Definitions

Fourier_analysis

Multiplier (Fourier analysis)

In Fourier analysis, a Fourier multiplier (or multiplier for short) is a kind of linear operator, or transformation of functions. These operators multiply the Fourier coefficients of a function by a specified function (known as the symbol), hence the name. Among the multipliers one can count some simple operators, such as translations and differentiation, but also some more complicated ones such as the convolutions, Hilbert transform, and others. Indeed, every translation-invariant operator on a group which obeys some very mild regularity conditions can be expressed as a Fourier multiplier, and conversely.

In signal processing, a Fourier multiplier is called a "filter", and the multiplier function (or "symbol") is the filter's frequency response (or transfer function).

Fourier multipliers are special cases of spectral multipliers, which arise from the functional calculus of an operator (or family of commuting operators). They are also special cases of pseudo-differential operators, and more generally Fourier integral operators.

Fourier multipliers are unrelated to Lagrange multipliers, except for the fact that they both involve the multiplication operation.

Mathematicians researching this field usually agree that this topic is not well understood. Many very natural questions are still open, especially those related to the role of arithmetic properties.

For the necessary background on Fourier series, see that page. Additional important background may be found on the pages operator norm and Lp space.

Definition

Multipliers can be defined on any group G for which the Fourier transform is also defined (in particular, on any locally compact amenable abelian group). The general definition is as follows. If f: G to C is a sufficiently regular function, let hat f: hat G to C denote its Fourier transform (where hat G is the Pontryagin dual of G). Let m: hat G to C denote another function, which we shall call the symbol. Then the Fourier multiplier T = T_m associated to this symbol m is defined via the formula

widehat{Tf}(xi) := m(xi) hat{f}(xi).
In other words, the Fourier transform of Tf at a frequency xi is given by the Fourier transform of f at that frequency, multiplied by the value of the symbol at that frequency. This explains the terminology "Fourier multiplier".

Note that the above definition only defines Tf implicitly; in order to recover Tf explicitly one needs to invert the Fourier transform. This can be easily done if both f and m are sufficiently smooth and integrable. One of the major problems in the subject is to determine, for any specified symbol m, whether the corresponding Fourier multiplier continues to be well-defined when f has very low regularity, for instance if it is only assumed to lie in an L^p space. See the discussion on the "boundedness problem" below. As a bare minimum, one usually requires the symbol m to be bounded and measurable; this is sufficient to establish boundedness on L^2 but is in general not strong enough to give boundedness on other spaces.

One can view the Fourier multiplier T as the composition of three operators, namely the Fourier transform, the operation of pointwise multiplication by m, and then the inverse Fourier transform. Equivalently, T is the conjugation of the pointwise multiplication operator by the Fourier transform. Thus one can think of Fourier multipliers as operators which are diagonalized by the Fourier transform.

We now specialize the above general definition to specific groups G. First consider the unit circle G = R / 2pi Z; functions on G can thus be thought of as 2pi-periodic functions on the real line. In this group, the Pontryagin dual is the integers hat G = Z, the Fourier transform (for sufficiently regular functions f) is given by

hat f(n) := frac{1}{2pi} int_0^{2pi} f(t) e^{-int} dt
and the inverse Fourier transform is given by
f(t) = sum_{n=-infty}^infty hat f(n) e^{int}.
A symbol in this setting is simply a sequence (m_n)_{n=-infty}^infty of numbers, and the multiplier T = T_m associated to this symbol is then given by the formula
(Tf)(t):=sum_{n=-infty}^{infty}m_n widehat{f}(n)e^{int},
at least for sufficiently well-behaved choices of the symbol (m_n)_{n=-infty}^infty and the function f.

Now let G be a Euclidean space G = R^n. Here the dual group is also Euclidean, hat G = R^n, and the Fourier and inverse Fourier transforms are given by the formulae

hat f(xi) := int_{R^n} f(x) e^{-2pi i x cdot xi} dx
f(x) = int_{R^n} hat f(xi) e^{2pi i x cdot xi} dxi.
A symbol in this setting is a function m: R^n to C, and the associated Fourier multiplier T = T_m is defined by
Tf(x) := int_{R^n} m(xi) hat f(xi) e^{2pi i x cdot xi} dxi,
again assuming sufficiently strong regularity and boundedness assumptions on the symbol and function.

In the sense of distributions, there is no difference between multipliers and convolution operators; every Fourier multiplier T can also be expressed in the form Tf = f * K for some distribution K, known as the convolution kernel of T. In this view, translation is convolution with the Dirac delta function δ, differentiation is convolution with δ', etc. People holding this view use the term multiplier in the specific sense of the problem of boundedness in L^p discussed below.

Diagrams

Examples

The following table shows some common examples of Fourier multipliers on the unit circle G = R/2pi Z.

