Badly approximable functions on curves and regions

8.6.

BADLY APPROXIMABLE

Let X be a compact closed,

separates

FUNCTIONS

Hausdorff

points,

ON CURVES AND REGIONS~

space and A a uniform algebra

and contains

the constants.

take A = P(X), the uniform limits on X of polynomials. is badly approximable (with respect to A) to mean

on X, i.e., A is uniformly

For example,

if

X c ~ "~ then we might

We say that a function

q

,q~C{X)

,

where II.[I~ is the superpremum norm over X. The problems discussed here concern finding concrete descriptions of the badly approximable functions for some classical function algebras. They are the functions that it is useless to try to approximate. In this section, we let G be a bounded domain in C , with boundary X, and let A*(X) be the algebra of boundary values of continuous functions on G U X that are analytic in G. In case G is the open unit disc, then'A*(X) is the "disc algebra" (regarded as consisting of functions on X and not on G). POREDA'S THEOREM [I]. If X consists of a simple closed Jordan curve, then ~ , ~ e C ( X ) , is badly approximable with respect to A*(X) if and only if ~ has nonzero constant modulus, and ind ~ < 0. Here,

ind ~

is the index of ~ , defined

THEOREM A [2]. If ~ , ~ C ~ ) badly approximable with respect

as the winding number

, has nonzero constant to A*(X).

modulus

on X of ~

and if ind

around

~ < 0, then

THEOREM B [2]. Each badly approximable (with respect A* (X)) function inC(X) stant modulus on the boundary of the complement of the closure of G. THEOREM C [2]. Suppose that X consists is badly approximable with respect to A*(X),

0. ~

is

has con-

of N + I disjoint closed Jordan curves. If then @ has constant modulus, and ind ~ < N.

An example was given in [2] to show that the range 0 ~ ind ~ < N is indeterminate, so that one cannot tell from the winding number alone, on such domains, whether or not ~ is badly approximable. Problem I. Find necessary and sufficient conditions for a function ~ to be badly approximable with respect to A*(X) if X is a finite union of disjoint Jordan curves. Note. In the case of the annulus, X = {z:Izl = r or Izl = I} where 0 < r < I, supposing ,~ is of modulus I on X, it is shown in [2] that ~ is badly approximable with respect to A*(X) if and only if either ind ~ < 0 or ind ~ = 0 and

Problem II. The analogue of Problem I for R*(X), which is the limits on X of rational functions with poles off G, where one permits G to have infinitely many holes. Problem III. Characterize X is any compact set in ~ . Problem

III'

the badly approximable

The same as problem

III but

functions

in the special

with respect

case X = clos

to P(X), where ~ .

Despite appearances, Problem III' is just about as general as Problem III. An answer to Problem III could be called a "co-Mergelyan theorem" since Mergelyan's theorem [3] characterizes the "well-approximable" functions on X.

fLEE A. RUBEL. bana, Illinois

2226

University 61801.

of Illinois

at Urbana--Champaign,

Department

of Mathematics,

Ur-

THEOREM I]4]. If ~ is badly approximable with respect to P(O~0$~) then U~II~)~4"II~, , where II'I~ is the supremum norm over D 9 The converse is false. Problem IV. Obtain, for sets X, X c ~ , ~ 2 , any significant result about badly approximable functions with respect to any algebra like P(X), A(X), or R(X). LITERATURE CITED ] ~

2. 3. 4. 5. 6.

S. J. Poreda, "A characterization of badly approximable functions," Trans. Am. Math. Soc., 169, 249-256 (1972). T. W. Gamelin, J. B. Garnett. L. A. Rubel, and A. L. Shields, "On badly approximable functions," J. Approx. Theory, 17, 280-296 (1976). W. Rudin, Real and Complex Analysis, New York (1966). Eric Kronstadt, Private Communication, September (1977). D. H. Luecking, "On badly approximable functions and uniform algebras," J. Approx. Theory, 22, 161-176 (1978). L. A. Rubel and A. L. Shields, "Badly approximable functions and interpolation by Blaschke products," Proc. Edinburgh Math. Soc., 20, 159-161 (1976).

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