Misc Academic Texts
by Ted Kaczynski
1967 - Boundary Functions
[doctoral dissertation]
Ann Arbor: University of Michigan
Let H denote the set of all points in the Euclidean plane having
positive y-coordinate, and let X denote the x-axis. If p is a point of X, then
by an arc at p we mean a simple arc v, having one endpoint at p, such that v -
{p} ( H. Let f be a function mapping H into the Riemann sphere. By a boundary
function for f we mean a function t defined on a set E ( X such that for each p
( E there exists an arc v at p for which
lim f(z) = t(p).
z -> p
z ( v
The set of curvilinear convergence of f is the largest set on which a boundary function for f can be defined; in other words, it is the set of all points p ( X such that there exists an arc at p along which f approaches a limit. A theorem of J.E. McMillan states that if f is a continuous function mapping H into the Riemann sphere, then the set of curvilinear convergence of F is of type F(sd). In the first of two chapters of this dissertation we give a more direct proof of this result than McMillan's, and we prove, conversely, that if A is a set of type F(sd) in X, then there exists a bounded continuous complex-valued function in H having A as its set of curvilinear convergence. Next, we prove that a boundary function for a continuous function can always be made into a function of Baire class 1 by changing its values on a countable set of points. Conversely, we show that if t is a function mapping a set E ( X into the Riemann sphere, and if t can be made into a function of Baire class 1 by changing its values on a countable set, then there exists a continuous function in H having t as a boundary function. (This is a slight generalization of a theorem of Bagemihl and Piranian.) In the second chapter we prove that a boundary function for a function of Baire class e > 1 in H is of Baire class at most e + 1. It follows from this that a boundary function for a Borel-measurable function is always Borel-measurable, but we show that a boundary function for a Lebesgue-measurable function need not be Lebesgue-measurable. The dissertation concludes with a list of problems remaining to be solved.
Boundary Functions and Sets of
Curvilinear Convergence For Continuous Functions
Trans. Amer. Math. Soc. 141 1969 107--125
The author completes the investigation, initiated by Bagemihl and Piranian, of boundary functions of continuous complex-valued functions defined in the open unit disk D. the set of curvilinear convergence A of such a function f is defined to be the set of those e^iT at which f has a finite or infinite limit along some open Jordan arc lying in the disk and having one endpoint at e^iT. A boundary function of f is a function t defined on A such that each t(e^iT) is one of these limit values. The author proved that t differs from some function of the first Baire class at at most countably many points, and McMillan proved that A is of type F(sd). By means of an intricate construction, the author proves that for any set A on the unit circle of type F(sd), and for any function t defined on A that differs from some function of the first Baire class at at most countably many points, there exists a continuous complex-valued function f defined in D having A as its set of curvilinear convergence and having t as its boundary function. The author points out the the problem remains open for real-valued functions.
On A Boundary Property of
Continuous Functions
Michigan Math. J. 13 1966 313--320
The author generalizes the result of McMillan (1966) to the effect that the set of curvilinear convergence of a continuous function f from D into Z is of type F(sd). The generalization considers f as a continuous function from D into a compact metric space E. Topologizing the set of closed sets C(E) of E with the Hausdorff metric and letting E be any closed set in C(E), it is shown that the set of all x ( C such that there is a boundary path v at x with the cluster set of f along v contained in some set of E is of type F(sd). Taking E to be the set of all singletons {y}, y ( E (which is closed in C(Z)) McMillan's theorem is obtained.
Various other corollaries are given by selecting appropriate closed sets E ( C(E).
The Set of Curvilinear
Convergence of A Continuous Function
Defined In The Interior of A Cube
Proc. Amer. Math. Soc. 23 1969 323--327
The set of points of the unit circle at which a continuous complex-valued function in the open unit disk has limits along curves (asymptotic values) is of type F(sd) and, in general, has no other properties. The author shows that for continuous complex-valued functions defined in a cube, this set of "curvilinear convergence" does not even need to be a Borel set. He asks whether such an example can be given for real-valued functions.
