Quasianalyticity in certain Banach function algebras

Let $X$ be a perfect, compact subset of the complex plane. We consider algebras of those functions on $X$ which satisfy a generalised notion of differentiability, which we call $\mathcal{F}$-differentiability. In particular, we investigate a notion of quasianalyticity under this new notion of differentiability and provide some sufficient conditions for certain algebras to be quasianalytic. We give an application of our results in which we construct an essential, natural uniform algebra $A$ on a locally connected, compact Hausdorff space $X$ such that $A$ admits no non-trivial Jensen measures yet is not regular. This construction improves an example of the first author (2001).

Let X be a perfect, compact subset of the complex plane C. We consider those normed algebras consisting of complex-valued, continuously complexdifferentiable functions on X, denoted D (1) (X).These algebras were introduced by Dales and Davie in [9] and further investigated, for example, in [2] and [10].The algebra D (1) (X) need not be complete, and the completion of D (1) (X) need not be a Banach function algebra in general.
Bland and the first author [2] introduced F -differentiation, which generalises the usual complex-differentiation, and considered normed algebras of F -differentiable functions, denoted D F (X).The algebra D (1) F (X) is complete and D (1) (X) ⊆ D Dales and Davie ( [9]) also considered those algebras of complex-valued functions which have continuous complex-derivatives of all orders, and introduced the Dales-Davie algebras D(X, M).They defined a notion of quasianalyticity for these algebras and gave sufficient conditions for the algebra D(X, M) to be quasianalytic.(For the classical definition of quasianalytic collections of functions, see [27,Chapter 19].) In this paper, we define a notion of quasianalyticity for infinitely Fdifferentiable functions (defined later) and give a sufficient condition for this new notion of F -quasianalyticity.In certain cases, this sufficient condition will improve that given by Dales and Davie in [9].We conclude the paper with the construction of a uniform algebra A on a locally connected, compact Hausdorff space X such that A is essential and A does not admit any nontrivial Jensen measures yet is not regular.(The relevant definitions are given in Section 4.) This construction improves an example of the first author from [13].

Definitions and Basic results
Throughout this paper we say compact plane set to mean a non-empty, compact subset of the complex plane C. We denote the set of non-negative integers by N 0 and the set of positive integers by N. Let X be a compact Hausdorff space.We denote the algebra (with pointwise operations) of all continuous, complex-valued functions on X by C(X).For E ⊆ X, we set With the norm | • | X , C(X) is a commutative, unital Banach algebra.Let S be a subset of C(X).We say that S separates the points of X if, for each x, y ∈ X with x = y, there exists f ∈ S such that f (x) = f (y).
Definition 1.1.Let X be a compact Hausdorff space.A normed function algebra on X a normed algebra (A, • ) such that A is a subalgebra of C(X), A contains all constant functions and separates the points of X, and, for each f ∈ A, f ≥ |f | X .A Banach function algebra on X is a normed function algebra on X which is complete.A uniform algebra is a Banach function algebra such that Let X be a compact Hausdorff space and let A be a Banach function algebra on X.We say that A is natural on X if every character on A is given by evaluation at some point of X.
We refer the reader to [8,Chapter 4] for further information on Banach function algebras and uniform algebras.
We are particularly interested in Banach function algebras consisting of continuous functions on a compact plane set which satisfy some notion of differentiability.
Definition 1.2.Let X be a perfect compact plane set and let f : X → C be a function.We say that f is complex differentiable at x ∈ X if the limit z − x exists.We say that f is complex differentiable on X if f is complex differentiable at each point x ∈ X and we call the function f ′ : X → C the derivative of f .We say that f is continuously complex differentiable if f ′ is continuous.
In the remainder of this paper, we shall say differentiable and continuously differentiable to mean complex differentiable and continuously complex differentiable, respectively.We refer the reader to [7], for example, for results from complex analysis.
Let X be a perfect compact plane set.We denote the algebra of all continuously differentiable functions on X by D (1) (X).For each n ∈ N, let D (n) (X) denote the algebra of all n-times continuously differentiable functions on X (defined inductively).Let D (∞) (X) := ∞ n=1 D (n) (X).Let n ∈ N and let f ∈ D (n) (X).We denote the nth derivative of f by f (n) , and we will often write f (0) for f .
n=0 be a sequence of positive real numbers.We say that M is an algebra sequence if M 0 = 1 and, for each j, k ∈ N 0 , we have We say that M is log-convex if, for each k ∈ N, we have M 2 k ≤ M k−1 M k+1 .We conclude this section with a discussion of paths in C. For the remainder of this section, let a, b ∈ R with a < b.
