Introduction

Harmonic Analysis is nowadays one of the most important branches of Mathematical Analysis. Some noticeable theoretic aspects are, e.g., Fourier Analysis on groups, or the study of classical function spaces (e.g. Hilbert spaces such as L^2(R) or L^2(T) by typical means as Fourier trasforms, or bases (and generalizations). But there are also many applications of Harmonic Analysis' concepts and tecniques: from the most classical Fourier trasform (used for decades by physicians, astronomers, engineers, geologists...) to the relatively new applications of Wavelets to signal processing.
The father of Harmonic Analysis can be considered, for both historical and intellectual reasons, Joseph Fourier. He conjectured, in the first years of the past century (around 1808, I guess, but he published his work on the theory of heating in 1822) that every (or almost every) function could be written as sum (either finite or infinite) of sines and cosines. He was brought to this conjecture while studying the heat equation (he had discovered), which he solved by writing the solutions as superposition of sinusoidal distributions of heat.
Did those series converge? This wasn't clear at all, and the study of the convergence of Fourier series wasn't indeed simple: great mathematicians tried to establish when (and in which sense) the Fourier series of a function converges to the function, and only around 1960 one of the last conjectures about such a topic (Lusin's: it stated that "the Fourier series of a continuous function converges to the function (pointwise) almost everywhere (with respect to Lebesgue measure)") was affirmatively proved (by Carleman).

The applications of concepts from Harmonic Analysis are important in many fields of physics, such as in the solution of differential equations, in the analysis of vibrations, images, sounds, earthquakes... and data compression.
Wavelets and the concept of Multi-Resolution Analysis (MRA) have born powerful tools in applied mathematics. The interesting and important applications of these concepts make of Wavelets one of the most studied mathematical objects by researchers in many fields.
Many good sites on them can be found on the Web, and below I've listed some.
In the following sections, I'd like to say two words about

Classical Fourier Analysis

Wavelets

Fourier Analysis on Groups

Haar wavelets

After this you will find useful (I do hope) links to other selected pages which contain material (such as introductory and/or advanced papers, software...) related to Fourier Analysis and Wavelets.
 
 

Classical Fourier Analysis

Fourier seems was one of the first mathematician to promote the idea that " every " function can be written as a finite or infinite sum of sines and cosines; D.Bernoulli had probably a similar idea, but he didn't exploit it. Fourier was trying to solve the heat equation (which he discovered!), and found solutions by writing them as superposition of an infinite number of sinusoidal waves. It wasn't very clear if those series really converged or defined a solution, but they worked in most real problems.
The theory, of course, came after, giving birth to that branch of Analysis today called Harmonic Analysis. For a while,I will adopt the language of signal analysis, since this may help imagination and since important applications of the mathematical theory are in this direction.
The basic idea of classical Fourier Analysis is the idea of "frequency content" of a signal (mathematically represented by a function with certain properties):  a superposition of sinusoidal waves with different frequencies can give birth to a complex signal (no more sinusoidal, maybe neither smooth) and, on the contrary, the frequencies of large classes of signals can be analyzed and then used to reconstruct the signal as a superposition of simple waves. It is a sort of bridge between the space domain in which the wave exists and the time domain in which the wave evolves.
From a mathematical point of view, we have two domains: the "space domain" G (e.g. an interval, or the whole real line), and the "frequency domain" G^ (which is in some manner related to that of the "space-domain"):
  

A function f (representing a signal) defined on the space-domain is analyzed and gives birth to a function f^ (its Fourier transform) on the frequency-domain. And again, from a function g on the frequency- domain we can get back (inverse Fourier transform) a function g~ defined on the space-domain. Moreover, if we make the inverse transform of the transform, we're back to our original signal: (f^)~ = f (this is the so called reconstruction formula, since it allows the reconstruction of a signal from its frequencies).
  

For large (at last from the standpoint of common applications) classes of functions it is possible to define the Fourier transform (a harder step is to find (characterize) the functions that are the transform of some function). Everything goes very well in special function spaces: the Hilbert spaces, such L^2 (R) or L^2 (R^n), which can be thought to contain all the functions that represent, from a physical point of view, signals with finite energy. Here this "energy" is given by 

