Multiplier (Fourier analysis)

In the setting of periodic functions defined on the unit circle, the Fourier transform of a function is simply the sequence of its Fourier coefficients. To see that differentiation can be realized as multiplier, consider the Fourier series for the derivative of a periodic function f(t). After using integration by parts in the definition of the Fourier coefficient we have that :\mathcal{F}(f')(n)=\int_{-\pi}^\pi f'(t)e^{-int}\,dt=\int_{-\pi}^\pi (i n) f(t)e^{-int}\,dt = in\cdot\mathcal{F}(f)(n). So, formally, it follows that the Fourier series for the derivative is simply the Fourier series for f multiplied by a factor i n . This is the same as saying that differentiation is a multiplier operator with multiplier i n . An example of a multiplier operator acting on functions on the real line is the Hilbert transform. It can be shown that the Hilbert transform is a multiplier operator whose multiplier is given by the m(\xi) = -i \operatorname{sgn}(\xi) , where sgn is the signum function. Finally another important example of a multiplier is the characteristic function of the unit cube in \R^n which arises in the study of "partial sums" for the Fourier transform (see Convergence of Fourier series). ==Definition==

Multiplier operators can be defined on any group G for which the Fourier transform is also defined (in particular, on any locally compact abelian group). The general definition is as follows. If f:G\to\Complex is a sufficiently regular function, let \hat f: \hat G \to \Complex denote its Fourier transform (where \hat G is the Pontryagin dual of G). Let m: \hat G \to \Complex denote another function, which we shall call the multiplier. Then the multiplier operator 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 ξ is given by the Fourier transform of f at that frequency, multiplied by the value of the multiplier at that frequency. This explains the terminology "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 multiplier m, whether the corresponding Fourier multiplier operator 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 multiplier 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 multiplier operator 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 multiplier operators as operators which are diagonalized by the Fourier transform. == Multiplier operators on common groups ==

We now specialize the above general definition to specific groups G. First consider the unit circle G = \R / 2\pi\Z; functions on G can thus be thought of as 2π-periodic functions on the real line. In this group, the Pontryagin dual is the group of 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 multiplier in this setting is simply a sequence (m_n)_{n=-\infty}^\infty of numbers, and the operator T = T_m associated to this multiplier is then given by the formula :(Tf)(t) := \sum_{n=-\infty}^{\infty}m_n \hat{f}(n)e^{int}, at least for sufficiently well-behaved choices of the multiplier (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 :\begin{align} \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. \end{align} A multiplier in this setting is a function m: \R^n \to \Complex, and the associated multiplier operator T = T_m 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 multiplier and function. In the sense of distributions, there is no difference between multiplier operators and convolution operators; every 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 by an amount x0 is convolution with a Dirac delta function δ(· − x0), differentiation is convolution with δ'. Further examples are given in the table below. ==Diagrams==

On the unit circle The following table shows some common examples of multiplier operators on the unit circle G = \R/2\pi \Z. {N}\right) 1_{-N \leq n \leq N} On the Euclidean space The following table shows some common examples of multiplier operators on Euclidean space G = \R^n. General considerations The map m \mapsto T_m is a homomorphism of C*-algebras. This follows because the sum of two multiplier operators T_m and T_{m'} is a multiplier operators with multiplier m+m', the composition of these two multiplier operators is a multiplier operator with multiplier mm', and the adjoint of a multiplier operator T_m is another multiplier operator with multiplier \overline{m}. In particular, we see that any two multiplier operators commute with each other. It is known that multiplier operators are translation-invariant. Conversely, one can show that any translation-invariant linear operator which is bounded on L2(G) is a multiplier operator. ==The Lp boundedness problem==