Name Symbol m_n Operator Tf(t) Kernel K(t)
Identity operator 1 f(t) Dirac delta function delta(t)
Multiplication by a constant c c cf(t) cdelta(t)
Translation by s e^{ins} f(t-s) delta(t-s)
Differentiation in f'(t) delta'(t)
k-fold differentiation (in)^k f^{(k)}(t) delta^{(k)}(t)
Constant coefficient differential operator P(in) P(frac{d}{dt}) f(t) P(frac{d}{dt}) delta(t)
Fractional derivative of order alpha >n|^alpha >frac{d}{dt}|^alpha f(t) >frac{d}{dt}|^alpha delta(t)
Mean value 1_{n = 0} frac{1}{2pi} int_0^{2pi} f(t) dt 1
Mean-free component 1_{n neq 0} f(t) - frac{1}{2pi} int_0^{2pi} f(t) dt delta - 1
Integration (of mean-free component) frac{1}{in} 1_{n neq 0} frac{1}{2pi} int_{0}^{2pi} (pi-s) f(t-s) ds Sawtooth function frac{1}{2}(1 - { frac{t}{2pi}})
Periodic Hilbert transform H 1_{ngeq 0} - 1_{n<0} Hf := p.v. frac{1}{pi} int_{-pi}^{pi} frac{f(s)}{e^{i(t-s)}-1} ds p.v. 2 frac{f(s)}{e^{i(t-s)}-1} ds
Dirichlet summation D_N 1_{-N leq n leq N} sum_{n=-N}^N hat f(n) e^{int} Dirichlet kernel sin((N+1/2)t) / sin(t/2)
Fejer summation F_N (1 - frac
{N}) 1_{-N leq n leq N}>
sum_{n=-N}^N (1 - frac
{N}) hat f(n) e^{int}>
Fejer kernel frac{1}{N} (sin(Nt/2) / sin(t/2))^2
General Fourier multiplier m_n sum_{n=-infty}^infty m_n hat f(n) e^{int} Tdelta = sum_{n=-infty}^infty m_n e^{int}
General convolution operator hat K(n) f*K(t) := frac{1}{2pi} int_0^{2pi} f(s) K(t-s) ds K(t)

The following table shows some common examples of Fourier multipliers on Euclidean space G = R^n.

Name Symbol m(xi) Operator Tf(x) Kernel K(x)
Identity operator 1 f(x) delta(x)
Multiplication by a constant c c cf(x) cdelta(x)
Translation by y e^{2pi iy cdot xi} f(x-y) delta(x-y)
Derivative d/dx (one dimension only) 2pi i xi frac{d f}{d x}(x) delta'(x)
Partial derivative partial/partial x_j 2pi i xi_j frac{partial f}{partial x_j}(x) frac{partial delta}{partial x_j}(x)
Laplacian Delta -4pi^2 >xi|^2 Delta f(x) Delta delta(x)
Constant coefficient differential operator P(nabla) P(ixi) P(nabla) f(x) P(nabla) delta(x)
Fractional derivative of order alpha (2pi >xi|)^alpha (-Delta)^{alpha/2} f(x) (-Delta)^{alpha/2} delta(x)
Fractional integral of order alpha (2pi >xi|)^{-alpha} (-Delta)^{-alpha/2} f(x) Riesz potential (-Delta)^{-alpha/2} delta(x) = c_{n,alpha} >x|^{alpha-n}
Inhomogeneous fractional integral of order alpha (1 + 4pi^2 >xi|^2)^{-alpha/2} (1-Delta)^{-alpha/2} f(x) Bessel potential (1-Delta)^{-alpha/2} delta(x)
Heat flow operator exp(tDelta) exp(-4pi^2 t >xi|^2) exp(tDelta) f(x) = frac{1}{(4pi t)^{n/2}} int_{R^n} e^{->x-y|^2/4t} f(y) dy Heat kernel frac{1}{(4pi t)^{n/2}} e^{->x|^2/4t}
Schrödinger equation evolution operator exp(itDelta) exp(-i4pi^2 t >xi|^2) exp(itDelta) f(x) = frac{1}{(4pi it)^{n/2}} int_{R^n} e^{i>x-y|^2/4t} f(y) dy frac{1}{(4pi it)^{n/2}} e^{i>x|^2/4t}
Hilbert transform H (one dimension only) -isgn(xi) Hf := p.v. frac{1}{pi} int_{-infty}^infty frac{f(y)}{x-y} dy p.v. frac{1}{pi s}
Partial Fourier integral S^0_R (one dimension only) 1_{-R leq xi leq R} int_{-R}^R hat f(xi) e^{2pi ixxi} dx sin(2pi R x) / pi x
Disk multiplier S^0_R 1_ leq R}>
int_ leq R} hat f(xi) e^{2pi ixxi} dx>
>x|^{-n/2} J_{n/2}(2pi |x|) (J is a Bessel function)
Bochner-Riesz operators S^delta_R (1 - >xi|^2/R^2)_+^delta int_ leq R} (1 - frac{|xi|^2}{R^2})^delta hat f(xi) dxi>
int_ leq R} (1 - frac{|xi|^2}{R^2})^delta dxi>
General Fourier multiplier m(xi) int_{R^n} m(xi) hat f(xi) e^{2pi i x cdot xi} dxi int_{R^n} m(xi) e^{2pi i x cdot xi} dxi
General convolution operator hat K(xi) f*K(x) := int_{R^n} f(y) K(x-y) dy K(x)

General considerations

The map m mapsto T_m is an homomorphism of C*-algebras, thus the sum of two multipliers T_m and T_{m'} is a multiplier with symbol m+m', the composition of these two multipliers is a multiplier with symbol mm', and the adjoint of a multiplier T_m is another multiplier with symbol overline{m}.

