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In contrast to previous notes, in this set of notes we shall focus exclusively on Fourier analysis in the one-dimensional setting for simplicity of notation, although all of the results here have natural extensions to higher dimensions. Depending on the physical context, one can view the physical domain as representing either space or time; we will mostly think in terms of the former interpretation, even though the standard terminology of “time-frequency analysis”, which we will make more prominent use of in later notes, clearly originates from the latter.

In previous notes we have often performed various localisations in either physical space or Fourier space , for instance in order to take advantage of the uncertainty principle. One can formalise these operations in terms of the functional calculus of two basic operations on Schwartz functions , the *position operator* defined by

and the *momentum operator* , defined by

(The terminology comes from quantum mechanics, where it is customary to also insert a small constant on the right-hand side of (1) in accordance with de Broglie’s law. Such a normalisation is also used in several branches of mathematics, most notably semiclassical analysis and microlocal analysis, where it becomes profitable to consider the semiclassical limit , but we will not emphasise this perspective here.) The momentum operator can be viewed as the counterpart to the position operator, but in frequency space instead of physical space, since we have the standard identity

for any and . We observe that both operators are formally self-adjoint in the sense that

for all , where we use the Hermitian inner product

Clearly, for any polynomial of one real variable (with complex coefficients), the operator is given by the spatial multiplier operator

and similarly the operator is given by the Fourier multiplier operator

Inspired by this, if is any smooth function that obeys the derivative bounds

for all and (that is to say, all derivatives of grow at most polynomially), then we can define the spatial multiplier operator by the formula

one can easily verify from several applications of the Leibniz rule that maps Schwartz functions to Schwartz functions. We refer to as the *symbol* of this spatial multiplier operator. In a similar fashion, we define the Fourier multiplier operator associated to the symbol by the formula

For instance, any constant coefficient linear differential operators can be written in this notation as

however there are many Fourier multiplier operators that are not of this form, such as fractional derivative operators for non-integer values of , which is a Fourier multiplier operator with symbol . It is also very common to use spatial cutoffs and Fourier cutoffs for various bump functions to localise functions in either space or frequency; we have seen several examples of such cutoffs in action in previous notes (often in the higher dimensional setting ).

We observe that the maps and are ring homomorphisms, thus for instance

and

for any obeying the derivative bounds (2); also is formally adjoint to in the sense that

for , and similarly for and . One can interpret these facts as part of the functional calculus of the operators , which can be interpreted as densely defined self-adjoint operators on . However, in this set of notes we will not develop the spectral theory necessary in order to fully set out this functional calculus rigorously.

In the field of PDE and ODE, it is also very common to study *variable coefficient* linear differential operators

where the are now functions of the spatial variable obeying the derivative bounds (2). A simple example is the quantum harmonic oscillator Hamiltonian . One can rewrite this operator in our notation as

and so it is natural to interpret this operator as a combination of both the position operator and the momentum operator , where the *symbol* this operator is the function

Indeed, from the Fourier inversion formula

for any we have

and hence on multiplying by and summing we have

Inspired by this, we can introduce the *Kohn-Nirenberg quantisation* by defining the operator by the formula

whenever and is any smooth function obeying the derivative bounds

for all and (note carefully that the exponent in on the right-hand side is required to be uniform in ). This quantisation clearly generalises both the spatial multiplier operators and the Fourier multiplier operators defined earlier, which correspond to the cases when the symbol is a function of only or only respectively. Thus we have combined the physical space and the frequency space into a single domain, known as phase space . The term “time-frequency analysis” encompasses analysis based on decompositions and other manipulations of phase space, in much the same way that “Fourier analysis” encompasses analysis based on decompositions and other manipulations of frequency space. We remark that the Kohn-Nirenberg quantization is not the only choice of quantization one could use; see Remark 19 below.