The endpoints of γ are the points γ(a) and γ(b), which we denote by γ − and γ + , respectively.We denote by γ * the image γ([a, b]) of γ.A subpath of γ is a path obtained by restricting γ to a non-degenerate, closed subinterval of Let γ : [a, b] → C be a path in C. We say that γ is a Jordan path if γ is an injective function.We denote the length of γ, as defined in [1,Chapter 6], by Λ(γ), and we say that γ is rectifiable if Λ(γ) < ∞ and γ is non-rectifiable otherwise.We say that γ is closed if γ + = γ − , and we say that γ is a closed Jordan path if γ is closed and γ(s) = γ(t), where s, t ∈ [a, b], implies that either s = t or s = a and t = b.We say that γ is admissible if γ is rectifiable and has no constant subpaths.The reverse of γ is the path Now suppose that γ is non-constant and rectifiable.We define the path length parametrisation γ pl : [0, Λ(γ)] → C of γ to be the unique path satisfying for all f ∈ C(γ * ).We shall use this fact implicitly throughout.
Definition 1.5.Let X be a perfect compact plane set and let F be a collection of paths in X. Define F * := {γ * : γ ∈ F }. We say that F is effective if F * is dense in X, each path γ ∈ F is admissible, and each subpath of a path in F belongs to F .
Let X be a compact plane set.We say that X is semi-rectifiable if the set of all Jordan paths in X is an effective collection of paths in X.We say that X is rectifiably connected if, for each x, y ∈ X, there exists a rectifiable path γ ∈ F such that γ − = x and γ + = y.We say that X is uniformly regular if there exists a constant C > 0 such that for each x, y ∈ X there exists a rectifiable path γ in X with γ − = x and γ + = y such that Λ(γ) ≤ C|x − y|.We say that X is pointwise regular if for each x ∈ X there exists a constant C x > 0 such that, for each y ∈ X, there exists a path γ in X with γ − = x and γ + = y such that Λ(γ) ≤ C x |x − y|.Note that each of the above conditions on X imply that X is perfect.

F-derivatives
In this section we discuss algebras of F -differentiable functions as investigated in [2] and [10].
Definition 2.1.Let X be a perfect compact plane set, let F be a collection of rectifiable paths in X, and let f ∈ C(X).A function g ∈ C(X) is an F -derivative for f if, for each γ ∈ F , we have If f has an F -derivative on X then we say that f is F -differentiable on X.
The following proposition is a list of the elementary properties of Fdifferentiable functions.Details can be found in [2] and [10].
Proposition 2.2.Let X be a semi-rectifiable compact plane set and let F be an effective collection of paths in X.
(a) Let f, g, h ∈ C(X) be such that g and h are F -derivatives for f .Then g = h.
(b) Let f ∈ D (1) (X).Then the usual complex derivative of f on X, f ′ , is an F -derivative for f .
Let X be a semi-rectifiable compact plane set, and let F be an effective collection of paths in X.By (a) of the above proposition, F -derivatives are unique.So, in this setting, we write f [1] for the unique F -derivative of an F -differentiable function.This will be the case considered throughout the remainder of this paper.We will often write f [0] for f .We write D (1) F (X) for the algebra of all F -differentiable functions on X.We note that, with the norm f F (X) is a Banach function algebra on X ([10, Theorem 5.6]).
For each n ∈ N, we define (inductively) the algebra and, for each f ∈ D (n) F (X), we write f [n] for the nth F -derivative for f .Note that, for each n ∈ N, D (n) F (X) is a Banach function algebra on X (see [2]) when given the norm In addition, we define the algebra F (X) of all functions which have Fderivatives of all orders; that is, We now describe a class of algebras of infinitely F -differentiable functions analogous to Dales-Davie algebras as introduced in [2] (see also [4]).
Definition 2.3.Let X be a semi-rectifiable, compact plane set and let F be an effective collection of paths in X.Let M = (M n ) ∞ n=0 be an algebra sequence.We define the normed algebra with pointwise operations and the norm

F-quasianalyticity
In this section, we discuss an F -differentiability version of quasianalyticity, and give a sufficient condition for a subalgebra of F (X) new notion of quasianalyticity.
We now introduce the following notion of F -quasianalyticity.
Definition 3.1.Let X be a semi-rectifiable compact plane set and let F be an effective collection of paths in X.Let A be a subalgebra of Let X be a semi-rectifiable compact plane set and let F be an effective collection of paths in X.