In particular, we see that any two Fourier multipliers commute with each other. Since every translation operator is a Fourier multiplier, we conclude that Fourier multipliers are translation-invariant. Conversely, one can show that any translation invariant linear operator (which is bounded on L^2(G)) is a Fourier multiplier.

The boundedness problem

The boundedness problem for any given group G is, stated simply, to identify the symbols m such that the corresponding multiplier is bounded from L^p(G) to L^p(G). Note that as multipliers are always linear, such operators are bounded if and only if they are continuous. This problem is considered extremely difficult in general, but many special cases can be treated. The problem depends greatly on p, however there is a duality relationship: if 1/p + 1/q = 1 and 1 leq p, q leq infty, then a Fourier multiplier is bounded on L^p if and only if it is bounded on L^q.

The Riesz-Thorin theorem shows that if a Fourier multiplier is bounded on two different L^p spaces, then they are also bounded on all intermediate spaces. Hence we get that the space is multipliers is smallest for L^1 and L and grows as one approaches L^2, which has the largest multiplier space.

Boundedness on L2.

This is the easiest case. Parseval's theorem allows to solve this problem completely and obtain that a multiplier T is bounded on L^2(G) if and only if the symbol m is bounded and measurable.

Boundedness on L1 or L

This case is more complicated than the Hilbertian case, but still relatively simple. The following is true:

Theorem: On the unit circle R/2pi Z, a symbol (m_n)_{n=-infty}^infty generates a bounded multiplier in L^1 (or L^infty) if an only if there exists a measure μ such that m_n is the n-th Fourier-Stieltjes coefficient of μ.

(the if part is a simple calculation. The only if part here is the interesting bit). While this might seem at first as a mere casting of the problem in different terms, in practice it turns out that measures are far simpler object. For example, this result allows a complete characterization of sequences of 0 and 1 giving rise to multipliers:

Theorem: A symbol (m_n)_{n=-infty}^infty consisting of zeros and ones generates a bounded Fourier multiplier in L^1 (or L^infty) if and only if it is periodic after modifying finitely many of the m_n.

Boundedness on Lp for 1 < p < ∞

For this case, one does not have general necessary and sufficient conditions for boundedness, even in the simplest case of the unit circle. However, several necessary conditions and several sufficient conditions are known. For instance it is known that in order for a multiplier to be bounded on even a single L^p space, the symbol must be bounded and measurable. However, this is not sufficient except when p=2.

Results that give sufficient conditions for boundedness are known as multiplier theorems. Two such results are given below.

Marcinkiewicz multiplier theorem. Let (m_n)_{n=-infty}^infty be a symbol which has uniformly bounded variation on the intervals 2^N leq n < 2^{N+1} and -2^{N+1} < n leq -2^N for all positive integers N. Then the multiplier associated to this symbol is bounded on L^p(R/2pi Z) for all 1 < p < infty. A similar statement holds when the group G is the real line, the only difference being that N now ranges over all the integers rather than just the positive ones.

Hormander-Mikhlin multiplier theorem. Let m be a symbol on R^n which is smooth except possibly at the origin, and such that the function |x|^k |nabla^k m| is bounded for all integers 0 leq k leq n/2+1. Then the multiplier associated to this symbol is bounded on L^p(R^n) for all 1 < p < infty.

The proof of these two theorems are fairly tricky, involving techniques from Calderon-Zygmund theory and the Marcinkiewicz interpolation theorem.

Examples

Translations are bounded operators on any L^p. Differentiation is not bounded on any L^p. The Hilbert kernel is bounded only for p different from 1 and ∞. The fact that it is unbounded on L is easy, since it is well known that the Hilbert transform of a step function is unbounded. Duality gives the same for p = 1. However, both the Marcinkiewicz and Hormander-Mikhlin multiplier theorems show that the Hilbert transform is bounded in L^p for all 1 < p < infty.

Another interesting case is when the sequence x_n is constant on the intervals [2^n,2^{n+1}-1] and [-2^{n+1}+1,-2^n]. From the Marcinkiewicz multiplier theorem we see that any such sequence (bounded, of course) is a multiplier for every 1 < p < ∞.

In one dimension, the disk multiplier S^0_R is bounded on L^p for every 1 < p < infty. However, in 1972, Charles Fefferman showed the surprising result that in two and higher dimensions the disk multiplier S^0_R is unbounded on L^p for every p neq 2. The corresponding problem for Bochner-Riesz multipliers is only partially solved; see the Bochner-Riesz conjecture.

A final result concerns a random m_n:

Theorem: Let (m_n)_{n=-infty}^infty be a symbol consisting of independent variables uniform on [0,1]. Then almost surely the Fourier multiplier corresponding to this symbol is bounded only L^2.

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