These spaces are infinite dimensional generalizations of the usual euclidean spaces R^n and C^n. Every point of such a space is a whole function, and we can sum two points and dilate every point by an arbitrary factor, as we can do in finite dimensional vector space, such as R^3.  In R^n we can choose a referiment (i.e. a basis, which is a set of n independent vectors) which allows us to associate to every vector n numbers (called components relative to the referiment chosen). These numbers can then be used to reconstruct the vector.
In Hilbert spaces we need infinite numbers to determine a vector of the space (which represents a function): we have bases with infinite vectors (for example the classical basis which is made up of all the sines and cosines with all integer frequencies) and an infinite set of numbers (called the Fourier coefficients of the function with respect to the basis) that are the components of the vector on the basis.
Moreover, in Hilbert space has still sense to speak of orthogonality, so we have infinite dimensional versions of Pitagora's theorem and intuitive geometric constructions and approximations. The sines and cosines are the analogue of an orthonormal basis in R^n, the Fourier coefficients are the projections of a function on the vectors of this basis, and the Fourier series and integrals are the formulas for the reconstruction of a function by means of its coefficients.
It turns out that a function of these spaces has a Fourier transform which is again in the same type of space (not necessarily the same: the space domain and the frequency domain may be different: remember the if G=[a,b] then G^=Z, but if G=R then G^=R). The Fourier transform becomes an isometric isomorphism (i.e. sends sum and scalar multiplication in a space in the same operations in the other space, and moreover conserves the total energy) between these spaces.
In particular the "self-duality" of R=G=G^ implies that we have two exact copies of the same space, and the Fourier transform can be regarded in a natural way as a transformation of the space itself. In fact, it can be considered as a rotation of such a space, or, conceptually, a change of point of of view (or of referiment).
  

The analysis in these spaces is achieved by writing the signal as a superposition of classical sinusoidal waves of all the possible frequencies, which is called " expanding" the signal on the "basis" of sinusoidal functions. The idea is that the more a signal is concentrated on a certain frequency, the larger will be the related coefficient we get in the expansion.
Also, once we've got the decomposition of the signal in frequencies, we could discard the coefficients less than a certain value, i.e. we could "forget" those frequencies that less contribute to the signal, and thus we would achieve a compression of the signal (however, by doing this, we lose a part of the signal a priori considered negligeable). Or maybe, for denoising purposes , we could discard high-frequency components. It is easy to imagine how many applications these ideas have found.
 
 

Wavelets and MultiResolution Analysis

We've seen that Fourier analysis is an useful mathematical approach to the analysis of the frequencies of a signal. Nevertheless, Fourier analysis is not the best way, for many aspects, to solve the problem of analysis and reconstruction.
Which is the main problem with Fourier analysis?
The answer can be summed up in a magic word: lack of localization.
Let me explain what is meant by localization. Sinusoidal waves are equal all over the space domain (they are periodic, hence translation-invariant): this means they are not localized. So, well, what does non-localization imply? Let me explain with a typical example. Suppose we have a simple signal which is quite flat all over the (time-)line, except for a sudden peak around a point, with a certain frequency phi. When we decompose this signal on the standard base of sinusoidal waves, we get not only large coefficients relative to frequencies near phi, but we get also significantly large coefficients in a wide spectrum. This is due to the fact that once we've got the frequencies around phi, we're with a sinusoidal wave, which maybe approximates well the peak, but certainly doesn't look flat far from it. This implies that, since, for the reconstruction formula, we must have back our original signal from the anti-trasform, the oscillations afar the peak are to be cancelled to recreate the flat signal. How does this happen? By adding higher and higher frequencies, each partially cancelling the others. So we get a trasform big over many frequencies, and it is not easy to read from it informations about the space-signal. This is why the classical Fourier transform is so ineffective in handling with peaks and sharp edges (in images). It is very useful, nevertheless, in studying signals that statistically show the same characteristics all over the time, whose behaviour doesn't vary abruptly in time. 
The problem of localization is solved by wavelets. These are a family of functions that, like sinusoidal waves, form a good (orthonormal and unconditional) basis in certain spaces (such Hilbert spaces, but also many others). But, unlike sines (optimally localized in frequency, but oscillating forever in time with the same amplitude), are localized both in time and frequency.

A family of wavelets (the classical ones, but there are many others with different charateristics and arising in different ways) is obtained by translations and dilations (generally by a factor of 2, but the case in which the factor is an integer greater than 2 is studied) of a single wavelet (often called the "mother wavelet").

Psi(x;j,m) = 2^(j/2)*Mother_Psi((2^j)x-m)

Index j gives the level of dilation, m the translation. The factor 2^(j/2) is introduced so that all the wavelets have the same energy. By translating we obtain localization in space, by dilating we obtain better and better resolutions. We can parametrize the family of wavelets with two parameters as above, one indicating where (in spatial sense) the wavelet is centered, the other indicating at what resolution the wavelet is analyzing. The two parameters are independent. By fixing the first and moving the second, we can analyze the signal as accurately as we wish around a certain position in space, in the other way we can analyze at a fixed resolution in different places.
A signal such as that described above can be easily analyzed with wavelets: we could start with inspecting the space domain, by varying the spatial parameter. By doing so, we would find a lot of nearly zero coefficients, except for a few ones, which are those relative to the wavelets centered near the peak. This enables us to know where a more accurate analysis is required (if desired): it is enough to fix the translation parameter and increase the resolution of the wavelets to get an accurate description of the peak.