We now aim to give some sufficient conditions for F -quasianalyticity for the algebras D F (X, M).Our method will follow the proof in [6] of the traditional Denjoy-Carleman theorem.
For the remainder of the section we fix an admissible path Γ.We also fix F to be the collection of all subpaths and reverses of subpaths of Γ.Let M = (M n ) ∞ n=0 be a sequence of positive real numbers satisfying We write ds for integrals with respect to the path length measure.Set F (Γ * ).We will require the following lemmas.The first lemma is a summary of the properties of the log-convex minorant of a sequence of positive real numbers.We refer the reader to [25, Chapter IV] and [26,Chapter 1] for details and properties of the log-convex minorants.The properties listed below are from [6].
n=0 be a sequence of positive real numbers such that lim inf n→∞ M 1/n n = +∞.Then there exists a log-convex sequence and a strictly increasing sequence (n j ) ∞ j=0 of integers with n 0 = 0 such that: Chapter IV] for details.We will require lemmas to prove the main result.The first lemma is standard; see, for example, [7, Proposition 1.17]. [1]| γ * .
Our next lemma is an F -differentiability analogue of [6, Lemma 2].
We now check an easy special case of our result.
Fix n ∈ N 0 .We prove our claim by induction on k ≤ n.First suppose that k = n so that n − k = 0. Let z ∈ γ * .We have and applying the claim to |f [k] (ζ)|, we have This proves the claim.We now see that |f (z)| ≤ M n s n /n! for all n ∈ N 0 and all z ∈ γ * .Now since lim inf n→∞ M 1/n n < ∞, there exists R > 0 such that M n < R n for infinitely many n ∈ N. Let (n k ) ∞ k=1 be a strictly increasing sequence in N such that M n k < R n k for all k ∈ N.Then, for each z ∈ γ * , we have This holds for all z ∈ Γ * , so the result follows.
Let 0 < α < 1.As in [6], we define B (α) j,k , for j, k ∈ N 0 with k ≥ j ≥ 0, as follows.For each j ∈ N 0 , let B Our main tool in the proof of the main theorem is the following lemma, which can be distilled from the proof of [6, Lemma 1] and Stirling's approximation.We omit the proof.Lemma 3.6.Let α ∈ (0, 1/4e).Then there exists a constant K > 0 such that B (α) j,k ≤ Kα < 1/2 for all j, k ∈ N with j < k.Moreover, for each n ∈ N, we have Note that, if α ∈ (0, 1/4e), then for all j, k ∈ N 0 with k ≥ j.
We now state and prove our main result.The proof is essentially the one used in [6], adapted for F -differentiation, and including additional details for the convenience of the reader.
Proof.If lim inf n→∞ M 1/n n < ∞ then the result follows from Lemma 3.5, so suppose that lim inf n→∞ M 1/n n = ∞.By Lemma 3.2, there exists a log convex sequence (M c n ) ∞ n=0 of positive real numbers and a strictly increasing sequence (n j ) ∞ j=0 with n 0 = 0 such that: By the comments following Lemma 3.2, we have Let z ∈ Γ * and let γ ∈ F such that γ − = z 0 , γ + = z and σ = γ pl .
