Multiplier (Fourier analysis)

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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.

Fourier multipliers are also used in signal processing, though they are usually referred to in that setting as filters, and the corresponding symbol is known as the frequency response.

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.


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[edit] 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 = Tm associated to this symbol m is defined via the formula

\widehat{Tf}(\xi) := m(\xi) f(\xi).

In other words, the Fourier transform of Tf at a frequency ξ 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 Lp 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 L2 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 / 2πZ; functions on G can thus be thought of as 2π-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}{2\pi} \int_0^{2\pi} 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 = Tm 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 = Rn. 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^{-2\pi i x \cdot \xi} dx
f(x) = \int_{R^n} \hat f(\xi) e^{2\pi i x \cdot \xi} d\xi.

A symbol in this setting is a function m: R^n \to C, and the associated Fourier multiplier T = Tm is defined by

Tf(x) := \int_{R^n} m(\xi) \hat f(\xi) e^{2\pi i x \cdot \xi} d\xi,

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 Lp discussed below.

[edit] Diagrams

[edit] Examples

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

Name Symbol mn Operator Tf(t) Kernel K(t)
Identity operator 1 f(t) Dirac delta function δ(t)
Multiplication by a constant c c cf(t) cδ(t)
Translation by s eins f(t-s) δ(t − s)
Differentiation in f'(t) δ'(t)
k-fold differentiation (in)k f(k)(t) δ(k)(t)
Constant coefficient differential operator P(in) P(\frac{d}{dt}) f(t) P(\frac{d}{dt}) \delta(t)
Fractional derivative of order α | n | α |\frac{d}{dt}|^\alpha f(t) |\frac{d}{dt}|^\alpha \delta(t)
Mean value 1n = 0 \frac{1}{2\pi} \int_0^{2\pi} f(t) dt 1
Mean-free component 1_{n \neq 0} f(t) - \frac{1}{2\pi} \int_0^{2\pi} f(t) dt δ − 1
Integration (of mean-free component) \frac{1}{in} 1_{n \neq 0} \frac{1}{2\pi} \int_{0}^{2\pi} (\pi-s) f(t-s) ds Sawtooth function \frac{1}{2}(1 - \{ \frac{t}{2\pi}\})
Periodic Hilbert transform H 1_{n\geq 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 DN 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 FN (1 - \frac{|n|}{N}) 1_{-N \leq n \leq N} \sum_{n=-N}^N (1 - \frac{|n|}{N}) \hat f(n) e^{int} Fejer kernel \frac{1}{N} (\sin(Nt/2) / \sin(t/2))^2
General Fourier multiplier mn \sum_{n=-\infty}^\infty m_n \hat f(n) e^{int} T\delta = \sum_{n=-\infty}^\infty m_n e^{int}
General convolution operator \hat K(n) f*K(t) := \frac{1}{2\pi} \int_0^{2\pi} f(s) K(t-s) ds K(t)