The concept of Multi-Resolution Analysis (MRA from now on) is strongly related to that of wavelets. The idea is to decompose the space of signals in a chain of particular nested subspaces {V(j)} with index j taking values in Z. They are "particular" in the sense that they are requested to posses the following properties:

the subspaces are nestes: each contains the precedent and is contained in the successive:

{0}<...<V(-2)<V(-1)<V(0)<V(1)<V(2)<...<L^2

the intersection of all these spaces is the only element 0, while their union is ("almost") the whole space,

there is a function f such that its integer traslates (which are in the form f(x-k) for some k in Z) generate V(0),

every space is obtaining by dilating the preceding one, in the sense that the functions in V(j) are exactly the functions in V(j+1) dilated by a factor of 2:

g(x) is in V(j) iff g(2x) is inV(j+1)

The idea underlying all this is as follows. If we have a function (signal) in the space, we can project it on a cerain subspace V(j). By doing so, we discard some informations in the original signal (which will lie in the orthogonal complement of that subspace), and remain with an approximation. The largest is j, the better this approximation will be, since a larger j means "more detailed functions" in V(j).

Where are the wavelets in the context of an MRA? A beautiful construction of Yves Meyer shows that if we consider the space W(j) that "fills the details" between V(j) and V(j+1) (i.e. the orthogonal complement of V(j) in V(j+1)), we find that each W(j+1) is a dilated W(j), all the W(j) are mutually orthogonal, and there exist a familty of wavelets functions Psi(x;j,m)=2^(j/2)*Mother_Psi((2^j)x-m) such that each W(j) is generated by traslated versions of Psi(x;j):

W(j) = < Psi(x;j,m) > for m varying in Z

So, for each level j of detail, the translated of a dilated f, i.e. the functions f(2^(j)*x-k), k in Z, generate V(j) and the translated of a dilated mother wavelets, i.e. the functions Mother_Psi((2^j)x-m), m in Z, generate W(j).

The technique for analyzing a signal is now the following: project on V(j), save the details you have in W(j), project on V(j-1), save the details that are in W(j-1)...repeat until the approximation on V(J) is meaningful to your purpose. To reconstruct, start from the coarsest approximation in V(J), add the details of W(J) to get the approximation in V(J+1),...,add the details in W(j-1) to obtain your original approximation in V(j).

Wavelets still present some lack of localization in the frequency domain, at last at high resolutions (small details). This problem is solved by wavelet packets, which allow a further splitting of the spaces W(j) in orthogonal subspaces some of which have better frequency localization than others. They are a sort of library of special wavelet-like functions, some better localized in frequency than others. A choice adapted to the signal currently being analyzed is also possible, in an efficent way, thus noticeably increasing the "power of frequency resolution" of the analysis. This tecnique is particularly useful when a great accuracy is required in recognizing different frequencies: such a need arises, for example, in speech recognition applications.

In these few lines I tried to give an idea on why wavelets can be so effectively used for accurate signal analysis, edge detection (where Fourier series fail), sound and image compression (only few larger coefficients are enough to get a good approximation of the signal, and moreover the algorithmic complexity is less than that of the fast Fourier transform!)...
 
 

Two words Fourier Analysis on Groups

I'll give here only the idea at the basis of this theory. Even this little bit is rather technical, and difficult to explain without using mathematical language.
This theory is a generalization of classical Fourier analysis. This was done by studying periodic functions (i.e. functions on T=finite interval on the real line), or functions on the real line R, and the analogues in many dimensions. The first (trivial, but important) observation is that all these domains are groups (i.e. they are endowed with a binary operation with good properties), so we will indicate them by G. On R we have the usual sum of real numbers, on T the sum modulus the length of the interval, on R^n the sum of vectors.
Let S be the unit circle in the complex plane C: it is a group with the usual multiplication of complex numbers.
Let us now consider the homomorphisms from G to S. These are maps that send the operation in G in the operation in S, i.e., if e is such a map, we have

e(a+b) = e(a)*e(b)

where + is the operation in G and * the complex moltiplication in S, for every a,b in G.
The complex exponentials of classical Fourier Analysis (which "real"ize in sines and cosines) are exactly such homomorphims for the torus, if the exponent is a relative integer, and for the real line, if the exponent is a real number. In fact the property we are interested in is exactly that