Define the points 0 = For each j ∈ {0, . . ., n} we claim that, for each k ∈ N 0 , with k ≤ n−j +1, The proof of the claim is by induction on j.Since f [k] (σ − ) = 0 for each k ∈ N 0 , ( * 0 k ) holds for all k ∈ N 0 with k ≤ n + 1. Fix j ∈ {1, . . ., n}.Assume now that ( * j ′ k ′ ) holds for all j ′ , k ′ ∈ N 0 with j ′ < j and k ′ ≤ n − j ′ + 1. Set i := n − j + 1.We now prove ( * j k ) holds for each k ∈ N 0 with k ≤ i by backwards induction on k.We first check the base case.Suppose that k = i.If i = n r for some r ∈ N 0 , then B (α) j,j = 1 and i for all t ∈ [0, x j ] by (7), and so ( * j k ) holds.Otherwise, i = n r for all r ∈ N 0 , in which case there exists r ∈ N 0 such that n r < i < n r+1 ≤ n.For each s ∈ [0, x j−1 ], by ( * j−1 i ) and ( 6), we have and so it remains to show that |f [i] (σ(s))| ≤ M c i for all s ∈ [x j−1 , x j ].As in [6], let m := n r+1 − i and let R := M c nr /M c nr+1 .Note that and, for each p ∈ N 0 with p ≤ m, we have and points s 0 = x j−m , . . ., s m−1 = x j−1 ), we have and, by applying ( * j−p−1 i+p ) for each 0 ≤ p ≤ m − 1, we obtain and so, by Lemma 3.6, we have This concludes the proof of the base case k = i.Now let k ∈ N 0 with k ≤ i − 1 and assume that ( * j k+1 ) holds, i.e., then, by applying ( * j−1 k ) and (6), we have Thus we may assume that s ∈ [x j−1 , x j ].By Lemma 3.3, we have Applying ( * j−1 k ) to the first term and applying ( * j k+1 ) to the second term we obtain Thus ( * j k ) holds, and both inductions may now proceed.Now, by Lemma 3.6, there exists a constant K > 0 such that, for all u, v ∈ N with v > u, we have B u,v < αK.Thus |f (σ(t))| ≤ KαM 0 for all t ∈ [0, Λ(σ)].It follows that f (σ(t)) = 0 for all t ∈ [0, Λ(σ)].In particular, |f (z)| = 0 and hence f (z) = 0. Since z ∈ Γ * was arbitrary, the above holds for all z ∈ Γ * .This completes the proof.
In the remainder of this paper we adopt the following convention.Let X be a semi-rectifiable compact plane set, let F be an effective collection of paths in X, and let f ∈ D Our first corollary will be used in the next section.
Our next corollary asserts the existence of an F -quasianalytic algebra of the form D F (X, M).Corollary 3.9.Let X be a semi-rectifiable compact plane set, let F be an effective collection of paths in X, and let M = (M n ) ∞ n=0 be an algebra sequence which satisfies (1).
We conclude this section with a note about F -analyticity, as introduced in [4] (see also [5]).Let X be a semi-rectifiable compact plane set, let F be an effective collection of paths in X, and let f ∈ D This is a generalisation of the term analytic used in [16,17], and is used to find sufficient conditions for maps to induce homomorphisms between the algebras D F (X, M). (Note that, in [15], the term analytic was used for those functions on X which extend to be analytic on a neighbourhood of X.This condition is stronger than in [16,17].)Let X be a semi-rectifiable compact plane set, let F be an effective collection of paths in X, and let f ∈ D (∞) F (X).Using Theorem 3.7, we can show that if f is F -analytic then, for each γ ∈ F and z ∈ γ * , there exist r > 0 and an analytic (in the usual sense) function g : B(z, r) → C such that g|(γ * ∩ B(z, r)) = f |(γ * ∩ B(z, r)).From this, it follows that in fact, for each γ ∈ F , there exist an open neighbourhood U of γ * and an analytic function h : U → C such that h|γ * = f |γ * .We wish to thank Prof. J. K. Langley and Dr. D. A. Nicks for showing us how to prove the latter implication.

Trivial Jensen measures without regularity
We conclude the paper with an application of the results from the previous sections.We construct a locally connected compact plane set X and an essential uniform algebra A on X such that A does not admit any non-trivial Jensen measures but is not regular.This example will improve an example of the first author ( [13]).
We begin with the relevant definitions.
Definition 4.1.Let X be a compact Hausdorff space, let A be a uniform algebra on X, and let ϕ be a character on A. A probability measure µ on X is a Jensen measure for ϕ (with respect to We say that A is regular on X if, for each closed set E ⊆ X and each point x ∈ X \ E, there exists f ∈ A such that f (x) = 0 and f (E) ⊆ {0}.We say that A is regular if the Gelfand transform of A is regular on the character space Φ A of A. We say that A is essential if there exists no proper closed subset E of X such that A contains every f ∈ C(X) such that f (y) = 0 for all y ∈ E.
In the above definition we adopt the convention that log(0) = −∞.Let X, A, ϕ be as in the above definition.It is standard that every Jensen measure for ϕ is a representing measure for ϕ.Moreover, for each ϕ ∈ Φ A , there is a Jensen measure on X for ϕ.Note that, for x ∈ X, the point-mass measure δ x is a Jensen measure for ε x , where (here, and for the remainder of the section) ε x is the evaluation character at x (with respect to A).We say that a Jensen measure µ on X for ε Let X be a compact plane set.Let R 0 (X) denote the set of restrictions to X of rational functions with no poles on X.Let R(X) denote the uniform closure of R 0 (X) in C(X).It is standard that R 0 (X) is an algebra and that R(X) is a natural uniform algebra on X.For the remainder of this section, all Jensen measures will be with respect to R(X) unless otherwise specified.
