If you want to know more about the Principle of Least Action and Hamilton's Principle, the following famous physics book kicks off with those instead of Newton's Equations of Motion:
Note that the conference webpage
created by my friend Lotfi Hermi (currently at Florida International Univ.)
contains a lot of quite interesting and useful information about the
isoperimetric problems and their applications.
For the historical study of isoperimetric problems conducted by Euler and
Lagrange, see:
Also, there is an amusing article on dogs and calculus of variations (
not necessarily the same dog swimming problem I discussed in the class though)
as follows:
There are a vast amount of literature on spline interpolation, spline approximation, and scattered data interpolation, as you can easily imagine. Below, I list a few interesting articles discussing the variational aspects of such problems:
The books on splines and scattered data interpolation are too numerous to list them here. I only list the following books that provide comprehensive,
multidimensional, and modern aspects for your reference:
I stated the Lagrange polynomial for the equispaced
points on an interval leads to Runge's phenomenon.
Then you may have several naturally occurring questions regarding algebraic
polynomial interpolations, e.g.,
How about the other point configuration? How to design a good point
configuration that are less detrimental than the equispaced points?
What is a good way to measure the goodness of the Lagrange polynomial
interpolation for a given set of points that are not necessarily equispaced?
The so-called Lebesgue constant provides such a measure.
The following references discuss this concept and attempts to answer the above
questions:
This new book by Nick Trefethen is an excellent resource for interpolation and
approximation in terms of both theoretical and numerical aspects, but
it is restricted on 1D problems.
Lectures 7-8: Basics of PDEs; String/Wave and Heat Equations
Other than Chap. II of the textbook, I would recommend the following literature:
I also briefly mentioned anisotropic diffusion equations and their use for image enhancement. The following references discuss these interesting ideas:
The potential theory (as well as the heat equation) is deeply connected to the theory of Brownian motion and random walks.
This was started by the following ground-breaking paper of Shizuo Kakutani:
I also used Laplace's and Poisson's equations for image processing
applications (e.g., approximation and compression), and got even patents
in US and Japan. If you are interested, check the following articles:
The backward heat equation (a.k.a. antidiffusion) is a typical example of ill-posed problems, yet there have been interest in many applications. One of typical applications is image deblurring/sharpening:
This equivalence has a practical importance: if you have a good numerical solver for Poisson's equation with, say, homogeneous Dirichlet boundary condition,
then you can use that solver to solve Laplace's equation with
inhomogeneous Dirichlet boundary condition.
I highly recommend the following articles, which describe
one of the nicest (i.e., very accurate and fast) Poisson solvers on a 2D
rectangle or a 3D rectangular cuboid:
H. Dym & H. P. McKean: Fourier Series & Integrals, Academic Press, 1972.
The last two books require some basic knowledge of the measure theory.
Of course, any serious person who is interested in Fourier series cannot
miss the following Bible:
Importance of Fourier Analysis according to Cornelius Lanczos (1893-1974)
On the occasion of the 120th anniversary of Cornelius Lanczos's birth
the NA group at Manchester made available online a series of video tapes
produced in 1972: http://www.maths.manchester.ac.uk/lanczos. I highly recommend you to watch this video series. In particular, in Tape 1, he talks about the most relevant subjects for this course. Don't miss the video segment around 24-30 minutes where he talks about the importance of Fourier analysis!
Lectures 15: Functions of Bounded Variations and Fourier Series
For the basics of Functions of Bounded Variation, see, e.g.:
The definition of BV in higher dimensions can be found in:
L. C.
Evans and R. F. Gariepy: Measure Theory and Fine Properties of
Functions, Revised Edition, CRC Press, 2015, Chap.5.
Lectures 16: Fourier Series on Intervals
For a general description on the Fourier Series on Intervals, Fourier Cosine and Sine Series, see, e.g.:
Folland: Fourier Analysis, Sec. 2.4.
For a given function supported on an interval, there are three standard
ways to extend it to the outside of the interval and make it periodic:
1) periodic extension; 2) even reflection followed by double periodic extension
(i.e., doubling the period); and 3) odd extension followed by double periodic
extension. Of course, in general, Choice 2 would be best in general, but
there are other ways to extend and periodize a given function on an interval
(in 1D) or on a rectangle (in higher dimensions). For 1D cases, see the
following works of Lanczos:
On the Gibbs (or more appropriately Gibbs-Wilbraham) phenomenon,
almost all books on Fourier analysis listed above, e.g.,
Dym-McKean, Folland, Hardy-Rogosinski, Körner, Lanczos,
Pinsky, Walker, Zygmund, describe it, yet the following interesting and
scholastic paper is clearly the best information source:
In higher dimensions, very peculiar things can happen. Gray and Pinsky
discovered a Gibbs-like phenomenon can occur at a point of continuity
in Rd for d > 2.
It is known as the Pinsky phenomenon. See the following articles
for the details.
I highly recommend both books. In particular, Young's book is one of my favorite books in the area of Hilbert space: extremely well written and motivating.
For the inclusion relationships between metric, normed, inner-product spaces and their complete versions, i.e., complete metric, Banach, and Hilbert spaces,
see the following math.stackexchange.com posting, in particular, the most voted answer:
E. Kreyszig: Introductory Functional Analysis with Applications, Chap.1, 2, 3, John Wiley & Sons, Inc., 1978.
I discussed the notion of nonlinear approximation along with
the usual best linear approximation in L2.
If you want to know more about nonlinear approximation, I would recommend
the following two articles (the article of DeVore may require more advanced
knowledge than this course yet it is a great resource and classic by now):
Here, I want to emphasize a bit more modern viewpoint on the
Sturm-Liouville theory based on some rudimentary operator theory,
as shown in the book of Young above.
For the history and the current status of the Sturm-Liouville theory,
the following book is quite informative:
As for compact operators and the notion of the compactness in general,
along with Chap. 8 of Young's book listed above, you will find the following article useful:
In my lectures, due to the lack of time, I could only discuss the Green's
functions of regular Sturm-Liouville (RSL) systems that do not have 0 eigenvalue.
But the Neumann or periodic boundary conditions lead to 0 eigenvalue.
To deal with such cases, the notion of Green's functions have been
extended, which are called generalized or modified Green's
functions. More information about them can be found in Sec. IX.2 of our
textbook as well as the appropriate sections of the following books:
After all, however, 0 eigenvalue is not an issue in the RSL side thanks to
the fact that we can always consider a new RSL system by shifting the
eigenvalues of the original RSL so that the new RSL system does not have 0
eigenvalue. Such shift is possible thanks to the fact that not every real
number is an eigenvalue of an RSL system!
Lecture 25: Eigenfunction Expansions
This lecture is in part based on the specified chapter of the
following book:
In particular, I strongly recommend to read this paper by Hajmirzaahmad and Krall, who also point out the importance of the Hilbert space setting.
We did not deal with the so-called weak solutions of differential equations in this course, which are under less stringent conditions compared to the classical solutions we have dealt with. For properly dealing with weak solutions, one needs the theory of Sobolev spaces. I would recommend that you consult the following books: