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| In [[mathematical analysis]], more precisely in [[microlocal analysis]], the '''wave front (set)''' WF(''f'') characterizes the [[Mathematical singularity|singularities]] of a [[generalized function]] ''f'', not only in [[space]], but also with respect to its [[Fourier transform]] at each point. The term "wave front" was coined by [[Lars Hörmander]] around 1970.
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| ==Introduction==
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| In more familiar terms, WF(''f'') tells not only ''where'' the function ''f'' is singular (which is already described by its [[singular support]]), but also ''how'' or ''why'' it is singular, by being more exact about the direction in which the singularity occurs. This concept is mostly useful in dimension at least two, since in one dimension there are only two possible directions. The complementary notion of a function being non-singular in a direction is ''microlocal smoothness''.
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| Intuitively, as an example, consider a function ƒ whose singular support is concentrated on a smooth curve in the plane at which the function has a jump discontinuity. In the direction tangent to the curve, the function remains smooth. By contrast, in the direction normal to the curve, the function has a singularity. To decide on whether the function is smooth in another direction ''v'', one can try to smooth the function out by averaging in directions perpendicular to ''v''. If the resulting function is smooth, then we regard ƒ to be smooth in the direction of ''v''. Otherwise, ''v'' is in the wavefront set.
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| Formally, in [[Euclidean space]], the '''wave front set''' of ƒ is defined as the [[Complement (set theory)|complement]] of the set of all pairs (''x''<sub>0</sub>,''v'') such that there exists a test function <math>\phi\in C_0^\infty</math> with φ(''x''<sub>0</sub>) ≠ 0 and an open cone Γ containing ''v'' such that the estimate
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| :<math>|(\phi f)^\wedge(\xi)| \le C_N(1+|\xi|)^{-N}\quad\mbox{for all }\ \xi\in\Gamma</math>
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| holds for all positive integers ''N''. Here <math>(\phi f)^\wedge</math> denotes the Fourier transform. Observe that the wavefront set is a [[cone (topology)|conical]] set, since if (''x'',''v'') ∈ Wf(ƒ), then (''x'',λ''v'') ∈ Wf(ƒ) for all λ > 0. In the example discussed in the previous paragraph, the wavefront set is the set-theoretic complement of the image of the tangent bundle of the curve inside the tangent bundle of the plane.
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| Because the definition involves cutoff by a compactly supported function, the notion of a wave front set can be transported to any [[differentiable manifold]] ''X''. In this more general situation, the wave front set is a closed conical subset of the [[cotangent bundle]] ''T''<sup>*</sup>(''X''), since the ξ variable naturally localizes to a [[covector]] rather than a vector. The wave front set is defined such that its projection on ''X'' is equal to the [[singular support]] of the function.
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| ==Definition==
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| In Euclidean space, the wave front set of a [[distribution (mathematics)|distribution]] ƒ is defined as
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| :<math>{\rm WF}(f) = \{ (x,\xi)\in \mathbb{R}^n\times\mathbb{R}^n \mid \xi\in\Sigma_x(f)\}</math>
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| where <math>\Sigma_x(f)</math> is the singular fibre of ƒ at ''x''. The singular fibre is defined to be the [[complement (set theory)|complement]] of all directions <math>\xi</math> such that the Fourier transform of ''f'', localized at ''x'', is sufficiently regular when restricted to an open cone containing <math>\xi</math>. More precisely, a direction ''v'' is in the complement of <math>\Sigma_x(f)</math> if there is a compactly supported smooth function φ with φ(''x'') ≠ 0 and an open cone Γ containing ''v'' such that the following estimate holds for each positive integer ''N'':
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| :<math>(\phi f)^\wedge(\xi) < c_N(1+|\xi|)^{-N}\quad{\rm for all}\ \xi\in\Gamma.</math>
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| Once such an estimate holds for a particular cutoff function φ at ''x'', it also holds for all cutoff functions with smaller support, possibly for a different open cone containing ''v''.
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| On a [[differentiable manifold]] ''M'', using local coordinates <math>x,\xi</math> on the [[cotangent bundle]], the wave front set WF(''f'')
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| of a distribution ƒ can be defined in the following general way:
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| : <math>{\rm WF}(f) = \{ (x,\xi)\in T^*(X) \mid \xi\in\Sigma_x(f) \}</math>
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| where the singular fibre <math>\Sigma_x(f)</math> is again the complement of all directions <math>\xi</math> such that the Fourier transform of ''f'', localized at ''x'', is sufficiently regular when restricted to a conical neighbourhood of <math>\xi</math>. The problem of regularity is local, and so it can be checked in the local coordinate system, using the Fourier transform on the ''x'' variables. The required regularity estimate transforms well under [[diffeomorphism]], and so the notion of regularity is independent of the choice of local coordinates.