In principle, the quantisations are potentially very useful for such tasks as inverting variable coefficient linear operators, or to localize a function simultaneously in physical and Fourier space. However, a fundamental difficulty arises: map from symbols to operators is now no longer a ring homomorphism, in particular

in general. Fundamentally, this is due to the fact that pointwise multiplication of symbols is a commutative operation, whereas the composition of operators such as and does not necessarily commute. This lack of commutativity can be measured by introducing the *commutator*

of two operators , and noting from the product rule that

(In the language of Lie groups and Lie algebras, this tells us that are (up to complex constants) the standard Lie algebra generators of the Heisenberg group.) From a quantum mechanical perspective, this lack of commutativity is the root cause of the uncertainty principle that prevents one from simultaneously localizing in both position and momentum past a certain point. Here is one basic way of formalising this principle:

Exercise 2 (Heisenberg uncertainty principle)For any and , show that(

Hint:evaluate the expression in two different ways and apply the Cauchy-Schwarz inequality.) Informally, this exercise asserts that the spatial uncertainty and the frequency uncertainty of a function obey the Heisenberg uncertainty relation .

Nevertheless, one still has the correspondence principle, which asserts that in certain regimes (which, with our choice of normalisations, corresponds to the high-frequency regime), quantum mechanics continues to behave like a commutative theory, and one can sometimes proceed as if the operators (and the various operators constructed from them) commute up to “lower order” errors. This can be formalised using the *pseudodifferential calculus*, which we give below the fold, in which we restrict the symbol to certain “symbol classes” of various orders (which then restricts to be pseudodifferential operators of various orders), and obtains approximate identities such as

where the error between the left and right-hand sides is of “lower order” and can in fact enjoys a useful asymptotic expansion. As a first approximation to this calculus, one can think of functions as having some sort of “phase space portrait” which somehow combines the physical space representation with its Fourier representation , and pseudodifferential operators behave approximately like “phase space multiplier operators” in this representation in the sense that

Unfortunately the uncertainty principle (or the non-commutativity of and ) prevents us from making these approximations perfectly precise, and it is not always clear how to even define a phase space portrait of a function precisely (although there are certain popular candidates for such a portrait, such as the FBI transform (also known as the Gabor transform in signal processing literature), or the Wigner quasiprobability distribution, each of which have some advantages and disadvantages). Nevertheless even if the concept of a phase space portrait is somewhat fuzzy, it is of great conceptual benefit both within mathematics and outside of it. For instance, the musical score one assigns a piece of music can be viewed as a phase space portrait of the sound waves generated by that music.

To complement the pseudodifferential calculus we have the basic *Calderón-Vaillancourt theorem*, which asserts that pseudodifferential operators of order zero are Calderón-Zygmund operators and thus bounded on for . The standard proof of this theorem is a classic application of one of the basic techniques in harmonic analysis, namely the exploitation of *almost orthogonality*; the proof we will give here will achieve this through the elegant device of the Cotlar-Stein lemma.

Pseudodifferential operators (especially when generalised to higher dimensions ) are a fundamental tool in the theory of linear PDE, as well as related fields such as semiclassical analysis, microlocal analysis, and geometric quantisation. There is an even wider class of operators that is also of interest, namely the Fourier integral operators, which roughly speaking not only approximately multiply the phase space portrait of a function by some multiplier , but also move the portrait around by a canonical transformation. However, the development of theory of these operators is beyond the scope of these notes; see for instance the texts of Hormander or Eskin.

This set of notes is only the briefest introduction to the theory of pseudodifferential operators. Many texts are available that cover the theory in more detail, for instance this text of Taylor.

Lars Hörmander, who made fundamental contributions to all areas of partial differential equations, but particularly in developing the analysis of variable-coefficient linear PDE, died last Sunday, aged 81.

I unfortunately never met Hörmander personally, but of course I encountered his work all the time while working in PDE. One of his major contributions to the subject was to systematically develop the calculus of Fourier integral operators (FIOs), which are a substantial generalisation of pseudodifferential operators and which can be used to (approximately) solve linear partial differential equations, or to transform such equations into a more convenient form. Roughly speaking, Fourier integral operators are to linear PDE as canonical transformations are to Hamiltonian mechanics (and one can in fact view FIOs as a quantisation of a canonical transformation). They are a large class of transformations, for instance the Fourier transform, pseudodifferential operators, and smooth changes of the spatial variable are all examples of FIOs, and (as long as certain singular situations are avoided) the composition of two FIOs is again an FIO.

