at least one side adjoining P k−1 , and at least one side disjoint from P k−1 (except at the vertices).
One may now extend F inductively, triangle by triangle. If F is already defined on P k−1 , then there exists a planar function on A k which agrees with F on the intersection A k ∩ P k−1 . The triangulation A ′ generates a triangulation A 0 of P 0 as follows. For each i, the set A i ∩ P 0 is a union of polygons, and thus can be written as a union of triangles. So F 0 = F|P 0 ∈ CTPP(P 0 , A 0 ) and F 0 | = f . ◻

Theorem 3.3 CTPP( ) is a vector space.
Proof It is clear that CTPP( ) is closed under scalar multiplication. Suppose that f , g ∈ CTPP( ), and that P is a polygon containing . By Lemma 3.2, there exist F, G ∈ CTPP(P) such that F| = f and G| = g. Clearly F + G is continuous and, using a common triangulation for F and G, is planar on polygonal regions of R.

Real CTPP functions
Some estimates are easier for real-valued functions, so we shall start by dealing with this case first, and then later give the corresponding results for complex-valued functions. We shall denote the real-valued functions in CTPP( ) and AC( ) by CTPP ℝ ( ) and AC ℝ ( ) , respectively. Suppose that is a subset of a polygon P, that f ∈ CTPP ℝ ( ), that F ∈ CTPP ℝ (P) is an extension of f to P. Since f is continuous, it attains a maximum value M and a minimum value m on . The function F ∶ P → ℝ is also an extension of f in CTPP ℝ (P) (with respect to a possibly refined triangulation). Thus, a suitable extension of f to P can always be chosen to have the same range and the same supremum norm. For a continuous function f ∶ → ℂ, let We begin by showing that all CTPP ℝ functions are Lipschitz, and hence of bounded variation. For a triangle A, let r A denote the inradius of A. Given a triangulation A = {A j } of a rectangle R, let r(A) = min j r A j .

Lemma 3.4 Suppose that A is a triangle and that F ∶ A → ℝ is planar. Then L A (F), the Lipschitz constant of F on A, satisfies
Proof Since F is planar, ∇F is constant and, for distinct x, x � ∈ A, It follows that there is a unit vector u so that L A (F) = |∇F ⋅ u|. Choose x, x ′ on the incircle of A so that x − x � = 2r A u. Then, ◻ Note that Lemma 3.4 can be improved slightly, but clearly the bound must depend on the smallest width of the triangle.

Theorem 3.5 CTPP ℝ ( ) ⊆ Lip( ) ⊆ BV( ).
Proof Let R be a rectangle that contains and suppose that f ∈ CTPP ℝ ( ). By Lemma 3.2, there exists a triangulation A = {A j } n j=1 of R and F ∈ CTPP ℝ (R, A) Suppose x, x � ∈ R are distinct. The line segment x x ′ joining x and x ′ can be written as a union of finitely many subsegments, denoted x j−1 x j (with j = 1, … , m ), with each subsegment entirely contained in a single triangle A j in A. Then, using Lemma 3.4,F and so L (f ) ≤ L R (F) = max j L A j (F). Thus, CTPP ℝ ( ) ⊆ Lip( ). The second inclusion is just Proposition 2.5. ◻ Note that the Lipschitz constant of f may be strictly smaller than the constant for an extension F. It is not always possible to choose an extension F with L (f ) = L R (F).
On the other hand, an elementary calculation shows that L (f ) = 2 ∕ √ 1 + 4 2 , which, for small , will be much smaller than L P (F).

◻
We also note that the bound that appears in Corollary 3.7 may be far from sharp.
Example 3.8 Let , f and be as in Example 3.6. Then, var (f , ) = 2 and so ‖f ‖ BV( ) = 2 2 . Let R be the rectangle with corners at (0, 0), ( , 0), ( , 2 ) and (0, 2 ), which can be split into two triangles along the diagonal from the origin to ( , 2 ), and let F(x, y) = y be the natural extension of f to R. For this triangulation, r(A) = 2 2( +1) , so calculating the bound above gives which will be much larger than ‖f ‖ BV( ) if is small.

CTPP ℝ () is contained in AC ℝ ()
Our next step is to show that every function in CTPP ℝ ( ) is not just of bounded variation, but is absolutely continuous. Suppose that f ∈ CTPP ℝ ( ) and that F is a piecewise planar extension of f to some rectangle R containing . If F is absolutely continuous, then so is its restriction f. Therefore, it will be sufficient to just consider CTPP functions defined on a rectangle. Suppose then that R is a rectangle in the plane and that f ∈ CTPP ℝ (R) with respect to a fixed triangulation A of R. Our aim to show that each point x ∈ R admits a compact neighbourhood U x on which f |U x is absolutely continuous and then to use the Patching Lemma (Theorem 2.4).
Note that since we can choose R as large as we like, it is sufficient to do this for each interior point of R as this will ensure that f is absolutely continuous on any compact subset of the interior of R. Doing this avoids some special cases in the argument below. Note that these three cases are exhaustive and mutually exclusive, and that the classification of x depends on the triangulation. Dealing with the first class of points is straightforward.
Suppose first that x is a planar point lying in the triangle A ∈ A. In this case x is an interior point of A and so U x = A is a compact neighbourhood of x. As f is planar on U x , it is absolutely continuous on this compact neighbourhood of x.

