%%  notes.tex
%%  Approximation Theory: Notes and Remarks
%%  Neal L. Carothers
%%  Bowling Green State University
%%  Bowling Green, Ohio  43403
%%  carother@math.bgsu.edu
%%  http://www.bgsu.edu/~carother/


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\line{\sc Math 680 \hfil 8/1/94}

\noindent
Here are a few corrections and
additions to certain of the handouts.
As a general rule, you can find further details
in the books by Cheney, DeVore and Lorentz, and 
Natanson.  For especially clear and elementary
proofs, Natanson has few peers; Cheney has
scrupulous references and a nice set of 
historical notes; DeVore and Lorentz are an
excellent source for modern viewpoints and
current research.  The canonical reference
for alternate versions of various classical 
theorems is Hobson, {\it The Theory of Functions 
of a Real Variable}, 2 vols., Cambridge, 1950.

\medskip

\line{\bf Preliminaries\hfil 6/27/94}

For more on strictly convex spaces (and their relatives),
see B.\ Beauzamy, {\it Introduction to Banach Spaces and their Geometry},
North-Holland Mathematics Studies, v.\ 68, North-Holland, 1982,
or R.\ B.\ Holmes, {\it Geometric Functional Analysis},
Graduate Texts in Mathematics 24, Springer-Verlag, 1975.

\medskip

\line{\bf Trigonometric Polynomials\hfil 7/5/94}

Almost all of this section is taken from de la Vall\'ee Poussin's
book {\it Le\c cons sur l'Approximation des Fonctions d'une Variable R\'eele},
Gauthier-Villars, 1919.  This book is also available as part of
the volume {\it L'Approximation}, Chelsea, 1952; also included
is a reprint of S.\ Bernstein's {\it Le\c cons sur les Propri\'et\'es
Extr\'emales et la Meilleure Approximation des Fonction Analytiques
d'une Variable R\'eele}, originally published in 1926.

\medskip

\line{\bf A Brief Introduction to Interpolation\hfil 7/12/94}

You can find out more about Vandermonde determinants from all
manner of sources; for example, I seem to recall that I first
saw them in B.\ L.\ van der Waerden's {\it Algebra}.

The estimate in the Theorem on p.\ 5 really should be written
as a pointwise estimate: 
$|f(x)-p(x)|\le {1\over{(n+1)!}}\,\Vert f^{\,(n+1)}\Vert\,|W(x)|$.

If I had thought of it at the time, I would have
included a discussion of Hermite interpolation;
that is, 
the possibility of simultaneously interpolating a
function and its first, say, $k$ derivatives.
You can find full details in Natanson, Vol.\ III.

\filbreak

\line{\bf A Brief Introduction to Fourier Series\hfil 7/18/94}

Here's a very useful observation, taken from Jackson's book
{\it Fourier Series and Orthogonal Polynomials}, MAA, 1941, p.\ 13:
If $f$ is a polygonal function (i.e., a piecewise linear,
continuous function) in $C^{2\pi}$, then the Fourier coefficients
for $f$ satisfy $|a_k|$, $|b_k|\le C/k^2$.  (Compare this to
the Theorem on p.\ 3 of my notes.)  In particular, each $2\pi$-periodic
polygonal function is the uniform limit of its Fourier series.  
Since the polygonal functions are clearly dense in $C^{2\pi}$,
this observation gives a quick proof of Weierstrass's second theorem!

For more on the applications, history, and ``culture'' of Fourier
series, see K\"orner's book, {\it Fourier Analysis}, Cambridge, 1988.

\medskip

\line{\bf Jackson's Theorems\hfil 7/18/94}

A detailed presentation in the spirit of Jackson's original 
proofs can be found in Natanson, Vol.\ I.  The version I've given 
is due to Korovkin; for more on this see Cheney.
Both Cheney and Natanson have more on inverse theorems to
share with you, too.

\medskip

\line{\bf Orthogonal Polynomials\hfil 7/20/94}

The recurrence formula for $n=1$ might look
better written $Q_1(x)=(x-a_0)\,Q_0(x)$.

I've left out a small detail from Example 1 on p.\ 3:
$b_1=1/2$ for the Chebyshev polynomials $T_n$.

\medskip

\line{\bf Gaussian Quadrature\hfil 7/26/94}

Both Natanson and Cheney have more to say about continued
fractions; Natanson also includes a proof of Hausdorff's
theorem on moment sequences (p.\ 11).  A detailed survey
of results of this type can be found in J.\ A.\ Shohat and 
J.\ D.\ Tamarkin, {\it The Problem of Moments}, AMS, 1943.

\medskip

\line{\bf The Stone-Weierstrass Theorem\hfil 8/1/94}

Two pertinent reference here are Marshall Stone's papers
``Applications of the theory of Boolean rings to
general topology,'' {\it Transactions of the American 
Mathematical Society}, Vol.\ 41, 1937, pp.\ 375--481,
and ``A generalized Weierstrass theorem,'' in 
{\it Studies in Modern Analysis}, R. C. Buck, ed., 
MAA, 1962.



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