In this paper, we show that holomorphic functions on ``wedge domains''
can be locally represented via an integral against a measure defined
on the ``edge of the wedge''
where
an ``edge'' is a submanifold M of C^{n} of real codimension greater
than 1.
Our edge
is required to have ``type 2'', which
means that the image of the Levi form sweeps out the normal cross section
of the wedge domain.

Our approach is to first study a certain ``model case'' where the edge is a nilpotent Lie group of step 2, and then show that the general case can be obtained by a three stage process which is inspired by Stein's work: (i) we pass from the original object of study to a ``free'' object by adding appropriate variables; (ii) we solve the problem on the freed object by approximating it suitably by the model case; (iii) we return to the orginal object by integrating out the extra variables.

We also apply these
local integral representation formulas to the
study of H^{p} functions on wedge domains with
a generic type 2 edge.
In particular, we prove that H^{p}
functions
have admissible limits almost everywhere along the edge, for 0 < p <= infinity,
and we obtain a necessary condition for a CR distribution on the
edge to be the boundary values of an H^{p} function.

Full postscript version is available.

In this paper, we discuss models that are appropriate for submanifolds of
C^{n} of high codimension that are
of higher type. The analysis of the most general submanifold of higher type
is complicated. Here, we focus on the simpler class of rigid submanifolds of
higher type where the dimension of the holomorphic tangent space at each point is one. For this class, we discuss three models: a tube-like model
that only depends on the real part of the holomorphic tangent coordinate; a
radial model that depends on the modulus of the holomorphic tangent coordinate;
and a free model which is the most general of our models. As we show, the
tube-like and free models have a Lie group structure that are generalizations
of the Lie group structure for the boundary of a Siegel domain.

We show that the hull of holomorphy of the tube-like model is its convex hull. We also show that the hull of holomorphy of the more general free model is strictly smaller than its convex hull. This leaves open the interesting question as to the exact nature of the hull of holomorphy of this basic model.

This paper is a proceedings conference version of a longer
manuscript entitled ``Local Reproducing Kernels on Wedge-Like Domains with Type 2 Edges''. This manuscript was delivered at a conference honoring
Eli Stein.
In this paper, we show that holomorphic functions on ``wedge domains''
can be locally represented via an integral against a measure defined
on the ``edge of the wedge''
where
an ``edge'' is a submanifold M of C^{n} of real codimension greater
than 1.
Our edge
is required to have ``type 2'', which
means that the image of the Levi form sweeps out the normal cross section
of the wedge domain.

Our approach is to first study a certain ``model case'' where the edge is a nilpotent Lie group of step 2, and then show that the general case can be obtained by a three stage process which is inspired by Stein's work: (i) we pass from the original object of study to a ``free'' object by adding appropriate variables; (ii) we solve the problem on the freed object by approximating it suitably by the model case; (iii) we return to the orginal object by integrating out the extra variables.

We also apply these
local integral representation formulas to the
study of H^{p} functions on wedge domains with
a generic type 2 edge.
In particular, we prove that H^{p}
functions
have admissible limits almost everywhere along the edge, for 0 < p <= infinity,
and we obtain a necessary condition for a CR distribution on the
edge to be the boundary values of an H^{p} function.

Full postscript version is available.

We consider the set of CR functions
on a connected tube submanifold of C^{n} satisfying a uniform bound
on the L^{p} - norm in the tube direction. We show that
all such CR functions holomorphically extend to H^{p} functions on
the convex hull of the tube (1 <= p <= infinity).
The H^{p} - norm of the extension is shown to be the same
as the uniform L^{p} - norm in the tube direction
of the CR function.

We show that a CR function of class C^{k}, 0 <= k <= infinity,
on a tube submanifold of C^{n} holomorphically extends to
the convex hull of the submanifold. The extension
and all its derivatives through order k are shown to have nontangential
pointwise boundary values on the original tube submanifold.
The C^{k} - norm of the extension is shown to be no bigger than the C^{k} - norm
of the original CR function.

We consider the space, CR^{p} (M), consisting of CR
functions which also lie in L^{p} (M)
on a quadric submanifold M of C^{n} of
codimension at least one. For 1 <= p <= infty, we prove
that each element in CR^{p}(M) extends uniquely
to an H^{p} function on the interior of the convex hull of M.
As part of the proof, we establish a semi-global version
of the CR approximation theorem of Baouendi and Treves
for submanifolds which are graphs and whose graphing functions
have polynomial growth. AMS Classification Numbers 32, 42 and 43.

Let M be a hypersurface in
C^{n} that is
the graph over a (2n-1)-real dimensional linear space.
The main result of the
paper is that any CR function on M can be uniformly approximated
on compact subsets
by entire functions on C^{n}.

This booklet is an outgrowth of summer workshops for Advanced Placement Calculus high school teachers that I taught in the summers of 1994 and 1995. I have two goals for this booklet. The first is to explain some of the highlights of calculus, using the graphing calculator, where appropriate, as a teaching aid. The second goal is to provide a set of problems, some of which use the graphing calculator as a tool to help with graphics or computations that may be cumbersome to do by hand.

This booklet is not intended to replace a standard calculus book, but rather to enhance it. My goal is to state the central ideas that are at the heart of some of the more important concepts of calculus at the level of the AB advanced placement calculus curriculum. Standard calculus books also contain the ideas in this booklet. However a published book includes additional material for completeness which sometimes muddles the central ideas at hand. In addition, some of the problems involving the calculator are somewhat novel and may not appear in many of the standard texts. The problems in the booklet vary from the routine to the quite difficult. Most of the more difficult problems are at the very end in the {\em Challenge Problems} section. In addition, there are some ideas for extended project ideas in section VIII for your students to work over a longer period of time than what would be required for routine homework problems.

- Functions
- Differentiation
- Applications of the Derivative
- The Relationship Between the First and Second Derivatives
- The Definition of the Integral
- Fundamental Theorem of Calculus
- Applications of Integration
- Project Ideas