The full theory of FIOs is quite extensive, occupying the entire final volume of Hormander’s famous four-volume series “The Analysis of Linear Partial Differential Operators”. I am certainly not going to try to attempt to summarise it here, but I thought I would try to motivate how these operators arise when trying to transform functions. For simplicity we will work with functions on a Euclidean domain (although FIOs can certainly be defined on more general smooth manifolds, and there is an extension of the theory that also works on manifolds with boundary). As this will be a heuristic discussion, we will ignore all the (technical, but important) issues of smoothness or convergence with regards to the functions, integrals and limits that appear below, and be rather vague with terms such as “decaying” or “concentrated”.

A function can be viewed from many different perspectives (reflecting the variety of bases, or approximate bases, that the Hilbert space offers). Most directly, we have the *physical space perspective*, viewing as a function of the physical variable . In many cases, this function will be concentrated in some subregion of physical space. For instance, a gaussian wave packet

where , and are parameters, would be physically concentrated in the ball . Then we have the *frequency space (or momentum space) perspective*, viewing now as a function of the frequency variable . For this discussion, it will be convenient to normalise the Fourier transform using a small constant (which has the physical interpretation of Planck’s constant if one is doing quantum mechanics), thus

For instance, for the gaussian wave packet (1), one has

and so we see that is concentrated in frequency space in the ball .

However, there is a third (but less rigorous) way to view a function in , which is the *phase space perspective* in which one tries to view as distributed simultaneously in physical space and in frequency space, thus being something like a measure on the phase space . Thus, for instance, the function (1) should heuristically be concentrated on the region in phase space. Unfortunately, due to the uncertainty principle, there is no completely satisfactory way to canonically and rigorously define what the “phase space portrait” of a function should be. (For instance, the Wigner transform of can be viewed as an attempt to describe the distribution of the energy of in phase space, except that this transform can take negative or even complex values; see Folland’s book for further discussion.) Still, it is a very useful heuristic to think of functions has having a phase space portrait, which is something like a non-negative measure on phase space that captures the distribution of functions in both space and frequency, albeit with some “quantum fuzziness” that shows up whenever one tries to inspect this measure at scales of physical space and frequency space that together violate the uncertainty principle. (The score of a piece of music is a good everyday example of a phase space portrait of a function, in this case a sound wave; here, the physical space is the time axis (the horizontal dimension of the score) and the frequency space is the vertical dimension. Here, the time and frequency scales involved are well above the uncertainty principle limit (a typical note lasts many hundreds of cycles, whereas the uncertainty principle kicks in at cycles) and so there is no obstruction here to musical notation being unambiguous.) Furthermore, if one takes certain asymptotic limits, one can recover a precise notion of a phase space portrait; for instance if one takes the *semiclassical limit* then, under certain circumstances, the phase space portrait converges to a well-defined classical probability measure on phase space; closely related to this is the *high frequency limit* of a fixed function, which among other things defines the wave front set of that function, which can be viewed as another asymptotic realisation of the phase space portrait concept.

If functions in can be viewed as a sort of distribution in phase space, then linear operators should be viewed as various transformations on such distributions on phase space. For instance, a pseudodifferential operator should correspond (as a zeroth approximation) to multiplying a phase space distribution by the symbol of that operator, as discussed in this previous blog post. Note that such operators only change the amplitude of the phase space distribution, but not the support of that distribution.

Now we turn to operators that alter the support of a phase space distribution, rather than the amplitude; we will focus on unitary operators to emphasise the amplitude preservation aspect. These will eventually be key examples of Fourier integral operators. A physical translation should correspond to pushing forward the distribution by the transformation , as can be seen by comparing the physical and frequency space supports of with that of . Similarly, a frequency modulation should correspond to the transformation ; a linear change of variables , where is an invertible linear transformation, should correspond to ; and finally, the Fourier transform should correspond to the transformation .

