Two-dimensional infrared spectroscopy

2013-12-13T08:42:47Z

129.125.7.208: /* Spectral interpretation */ Reference to freely available simulation packages made

In the [[mathematics|mathematical]] theory of [[harmonic analysis]], the '''Riesz transforms''' are a family of generalizations of the [[Hilbert transform]] to [[Euclidean space]]s of dimension ''d'' > 1. They are a type of [[singular integral operator]], meaning that they are given by a [[convolution]] of one function with another function having a singularity at the origin. Specifically, the Riesz transforms of a complex-valued function ƒ on '''R'''''d'' are defined by
{{NumBlk|:|<math>R_jf(x) = c_d\lim_{\epsilon\to 0}\int_{\mathbf{R}^d\backslash B_\epsilon(0)}\frac{(t_j-x_j)f(t)}{|x-t|^{d+1}}\,dt</math>|{{EquationRef|1}}}}
for ''j'' = 1,2,...,''d''. The constant ''c''''d'' is a dimensional normalization given by

:<math>c_d = \frac{1}{\pi\omega_{d-1}} = \frac{\Gamma[(d+1)/2]}{\pi^{(d+1)/2}}.</math>

where ω''d''−1 is the volume of the (''d'' − 1)-ball. The limit is written in various ways, often as a [[Cauchy principal value|principal value]], or as a [[convolution]] with the [[tempered distribution]]

:<math>K(x) = \frac{1}{\pi\omega_{d-1}} \, p.v. \frac{x_j}{|x|^{d+1}}.</math>

The Riesz transforms arises in the study of differentiability properties of harmonic potentials in [[potential theory]] and [[harmonic analysis]]. In particular, they arise in the proof of the [[Calderón-Zygmund inequality]] {{harv|Gilbarg|Trudinger|1983|loc=§9.4}}.

==Multiplier properties==

The Riesz transforms are given by a [[Fourier multiplier]]. Indeed, the [[Fourier transform]] of ''R''''j''ƒ is given by

:<math>\mathcal{F}(R_jf)(x) = i\frac{x_j}{|x|}(\mathcal{F}f)(x)</math>

(up to an overall positive constant depending on the normalization of the Fourier transform). In this form, the Riesz transforms are seen to be generalizations of the Hilbert transform. The kernel is a [[distribution (mathematics)|distribution]] which is [[homogeneous function|homogeneous]] of degree zero. A particular consequence of this last observation is that the Riesz transform defines a [[bounded linear operator]] from ''L''2('''R'''''d'') to itself.<ref>Strictly speaking, the definition ({{EquationNote|1}}) may only make sense for [[Schwartz function]] ''f''. Boundedness on a dense subspace of ''L''2 implies that each Riesz transform admits a continuous linear extension to all of ''L''2.</ref>

This homogeneity property can also be stated more directly without the aid of the Fourier transform. If σ''s'' is the [[homothety|dilation]] on '''R'''''d'' by the scalar ''s'', that is σ''s''''x'' = ''sx'', then σ''s'' defines an action on functions via [[pullback (differential geometry)|pullback]]:

:<math>\sigma_s^* f = f\circ\sigma_s.</math>

The Riesz transforms commute with σ''s'':

:<math>\sigma_s^* (R_jf) = R_j(\sigma_x^*f).</math>

Similarly, the Riesz transforms commute with translations. Let τ''a'' be the translation on '''R'''''d'' along the vector ''a''; that is, τ''a''(''x'') = ''x'' + ''a''. Then

:<math>\tau_a^* (R_jf) = R_j(\tau_a^*f).</math>

For the final property, it is convenient to regard the Riesz transforms as a single [[Vector (geometric)|vectorial]] entity ''R''ƒ = (''R''1ƒ,…,''R''''d''ƒ). Consider a [[rotation]] ρ in '''R'''''d''. The rotation acts on spatial variables, and thus on functions via pullback. But it also can act on the spatial vector ''R''ƒ. The final transformation property asserts that the Riesz transform is [[equivariant]] with respect to these two actions; that is,

:<math>\rho^* R_j [(\rho^{-1})^*f] = \sum_{k=1}^d \rho_{jk} R_kf.</math>

These three properties in fact characterize the Riesz transform in the following sense. Let ''T''=(''T''''1'',…,''T''''d'') be a ''d''-tuple of bounded linear operators from ''L''2('''R'''''d'') to ''L''2('''R'''''d'') such that

* ''T'' commutes with all dilations and translations.
* ''T'' is equivariant with respect to rotations.

Then, for some constant ''c'', ''T'' = ''cR''.

==Relationship with the Laplacian==
Somewhat imprecisely, the Riesz transforms of ƒ give the first [[partial derivative]]s of a solution of the equation

:<math>{(-\Delta)^{\frac{1}{2}} u = f},</math>

where Δ is the Laplacian. Thus the Riesz transform of ƒ can be written as:

:<math>{R f = \nabla (-\Delta)^{-\frac{1}{2}}f}</math>

In particular, one should also have
:<math>R_iR_j\Delta u = -\frac{\partial^2u}{\partial x_i\partial x_j},</math>
so that the Riesz transforms give a way of recovering information about the entire [[Hessian matrix|hessian]] of a function from knowledge of only its Laplacian.

This is now made more precise. Suppose that ''u'' is a [[Schwartz function]]. Then indeed by the explicit form of the Fourier multiplier, one has
:<math>R_iR_j(\Delta u) = -\frac{\partial^2u}{\partial x_i\partial x_j}.</math>
The identity is not generally true in the sense of [[distribution (mathematics)|distributions]]. For instance, if ''u'' is a [[tempered distribution]] such that Δ''u'' ∈ ''L''2('''R'''''d''), then one can only conclude that
:<math>\frac{\partial^2u}{\partial x_i\partial x_j} = -R_iR_j\Delta u + P_{ij}(x)</math>
for some polynomial ''P''''ij''.

==See also==
* [[Poisson kernel]]
* [[Riesz potential]]

==References==
{{Reflist}}
* {{citation|first1=D.|last=Gilbarg|first2=Neil|last2=Trudinger|authorlink2=Neil Trudinger|title=Elliptic Partial Differential Equations of Second Order|publisher=Springer|publication-place=New York|year=1983|isbn=3-540-41160-7}}.
* {{citation|first=Elias|last=Stein|authorlink=Elias Stein|title=Singular integrals and differentiability properties of functions|publisher=Princeton University Press|year=1970}}.
* {{citation|first1=Elias|last1=Stein|authorlink1=Elias Stein|first2=Guido|last2=Weiss|title=Introduction to Fourier Analysis on Euclidean Spaces|publisher=Princeton University Press|year=1971|isbn=0-691-08078-X}}.
* {{citation|first1=N.|last=Arcozzi|title=Riesz Transform on spheres and compact Lie groups|publisher=Springer|publication-place=New York|year=1998|issn=0004-2080}}.

[[Category:Harmonic analysis]]
[[Category:Integral transforms]]
[[Category:Potential theory]]
[[Category:Singular integrals]]

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Two-dimensional infrared spectroscopy