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| ===Generalizations===
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| The notion of a wave front set can be adapted to accommodate other notions of regularity of a function. Localized can here be expressed by saying that ''f'' is truncated by some [[Smooth function#Smoothness|smooth]] [[cutoff function]] not vanishing at ''x''. (The localization process could be done in a more elegant fashion, using [[germ (mathematics)|germ]]s.) | |
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| More concretely, this can be expressed as
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| : <math>\xi\notin\Sigma_x(f) \iff \exists\phi\in\mathcal D_x,\ \exists V\in\mathcal V_\xi: \widehat{\phi f}|_V\in O(V) </math> (or <math>\xi=o</math>, never in <math>\Sigma_x(f)</math>)
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| where
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| *<math>\mathcal D_x</math> are [[compactly supported]] [[smooth function]]s not vanishing at ''x'',
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| *<math>\mathcal V_\xi</math> are ''conical neighbourhoods'' of <math>\xi</math>, i.e. [[neighbourhood (mathematics)|neighbourhood]]s ''V'' such that <math>c\cdot V\subset V</math> for all <math>c > 0</math>,
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| *<math>\widehat u|_V</math> denotes the [[Fourier transform]] of the (compactly supported generalized) function ''u'', restricted to ''V'',
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| *and finally <math>O: \Omega\to O(\Omega)</math> is the [[Presheaf#Presheaves|presheaf]] characterizing the regularity of the Fourier transform.
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| Typically, sections of ''O'' are characterized by some growth (or decrease) condition at infinity, e.g. such that <math>(1+|\xi|)^s v(\xi)</math> belong to some [[Lp space|L<sup>p</sup> space]].
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| This definition makes sense, because the Fourier transform becomes more
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| regular (in terms of growth at infinity) when ''f'' is truncated with the smooth cutoff <math>\phi</math>.
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| The most difficult "problem", from a theoretical point of view,
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| is finding the adequate sheaf ''O'' characterizing functions belonging to a given subsheaf ''E'' of the space ''G'' of generalized functions.
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| ===Example===
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| If we take ''G'' = ''D''′ the space of [[Schwartz distribution]]s and want to characterize distributions which are locally <math>C^\infty</math> functions, | |
| we must take for ''O''(Ω) the classical function spaces called ''O''′<sub>''M''</sub>(Ω) in the literature.
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| Then the projection on the first component of a distribution's wave front set is nothing else than its classical [[singular support]], i.e. the complement of the set on which its restriction would be a [[smooth function]].
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| ===Applications===
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| The wave front set is useful, among others, when studying [[wave propagation|propagation]] of [[Mathematical singularity|singularities]] by [[pseudodifferential operator]]s.
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| ==See also==
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| * [[FBI transform]]
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| * [[Singular spectrum]]
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| * [[Essential support]]
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| ==References==
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| * [[Lars Hörmander]], ''Fourier integral operators I'', Acta Math. '''127''' (1971), pp. 79-183.
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| *{{citation|last=Hörmander|first=Lars|authorlink=Lars Hörmander|pages=251–279|title=The Analysis of Linear Partial Differential Equations I: Distribution Theory and Fourier Analysis|edition=2nd|publisher=Springer|series=Grundlehren der mathematischen Wissenschaften|volume= 256|year=
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| 1990|id=ISBN 0-387-52345-6}} Chapter VIII, Spectral Analysis of Singularities
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| [[Category:Mathematical analysis]]
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| [[Category:Generalized functions]]
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So in case you are asking yourself if you're able to still be in good shape after 40 the answer is yes, nevertheless you must understand that if you wish to stay in this shape you'll need exercise. The hypnotist would probably create some recommendations that concentrate on this anxiety to discover the invisible problem. Males with erectile problems are the majority of the time attempting to reach orgasm inside a hurry because they're afraid of losing an erection before intercourse is done, deeply disturbed, these men ejaculate faster than hoped. He has snubbed the world's most powerful spy agency and government with abandon. Levitra takes it easy the effortless muscles of the penis and advances the flow of the blood into the penis, when men are sexually aroused.
In most cases the cause for male impotence is a physical problem. Therefore, if you are considering boosting your testosterone either by using a natural testosterone supplement or by visiting your doctor for testosterone injections or creams, you should look online to get a feel for what products are working for other men. In fact, we would love to get you started today with Top Ten Tips to Naturally Cure Male Impotence. If these key treatments do not assist you, your specialist may recommend you impotence drugs such as Viagra, Cialis and Levitra. By witnessing the success of the anti-impotence pill, many manufactures started to produce drug similar to the innovator anti-impotence which would be more appealing to the general public.
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