Based on these examples, one may hope that given any diffeomorphism of phase space, one could associate some sort of unitary (or approximately unitary) operator , which (heuristically, at least) pushes the phase space portrait of a function forward by . However, there is an obstruction to doing so, which can be explained as follows. If pushes phase space portraits by , and pseudodifferential operators multiply phase space portraits by , then this suggests the intertwining relationship

and thus is approximately conjugate to :

The formalisation of this fact in the theory of Fourier integral operators is known as Egorov’s theorem, due to Yu Egorov (and not to be confused with the more widely known theorem of Dmitri Egorov in measure theory).

Applying commutators, we conclude the approximate conjugacy relationship

Now, the pseudodifferential calculus (as discussed in this previous post) tells us (heuristically, at least) that

and

where is the Poisson bracket. Comparing this with (2), we are then led to the compatibility condition

thus needs to preserve (approximately, at least) the Poisson bracket, or equivalently needs to be a symplectomorphism (again, approximately at least).

Now suppose that is a symplectomorphism. This is morally equivalent to the graph being a Lagrangian submanifold of (where we give the second copy of phase space the negative of the usual symplectic form , thus yielding as the full symplectic form on ; this is another instantiation of the closed graph theorem, as mentioned in this previous post. This graph is known as the *canonical relation* for the (putative) FIO that is associated to . To understand what it means for this graph to be Lagrangian, we coordinatise as suppose temporarily that this graph was (locally, at least) a smooth graph in the and variables, thus

for some smooth functions . A brief computation shows that the Lagrangian property of is then equivalent to the compatibility conditions

for , where denote the components of . Some Fourier analysis (or Hodge theory) lets us solve these equations as

for some smooth potential function . Thus, we have parameterised our graph as

A reasonable candidate for an operator associated to and in this fashion is the oscillatory integral operator

for some smooth amplitude function (note that the Fourier transform is the special case when and , which helps explain the genesis of the term “Fourier integral operator”). Indeed, if one computes an inner product for gaussian wave packets of the form (1) and localised in phase space near respectively, then a Taylor expansion of around , followed by a stationary phase computation, shows (again heuristically, and assuming is suitably non-degenerate) that has (3) as its canonical relation. (Furthermore, a refinement of this stationary phase calculation suggests that if is normalised to be the *half-density* , then should be approximately unitary.) As such, we view (4) as an example of a Fourier integral operator (assuming various smoothness and non-degeneracy hypotheses on the phase and amplitude which we do not detail here).

Of course, it may be the case that is not a graph in the coordinates (for instance, the key examples of translation, modulation, and dilation are not of this form), but then it is often a graph in some other pair of coordinates, such as . In that case one can compose the oscillatory integral construction given above with a Fourier transform, giving another class of FIOs of the form

This class of FIOs covers many important cases; for instance, the translation, modulation, and dilation operators considered earlier can be written in this form after some Fourier analysis. Another typical example is the half-wave propagator for some time , which can be written in the form

This corresponds to the phase space transformation , which can be viewed as the classical propagator associated to the “quantum” propagator . More generally, propagators for linear Hamiltonian partial differential equations can often be expressed (at least approximately) by Fourier integral operators corresponding to the propagator of the associated *classical* Hamiltonian flow associated to the symbol of the Hamiltonian operator ; this leads to an important mathematical formalisation of the correspondence principle between quantum mechanics and classical mechanics, that is one of the foundations of microlocal analysis and which was extensively developed in Hörmander’s work. (More recently, numerically stable versions of this theory have been developed to allow for rapid and accurate numerical solutions to various linear PDE, for instance through Emmanuel Candés’ theory of curvelets, so the theory that Hörmander built now has some quite significant practical applications in areas such as geology.)

In some cases, the canonical relation may have some singularities (such as fold singularities) which prevent it from being written as graphs in the previous senses, but the theory for defining FIOs even in these cases, and in developing their calculus, is now well established, in large part due to the foundational work of Hörmander.

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