The Lp boundedness problem (for any particular p) for a given group G is, stated simply, to identify the multipliers m such that the corresponding multiplier operator is bounded from Lp(G) to Lp(G). Such multipliers are usually simply referred to as "Lp multipliers". Note that as multiplier operators are always linear, such operators are bounded if and only if they are continuous. This problem is considered to be extremely difficult in general, but many special cases can be treated. The problem depends greatly on p, although there is a duality relationship: if 1/p + 1/q = 1 and 1 ≤ p, q ≤ ∞, then a multiplier operator is bounded on Lp if and only if it is bounded on Lq. The Riesz-Thorin theorem shows that if a multiplier operator is bounded on two different Lp spaces, then it is also bounded on all intermediate spaces. Hence we get that the space of multipliers is smallest for L1 and L∞ and grows as one approaches L2, 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 function m is an L2(G) multiplier if and only if it is bounded and measurable. Boundedness on L1 or L∞ This case is more complicated than the Hilbertian (L2) case, but is fully resolved. The following is true: Theorem: In the euclidean space \R^n a function m(\xi) is an L1 multiplier (equivalently an L∞ multiplier) if and only if there exists a finite Borel measure μ such that m is the Fourier transform of μ. (The "if" part is a simple calculation. The "only if" part here is more complicated.) ===Boundedness on Lp for 1 p space, the multiplier must be bounded and measurable (this follows from the characterisation of L2 multipliers above and the inclusion property). However, this is not sufficient except when p'' = 2. Results that give sufficient conditions for boundedness are known as multiplier theorems. Three such results are given below. Marcinkiewicz multiplier theorem Let m: \R \to \R be a bounded function that is continuously differentiable on every set of the form \left(-2^{j+1}, -2^j\right) \cup \left(2^j, 2^{j+1}\right) for j \in \Z and has derivative such that :\sup_{j \in \Z} \left( \int_{-2^{j+1}}^{-2^j} \left|m'(\xi)\right| \, d\xi + \int_{2^j}^{2^{j+1}} \left|m'(\xi)\right| \, d\xi \right) Then m is an Lp multiplier for all 1 \R^n which is smooth except possibly at the origin, and such that the function |x|^k \left|\nabla^k m\right| is bounded for all integers 0 \leq k \leq \frac{n}{2} + 1: then m is an Lp multiplier for all . This is a special case of the Hörmander-Mikhlin multiplier theorem. The proofs of these two theorems are fairly tricky, involving techniques from Calderón–Zygmund theory and the Marcinkiewicz interpolation theorem: for the original proof, see or . Radial multipliers For radial multipliers, a necessary and sufficient condition for L^p\left(\mathbb{R}^n\right) boundedness is known for some partial range of p. Let n \geq 4 and 1 . Suppose that m is a radial multiplier compactly supported away from the origin. Then m is an L^p\left(\mathbb{R}^n\right) multiplier if and only if the Fourier transform of m belongs to L^p\left(\mathbb{R}^n\right). This is a theorem of Heo, Nazarov, and Seeger. They also provided a necessary and sufficient condition which is valid without the compact support assumption on m. Examples Translations are bounded operators on any Lp. Differentiation is not bounded on any Lp. The Hilbert transform is bounded only for p strictly between 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 . However, both the Marcinkiewicz and Mikhlin multiplier theorems show that the Hilbert transform is bounded in Lp for all . Another interesting case on the unit circle is when the sequence (x_n) that is being proposed as a multiplier is constant for n in each of the sets \left\{2^n, \ldots, 2^{n+1} - 1\right\} and \left\{-2^{n+1} + 1, \ldots, -2^n\right\}. From the Marcinkiewicz multiplier theorem (adapted to the context of the unit circle) we see that any such sequence (also assumed to be bounded, of course) is a multiplier for every . In one dimension, the disk multiplier operator S^0_R(see table above) is bounded on Lp for every . However, in 1972, Charles Fefferman showed the surprising result that in two and higher dimensions the disk multiplier operator S^0_R is unbounded on Lp for every . The corresponding problem for Bochner–Riesz multipliers is only partially solved; see also Bochner–Riesz conjecture. == See also ==