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

Name Symbol m(ξ) Operator Tf(x) Kernel K(x)
Identity operator 1 f(x) δ(x)
Multiplication by a constant c c cf(x) cδ(x)
Translation by y e^{2\pi iy \cdot \xi} f(x-y) δ(x − y)
Derivative d / dx (one dimension only) 2πiξ \frac{d f}{d x}(x) δ'(x)
Partial derivative \partial/\partial x_j 2πiξj \frac{\partial f}{\partial x_j}(x) \frac{\partial \delta}{\partial x_j}(x)
Laplacian Δ − 4π2 | ξ | 2 Δf(x) Δδ(x)
Constant coefficient differential operator P(\nabla) P(iξ) P(\nabla) f(x) P(\nabla) \delta(x)
Fractional derivative of order α (2π | ξ | )α ( − Δ)α / 2f(x) ( − Δ)α / 2δ(x)
Fractional integral of order α (2π | ξ | ) − α ( − Δ) − α / 2f(x) Riesz potential ( − Δ) − α / 2δ(x) = cn,α | x | α − n
Inhomogeneous fractional integral of order α (1 + 4π2 | ξ | 2) − α / 2 (1 − Δ) − α / 2f(x) Bessel potential (1 − Δ) − α / 2δ(x)
Heat flow operator exp(tΔ) exp( − 4π2t | ξ | 2) \exp(t\Delta) f(x) = \frac{1}{(4\pi t)^{n/2}} \int_{R^n} e^{-|x-y|^2/4t} f(y) dy Heat kernel \frac{1}{(4\pi t)^{n/2}} e^{-|x|^2/4t}
Schrödinger equation evolution operator exp(itΔ) exp( − i4π2t | ξ | 2) \exp(it\Delta) f(x) = \frac{1}{(4\pi it)^{n/2}} \int_{R^n} e^{i|x-y|^2/4t} f(y) dy Schrödinger kernel \frac{1}{(4\pi it)^{n/2}} e^{i|x|^2/4t}
Hilbert transform H (one dimension only) − isgn(ξ) 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^{2\pi ix\xi} dx sin(2πRx) / πx
Disk multiplier S^0_R 1_{|\xi| \leq R} \int_{|\xi| \leq R} \hat f(\xi) e^{2\pi ix\xi} dx | x | − n / 2Jn / 2(2π | x | ) (J is a Bessel function)
Bochner-Riesz operators S^\delta_R (1 - |\xi|^2/R^2)_+^\delta \int_{|\xi| \leq R} (1 - \frac{|\xi|^2}{R^2})^\delta \hat f(\xi)\ d\xi \int_{|\xi| \leq R} (1 - \frac{|\xi|^2}{R^2})^\delta\ d\xi
General Fourier multiplier m(ξ) \int_{R^n} m(\xi) \hat f(\xi) e^{2\pi i x \cdot \xi} d\xi \int_{R^n} m(\xi) e^{2\pi i x \cdot \xi}\ d\xi
General convolution operator \hat K(\xi) f*K(x) := \int_{R^n} f(y) K(x-y) dy K(x)

In signal processing, a multiplier whose symbol is close to 1 for low frequencies and close to 0 for high frequencies is known as a low pass filter; conversely, a multiplier whose symbol is close to 0 for low frequencies and close to 1 for high frequencies is known as a high pass filter. In Fourier analysis, such multipliers are sometimes known as Littlewood-Paley multipliers. The Littlewood-Paley square function can also be viewed as a Fourier multiplier, but the symbol is vector-valued rather than scalar-valued.

[edit] General considerations

The map m \mapsto T_m is an homomorphism of C*-algebras, thus the sum of two multipliers Tm and Tm' 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 Tm 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 L2(G)) is a Fourier multiplier.

[edit] 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 Lp(G) to Lp(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 Lp if and only if it is bounded on Lq.

The Riesz-Thorin theorem shows that if a Fourier multiplier is bounded on two different Lp spaces, then they are also bounded on all intermediate spaces. Hence we get that the space is multipliers is smallest for L1 and L∞ and grows as one approaches L2, which has the largest multiplier space.

[edit] 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 L2(G) if and only if the symbol m is bounded and measurable.

[edit] 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 / 2πZ, a symbol (m_n)_{n=-\infty}^\infty generates a bounded multiplier in L1 (or L^\infty) if an only if there exists a measure μ such that mn 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 L1 (or L^\infty) if and only if it is periodic after modifying finitely many of the mn.

[edit] 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 Lp 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 Lp(R / 2πZ) 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 Rn 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 Lp(Rn) 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.

[edit] Examples

Translations are bounded operators on any Lp. Differentiation is not bounded on any Lp. 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 Lp for all 1 < p < \infty.

Another interesting case is when the sequence xn is constant on the intervals [2n,2n + 1 − 1] and [ − 2n + 1 + 1, − 2n]. 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 Lp 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 Lp 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 mn:

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 L2.

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