Let x ∈ X.Let J x denote the ideal in R(X) of all functions which vanish on a neighbourhood of x.Let M x denote the ideal in R(X) of all functions which vanish at x. Clearly J x ⊆ M x .We say that x is a point of continuity (for R(X)) if, for all y ∈ X \ {x} we have J y M x .We say that x is an R-point if, for all y ∈ X \ {x}, we have J x M y .For further information see [12,13,18,19].(Note that in [12] points of continuity are referred to as regularity points of type one and R-points are referred to as regularity points of type two.) It is standard that R(X) is regular if and only if every point of X is a point of continuity, and this holds if and only if every point of X is an R-point.It is also standard that if x is a point of continuity then the only Jensen measure for ε x is the point mass measure.
Let X be a topological space and let E be a subset of X.We denote by int X (E) the interior of E with respect to the topological space X.In particular, if E ⊆ C then int C E coincides with the usual interior of E.
For the remainder of this section, we denote the set of non-negative real numbers by R + .Let X be a metric space, let x ∈ X, and let r > 0. We denote the open ball in X with centre x and radius r by B X (x, r).We the denote the corresponding closed ball by BX (x, r).
In the special case where X = C, for each a ∈ C and r > 0, we write B(a, r) = B C (a, r) and B(a, r) = BC (a, r).For each a ∈ C, we set B(a, 0) = ∅ and B(a, 0) = {a}.
Lemma 4.2.Let X, Y be compact plane sets with Y ⊆ X. Suppose that R(Y ) is regular.Then each point y ∈ int X (Y ) is a point of continuity for R(X).
Since R(Y ) is regular, it follows (from [14, lemma 1.6], for example) that R(Y ∩ B(y, r)) is regular.Set E := BX (y, r) \ B X (y, δ).Then there exists a function g ∈ R(Y ∩ BX (y, r)) such that f (y) = 1 and f (x) = 0 for all x ∈ E. Let f ∈ C(X) be given by f (z) = g(z) for all z ∈ X ∩ BX (y, r) and f (z) = 0 for all z ∈ X \ BX (y, r).It follows from [24,Corollary II.10.3] that f ∈ R(X) and clearly f vanishes on a neighbourhood of x.Thus J x M y (where these are the ideals in R(X)).It follows that y is a point of continuity for R(X) and so the proof is complete.
Lemma 4.3.Let X be a compact plane set and let x ∈ X. Suppose that there exists a neighbourhood U of x in X such that every point in U \ {x} is a point of continuity.Then x is an R-point.
4. An abstract Swiss cheese is a sequence n=0 be an abstract Swiss cheese.Set and set ρ(A) = ∞ n=1 r n .We say that A is classical if ρ(A) < ∞, r 0 > 0 and for all k ∈ N with r k > 0 the following hold: We say that a compact plane set X is a Swiss cheese set if there exists an abstract Swiss cheese A such that X = X A .We say that a Swiss cheese set X is classical if there exists a classical abstract Swiss cheese A with X = X A .If X is a classical Swiss cheese set then X is a uniformly regular (see the proof of [10,Theorem 8.3]) and R(X) is essential (see [3, p. 167] or [14,Theorem 1.8]).It follows that if A is classical then X A is also connected and locally connected.
In [13], the first author gave an example of a non-classical Swiss cheese set X such that R(X) has no non-trivial Jensen measures, but such that R(X) is not regular.We shall show that there is a classical Swiss cheese set with these properties; this is the content of the following theorem, which is the main theorem of this section.Theorem 4.5.There exists a classical abstract Swiss cheese A = ((a n , r n )) such that R(X A ) is not regular and does not admit any non-trivial Jensen measures.
Most of the remainder of this section is devoted to the proof of this theorem.We require some preliminary results.The following proposition is [19,Lemma 4.1].Proposition 4.6.Let X be a compact plane set, let Y be a non-empty closed subset of X, and let x ∈ Y .Suppose that no bounded component of C \ Y is contained in X, and that there exists a non-trivial representing measure µ for ε x with respect to R(X) such that supp µ ⊆ Y .Then µ is a non-trivial representing measure for ε x with respect to R(Y ), and R(Y ) = C(Y ).
Note that, if int C X = ∅ then the condition on bounded components of C \ Y is automatically satisfied.