exp(i(a+b)) =exp(ia)*exp(ib)

which means they are homomorphism from T or R in the unit circle of C. Note the the +is the operation in T or R and * is the operation in S.
It is a fact that on many groups it is possible to define a topology, which gives the notion of open set and hence of a continuous functions (it also required for the group operation to be continuous with respect to such a topology: this makes the group a so called topological group). But it also possible to endow a large family of groups with a measure (called the Haar measure, which furthermore is invariant under the operation of the group), which let us talk of measurable and integrable functions, and of function spaces on these groups.
The characters are exactly those complex homomorphisms from the group to S which are also continuous. They form a group (with pointwise product), which is called the dual group G^. We can also define the dual of the dual, but this turns out to be the original G. The dual of T is Z, the dual of R is R again. This is why we use "discrete frequencies" to analyze functions on T, but we pass to a "continuous parameter" when we pass to functions on the real line.
This way of looking at Fourier Analysis makes many things clearer, and this is why it is appreciated and adopted by many mathematicians. It also offers powerful and general tools which could hardly have been discovered by other means.
 
 

Links

Here are some links I consider interesting. I hope you'll find them good and that they will make you save your time in searching the Web for what you really are looking for. There are both a section with introductions to wavelets and research papers, neatly divided in categories according to their topic.

• Wavelets Resources at MathSoft
  Good and vast bibliography, lots of papers (pre-prints) of great authors, subdivided by argument. Also links to wavelets-related software are available. A very good page!

• Rice DSP Publication Archives
  Lots (really lots) of papers on Wavelets, FFT, software...Rice University site is well organized and rich in materials. If are interested in latest research, you can also find the Techincal Reports of the past years, all available in electronic format (see CML). It is generally quite fast in download. Reserve a bookmark for this!

• I. Daubechies' home page
  The home page of the great researcher in Wavelets. Daubechies' studies in wavelet theory have brought to many important achievements. One of the most important is her study and characterization of compactly supported (i.e. with "finite length in time") wavelets, which are , most rightly, named after her.

• An Introduction to Wavelets
  A simple introductory article (by Amara Grasp) which presents basic ideas in Wavelet theory and some of their applications (did you know that FBI fingerprint archives are compressed with wavelets?). The home page is at Amara's page.

• Wavelet.org
  This is a really good site on wavelets, featuring sections on the latest news in wavelets' world and on the current research. Here you will be also given the opportunity to subscribe (free of charge) to the Wavelet Digest, an electronic journal you can periodically receive by e-mail: it contains links to papers available on the web, current lines of research in related topics, calendars of wavelet-related events, questions and answers from other waveleteers.
  On the site you will also find pages of links to other sites, to papers, books, software...
  Don't miss this!

• Numerical Harmonic Analysis Group (NUHAG)
  This group's research interests are principally in Gabor Analysis, Banach algebras, and applications. You can also subscribe (free of charge) the electronic journal of the group, dealing mainly with current lines of development and research in Gabor Analysis, group representation, Wiener algebras...and much, much more. A wide collection of papers (available in electronic format) of the NUHAG members is also available.

• Wavelet Papers
  A site collecting some papers on wavelets.

• Wim Sweldens' Selected publications Many good papers of recognized researchers in Wavelets and new lines of research in related sectors.

• Wavelets at MIT Researching group in Wavelets at MIT. Applications of wavelets to image compression and to algorithms for solving partial differential equations are considered. Some papers are also available.

• Wavelets References and Homepages
  Page containing a list of links to homepages on wavelets and to other wavelet related sites on the Web.

• Wavelets - Internet Resources
  Lots of good links to available papers, homepages, courses on Wavelets and their applications. Last time I visited it, some suggested links were broken, but here you can find so many of them that you won't worry if some aren't up to date.

• Wavelet links at WashU of St.Louis
  List of Web resources on wavelets and related topics.

• Papers on Wavelets

• Wavelet Image Compression
  Application of Wavelets to image compression has given very good results, and the studies for improving the efficiency and speed of these methods are making them always more and more widespread. You won't be surprised if in a year you digital photo camera will use wavelet compression to contain more and more films and photos.
  At last, the high end of compression seems to be in joining Wavelet and Fractal techniques. See for example Fractal Image Compression (site and introductory article ) or Fractal Image Compression Bibliography for a (quite long) bibliography (without links: only books and papers).

• Wavelets for retarded
  A simple introduction to wavelets.

• Bell Labs Wavelet Group

• Yale's University Wavelet Resources

• Dragon Applet
  Applet that illustrates the links between iterative methods and wavelets.

• Wavelet Image Compression Theory

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