Lemma 4.2
Suppose that x is an edge point for f ∈ CTPP ℝ (R) with respect to the triangulation A. Then, there exists a compact neighbourhood U x of x such that f |U x ∈ AC(U x ).
Proof By affine invariance, we may assume that x = (0, 0). Moreover, we may assume that the two triangles that x lies in intersect on the line x = 0. Thus, there exists some small square R 0 centred at the origin such that if (x, y) ∈ R ∩ R 0 , then The most difficult case is that of vertex points. Let Q = Q t = [−t, t] 2 ⊂ ℝ 2 . We shall say that a function f ∶ Q → ℝ is star-planar on Q, if there exists a partitioning A = {A i } n i=1 of Q given by n ≥ 2 rays starting at the origin such that f is planar on each set A i . (If desired, one may enforce that each A i is triangular by adding the rays from the origin to the four corners of Q.)

Lemma 4.3 If f is star-planar on Q with respect to
Proof First note that if f is star-planar on Q with respect to {A i } n i=1 , then it is starplanar with respect to a finer partition {A � i } m i=1 with m ≤ 2n formed by extending the rays to full lines. The illustration below provides an example of such an extension.

Without loss, assume that
The finer partition and ordering of A ′ i s has the property that B k = ⋃ k i=1 A � i is convex for each 1 ≤ k ≤ m 2 . By using Theorem 2.3 inductively, one deduces that we find that f | C k ∈ BV(C k ). Since Q = B m∕2 ∪ C m∕2 is convex, using Theorem 2.3 one last time gives f ∈ BV(Q) with the claimed bound. ◻

Lemma 4.4
Suppose that x is a vertex point for f ∈ CTPP ℝ (R) with respect to the triangulation A. Then, there exists a compact neighbourhood U x of x such that f |U x ∈ AC(U x ).
Proof By affine invariance, we may assume that x = (0, 0). Also, by replacing f with f − f (0, 0), we may assume that f (0, 0) = 0. Let h = f , let B = Q �(−s∕3, s∕3) 2 , and suppose that w ∈ B. As Q contained only one vertex point, namely the origin, then w is either a planar point or an edge point for f. As shown earlier, there is a compact neighbourhood V w of w such that f |V w ∈ AC(V w ). Clearly |V w ∈ AC(V w ), and so h|V w ∈ AC(V w ). On the other hand, if w ∈ (−s∕3, s∕3) 2 , then h = 0 on an open neighbourhood of w, and so again we can choose a compact neighbourhood V w of w such that h|V w ∈ AC(V w ). It follows from the Patching Lemma (Theorem 2.4) that h ∈ AC(Q ). Moreover, we have that Combining these results with the Patching Lemma gives the following corollary.  Note that each triangle has diameter less than or equal to . Let g be the function in CTPP(R, A) which agrees with f at all vertices in the above triangulation.

C 1 functions are absolutely continuous
Fix a triangle A ∈ A. The bounds in (5.1) imply that there exist m, M such that M − m < and m ≤ f ≤ M on A. As g is planar on A (so its extrema occur at the vertices of A) and g agrees with f at the vertices of A, we also have that m ≤ g ≤ M. So for any x ∈ A, We will now estimate the Lipschitz constant of d = f − g. Suppose that x, y ∈ A are distinct, and let denote the line segment from x to y. By the Mean Value Theorem, there exists q ∈ such that The planarity of g implies that ∇g is constant. Using the fact that f = g on the vertices of A, applying Rolle's Theorem to d along the sides of A parallel to the coordinate axes yields and on the boundary of A such that Thus, we can estimate ‖∇d(q)‖ as follows: by (5.1), and so we have that One may extend this last inequality to general distinct x, y ∈ R by splitting the line segment between x and y into segments which lie entirely in a single triangle in the spirit of the proof of Theorem 3.5. Thus, using Proposition 2.5, Thus, C 1 ℝ (R) ⊆ cl(CTPP ℝ (R)) ⊆ AC ℝ (R). ◻ Theorem 5.2 Suppose that is a nonempty compact subset of the plane. Then C 1 ℝ ( ) is a dense subset of AC ℝ ( ).
Proof Suppose that f ∈ C 1 ( ), and thus admits a C 1 extension (also denoted by f) on some open neighbourhood V of . Suppose that x ∈ . Then, there exists a closed rectangle R x centred at x which lies inside V. By Theorem 5.1, f |R x ∈ AC(R x ). The set U x = R x ∩ is a compact neighbourhood of x such that f |U x ∈ AC(U x ). By the Patching Lemma, one deduces that f ∈ AC( ). The density of C 1 ( ) follows from the fact that the polynomials are contained in C 1 ( ). ◻