Let X be a compact plane set with int C X = ∅, let x ∈ X, and let µ be a non-trivial representing measure for ε x .Let Y = supp µ ∪ {x}, where supp µ denotes the closed support of µ.Then, by the above, we must have R(Y ) = C(Y ).In particular, as noted in [19], Y must have positive area.(See also the Hartogs-Rosenthal theorem [22,Corollary II.8.4].) Combining these observations with Proposition 4.6 gives the following corollary, which we use below.
Corollary 4.7.Let X be a compact plane set with int C (X) = ∅, let E be a closed subset of X, and let x ∈ E. Suppose that E has area 0, that µ is a Jensen measure for ε x with respect to R(X), and µ is supported on E. Then µ is trivial.
We also require the following lemma, which is a special case of [19, Lemma 2.1].Lemma 4.8.Let X be a compact plane set and let x ∈ X. Suppose that µ is a non-trivial Jensen measure for x, and let F be the closed support of µ.Then, for all y ∈ F \ {x}, we have J y ⊆ M x .Thus x is not a point of continuity and no point of F \ {x} is an R-point.
The following estimates on derivatives are standard.See, for example [19,Lemma 4.4].(This result also appears in [13] but with some typographical errors.)Lemma 4.9.Let A = ((a n , r n )) ∞ n=1 be an abstract Swiss cheese, and let z ∈ C. For each n ∈ N, let d n denote the distance from B(a n , r n ) to z.Let d 0 = r 0 − |z − a 0 |.Suppose that d n > 0 for all n ∈ N 0 .Then z ∈ X A and, for all f ∈ R 0 (X A ) and k ∈ N 0 , we have Our construction will use the following proposition, which is a combination of [21, Lemma 8.5] and, for example, [14, Example 2.9].Proposition 4.10.Let a ∈ C, λ 1 ≥ 0, λ 0 > λ 1 , and ε > 0. Then there exists a classical abstract Swiss cheese A = ((a n , r n )) ∞ n=0 such that a 0 = a 1 = a, r 0 = λ 0 and r 1 = λ 1 such that R(X A ) is regular and ∞ n=2 r n < ε.Note that, since R(X A ) is regular, we must have int C X A = ∅.We now give the details of the construction.Proof.Our abstract Swiss cheese A will be obtained by combining a certain pair of sequences (A n ), (B n ) of abstract Swiss cheeses in a suitable way.We first construct the sequences (A n ), (B n ).Choose a positive integer n 0 large enough so that r + 2 1−n 0 < 1 and r − 2 1−n 0 > 0. As in [19], choose a sequence (γ n ) of positive real numbers such that, for each n, k ∈ N, we have It is also possible to show that R(X A ) admits no non-trivial Jensen measures by appealing to the theory of Jensen interior.(See, for example, [24, p. 319].)This is the approach used in [13].
Our final corollary follows immediately from Theorem 4.5.
Corollary 4.12.There exists a locally connected compact plane set X such that R(X) is essential, non-trivial and non-regular and yet R(X) admits no non-trivial Jensen measures.
We conclude with some open questions.
Question 4.13.Is the uniform algebra R(X A ) constructed in Theorem 4.5 necessarily antisymmetric?If not, can the construction be modified to yield an example which has the properties in that theorem and is also antisymmetric?
Question 4.14.Let A be a uniform algebra on a compact Hausdorff space X, and let M i (i ∈ I) be the decomposition of X into maximal A-antisymmetric subsets.
(a) Suppose that A|M i is regular on M i for all i ∈ I. Must A be regular on X? What if we assume the stronger condition that A|M i is regular (so natural and regular on M i ) for all i ∈ I?
(b) What is the answer to (a) in the special case where X is a compact plane set and A = R(X)?
= 0 and, for each j ∈ N, let B (α) j,j = 1.For each j, k ∈ N 0 with k > j, define B

Theorem 3 . 7 .
Let Γ : [a, b] → C be an admissible path and let F denote the collection of all subpaths and reverses of subpaths of Γ

Lemma 4 . 11 .
Let 0 < r < 1 be given, let C r denote the circle of radius r centred at 0, and let ε > 0. Then there exists a classical abstract Swiss cheeseA = ((a n , r n )) such that (a) ρ(A) < ε, int C (X A ) = ∅, and C r ⊆ X A , (b) there is a dense open subset U of X A such that X A \ U has area zero and each point z ∈ U is a point of continuity for R(X A ), (c) for each f ∈ R(X), we have f |C r ∈ D (∞) (C r ) and ∞ k=1 |f (k) | −1/k Cr = ∞.