Boundary Functions For Bounded
Harmonic Functions
Trans. Amer. Math. Soc. 137 1969 203--209
A function p(e) defined on the unit circle is a boundary function for a function f(z) defined in the unit disk provided for each e, f(z) has the limit p(e) at e along some curve lying in the unit disk and having one endpoint at e. Any two boundary functions for the same function f differ at only countably many points by the ambiguous-point theorem of Bagemihl; and a boundary function for a continuous function differs from some function in the first Baire class at only countably many points. In answer to a question of Bagemihl and Piranian, the author constructs a bounded harmonic function having a boundary function that is not in the first Baire class. He shows that nevertheless the set of points of discontinuity of such a boundary function is a set of the first Baire category.
Boundary Functions For Function
Defined In A Disk
1965 589--612. . J. Math. Mech.
Let D denote the unit disk |z| < 1, C its boundary, and let f(z) be any
function that is defined in D and takes its values in some metric space S. Then
a boundary function for f is a function t on C such that for every x ( C there
exists an arc v at x with
lim f(z) = t(x).
z -> x
z ( v
The author proves several theorems on boundary functions in the following four cases: (1) f(z) a homeomorphism of D onto D, (2) f(z) a continuous function, (3) f(z) a Baire function and (4) f(z) a measurable function. These theorems include answers to two questions raised by Bagemihl and Piranian.
Theorem 1 states that if f(z) is a homeomorphism of D onto D, then there exists a countable set N such that t|C - N is continuous.
In the case of continuous functions, one needs some definitions. Let S and T be metric spaces. f is said to be of Baire class 1(S, T) if and only if (i) domain f = S, (ii) range f ( T and (iii) there exists a sequence {f(n)} of continuous functions, each mapping S into T, such that f(n) -> f pointwise on S. g is of honorary Baire class 2(S, T) if and only if (i) domain g = S, (ii) range g ( T and (iii) there exists a function f of Baire class 1(S, T) and a countable set N such that f|S - N = g|S - N. Using these defnitions, Theorems 2 and 3 read as follows. Theorem 2: Let f be a continuous real-valued function in D and let t be a finite-valued boundary function for f. Then t is of honorary Baire class 2(C, R), where R is the set of real numbers. Theorem 3: Let f be a continuous function mapping D into the Riemann sphere S and let t be a boundary function for f. Then t is of honorary Baire class 2(C, S).
In the cases of Baire functions and measurable functions, for the sake of convenience consider the open upper half-plane D^0: I(z) > 0, and its boundary C^0: I(z) = 0, instead of D and C, respectively. Theorem 4 states that if f is a real-valued function of Baire class a > 1 in D^0, and t is a finite-valued boundary function, then t is of Baire class a + 1. As an immediate consequence of Theorem 4, one has Theorem 5: Let f be a real-valued Borel-measurable function in D^0 and let t be a finite-valued boundary function for f; then t is Borel-measurable.
Next, the author proves that for an arbitrary function t on C^0, there exists a function f on D^0 such that f(z) = 0 almost everywhere and t is a boundary function for f. The paper concludes with some remarks concerning extensions of these theorems into three dimensions.
Also See:
Kaczynski's 1967 80 page doctoral thesis on "boundary functions" won "best
thesis of year" in the math department at U Michigan.
Kaczynski, T.J. 1964. Another proof of Wedderburn's theorem. Am.
Math. Month. 71:652-653.
Kaczynski, T.J. 1964. Distributivity and (-1)x = -x. Am. Math. Month.
71:689.
Kaczynski, T.J. 1965. Distributivity and (-1)x = -x [with solution by Bilyeau,
R.G.]. Am. Math. Month. 72:677-678.
Kaczynski, T. J. Note on a problem of Alan Sutcliffe. Math. Mag. 41
1968 84--86. (Reviewer: B. M. Stewart) 10.05
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