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\title{Successive Projection May not be a Good Method}
%projec[f88,jmc]		Successive projection is often not so good

	My 1951 Princeton PhD thesis, ``Projection Operators and
Partial Differential Equations'' proposed to solve first order
differential equations by successive projection between the
space of vector fields satisfying an algebraic condition and
the space of gradients.  Consider the partial differential equation
%
$$a(x,y){{∂f}\over{∂x}} + b(x,y){{∂f}\over{∂y}} = 0.$$
%
Consider further a vector field $(u,v) = (u(x,y),v(x,y))$.  Such
a vector field will be the gradient of a solution $f$ of (1),
provided $(u,v)$ is a gradient and it satisfies the linear
condition
%
$$a(x,y)u(x,y)+b(x,y)v(x,y) = 0$$
%
for each $x$ and $y$.
The gradients form a linear subspace of the space of vector fields
and so do those vector fields satisfying (2).  Let $P↓g$ be the
orthogonal projection on the space of gradients, and let $P↓a$ be
the orthogonal projection on the space of vector fields satisfying
the pointwise algebraic condition (2).  The method of (McCarthy 1951)
proposed to find a solution of (1) by starting with an arbitrary
initial vector field $(u↓0,v↓0)$ and taking the limit of
%
$$(P↓gP↓a)↑n$$
%
as $n$ goes to infinity.  This should converge to the projection $P↓{win}$
onto the space of vector fields of gradients of solutions of (1).

	What Hilbert space should be chosen for the purpose of
defining the projections?  I chose the integral of the square
of the vector over some finite region.  In this case the projection
onto the space of gradients turned out to be a certain integral operator
involving the Neumann's function of the region.

	In the thesis I proved a theorem about the convergence to the projection
process.  However, at the time I couldn't prove anything about the speed
of convergence.  It can be treated as a question of the eigenvalues of the operator
$P↓gP↓aP↓g$ which is self-adjoint.

	The object of the present note is to give an example in
which the operators and their convergence can be computed analytically.
I should have gotten this example in 1951, but I was too focussed
on the general problem to search for a simple example.

	Consider the differential equation
%
$${{∂f}\over{∂y}} = 0$$
%
on the domain bounded by the lines $x = \pm π$, $y = \pm π$.
The projection $P↓a$ therefore has the effect of removing the $y$-component
of a vector field $(u(x,y),v(x,y))$.
As for $P↓g$, consider the expansion of functions and of the components
of vector fields in Fourier series.  The gradient of a component
$e↑{i(mx+ny)}$ is the vector field $i(m,n)e↑{i(mx+ny)}$.  Thus the
gradients are vector fields such that their components have Fourier
co-efficients $c↓{mn}$ and $d↓{mn}$ related by
%
$${c↓{mn}\over {m}} = {d↓{mn}\over {n}}.$$
%
	The orthogonal complement of the gradients is easily found on
this domain.  It consists of those vector fields whose Fourier components
satisfy
%
$${c↓{mn}\over {n}} = -{d↓{mn}\over {m}}.$$
%
Therefore, the operator $P↓g$ operates on the Fourier components
of the vector field separately, i.e.
%
$$P↓g\Sigma(c↓{mn},d↓{mn})e↑{i(mx+ny)}=\Sigma[P↓{mn}(c↓{mn},d↓{mn})]e↑{i(mx+ny)},$$
%
where
%
$$P↓{mn}(c,d) = {{mc+nd}\over{m↑2+n↑2}}(m,n).$$

	Now let $(u,v) = (e↑{i(mx+ny)},0)$, i.e. $(u,v)$ is a unit vector
parallel to the $x$-axis.  We then have
%
$$P↓aP↓g(u,v) = P↓a {m\over{m↑2+n↑2}}(m,n)e↑{i(mx+ny)} = {m\over{m↑2+n↑2}}(m,0) =
{{m↑2}\over{m↑2+n↑2}}(u,v).$$

Thus $(e↑{i(mx+ny)},0)$ is an eigenvector of $P↓aP↓g$ with eigenvalue
${{m↑2}\over{m↑2+n↑2}}$.  If $n=0$, the eigenvalue is 1 as it should be,
because we are then dealing with a the gradient of a function of $x$.
Similarly, if $m=0$, the eigenvalue is 0, since we have the gradient
of a function of $y$.  Otherwise, the eigenvalue can be close to 1
or far from it.  When $m$ is large compared to $n$, we are dealing
with a function that varies rapidly with $x$ and slowly with $y$.
Such a function is only slowly diminished by the operator $P↓aP↓g$.
Because such components of the initial vector field only diminish
slowly, the method proposed in (McCarthy 1951) doesn't converge rapidly.

While this is just a single example, I would expect this behavior
to be typical.

Perhaps some polynomials in $P↓g$ and $P↓a$ would behave better.
\smallskip\centerline{Copyright \copyright\ \number\year\ by John McCarthy}
\smallskip\noindent{This draft of \jobname\ TEXed on \jmcdate\ at \theTime}
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