Fourier inversion theorem

Fourier inversion theorem In mathematics, the Fourier inversion theorem says that for many types of functions it is possible to recover a function from its Fourier transform. Intuitively it may be viewed as the statement that if we know all frequency and phase information about a wave then we may reconstruct the original wave precisely.

The theorem says that if we have a function {stile di visualizzazione f:mathbb {R} a matematicabb {C} } satisfying certain conditions, and we use the convention for the Fourier transform that {stile di visualizzazione ({matematico {F}}f)(xi ):=int _{mathbb {R} }e^{-2pi iycdot xi },f(y),dio,} poi {stile di visualizzazione f(X)=int _{mathbb {R} }e^{2pi ixcdot xi },({matematico {F}}f)(xi ),dxi .} In altre parole, the theorem says that {stile di visualizzazione f(X)= non _{mathbb {R} ^{2}}e^{2pi io(x-y)cdot xi },f(y),dio,dxi .} This last equation is called the Fourier integral theorem.

Another way to state the theorem is that if {stile di visualizzazione R} is the flip operator i.e. {stile di visualizzazione (Rf)(X):=f(-X)} , poi {stile di visualizzazione {matematico {F}}^{-1}={matematico {F}}R=R{matematico {F}}.} The theorem holds if both {stile di visualizzazione f} and its Fourier transform are absolutely integrable (nel senso di Lebesgue) e {stile di visualizzazione f} is continuous at the point {stile di visualizzazione x} . Tuttavia, even under more general conditions versions of the Fourier inversion theorem hold. In these cases the integrals above may not converge in an ordinary sense.

Contenuti 1 Dichiarazione 1.1 Inverse Fourier transform as an integral 1.2 Fourier integral theorem 1.3 Inverse transform in terms of flip operator 1.4 Two-sided inverse 2 Conditions on the function 2.1 Schwartz functions 2.2 Integrable functions with integrable Fourier transform 2.3 Integrable functions in one dimension 2.4 Square integrable functions 2.5 Tempered distributions 3 Relation to Fourier series 4 Applicazioni 5 Properties of inverse transform 6 Prova 7 Appunti 8 References Statement In this section we assume that {stile di visualizzazione f} is an integrable continuous function. Use the convention for the Fourier transform that {stile di visualizzazione ({matematico {F}}f)(xi ):=int _{mathbb {R} ^{n}}e^{-2pi iycdot xi },f(y),dio.} Inoltre, we assume that the Fourier transform is also integrable.

Inverse Fourier transform as an integral The most common statement of the Fourier inversion theorem is to state the inverse transform as an integral. For any integrable function {stile di visualizzazione g} e tutto {displaystyle xin mathbb {R} ^{n}} impostare {stile di visualizzazione {matematico {F}}^{-1}g(X):=int _{mathbb {R} ^{n}}e^{2pi ixcdot xi },g(xi ),dxi .} Then for all {displaystyle xin mathbb {R} ^{n}} noi abbiamo {stile di visualizzazione {matematico {F}}^{-1}({matematico {F}}f)(X)=f(X).} Fourier integral theorem The theorem can be restated as {stile di visualizzazione f(X)=int _{mathbb {R} ^{n}}int _{mathbb {R} ^{n}}e^{2pi io(x-y)cdot xi },f(y),dio,dxi .} If f is real valued then by taking the real part of each side of the above we obtain {stile di visualizzazione f(X)=int _{mathbb {R} ^{n}}int _{mathbb {R} ^{n}}cos(2pi (x-y)cdot xi ),f(y),dio,dxi .} Inverse transform in terms of flip operator For any function {stile di visualizzazione g} define the flip operator[Nota 1] {stile di visualizzazione R} di {displaystyle Rg(X):=g(-X).} Then we may instead define {stile di visualizzazione {matematico {F}}^{-1}f:=R{matematico {F}}f={matematico {F}}Rf.} It is immediate from the definition of the Fourier transform and the flip operator that both {stile di visualizzazione R{matematico {F}}f} e {stile di visualizzazione {matematico {F}}Rf} match the integral definition of {stile di visualizzazione {matematico {F}}^{-1}f} , and in particular are equal to each other and satisfy {stile di visualizzazione {matematico {F}}^{-1}({matematico {F}}f)(X)=f(X)} .

Da {displaystyle Rf=R{matematico {F}}^{-1}{matematico {F}}f=RR{matematico {FF}}f} noi abbiamo {stile di visualizzazione R={matematico {F}}^{2}} e {stile di visualizzazione {matematico {F}}^{-1}={matematico {F}}^{3}.} Two-sided inverse The form of the Fourier inversion theorem stated above, as is common, is that {stile di visualizzazione {matematico {F}}^{-1}({matematico {F}}f)(X)=f(X).} In altre parole, {stile di visualizzazione {matematico {F}}^{-1}} is a left inverse for the Fourier transform. However it is also a right inverse for the Fourier transform i.e.

{stile di visualizzazione {matematico {F}}({matematico {F}}^{-1}f)(xi )=f(xi ).} Da {stile di visualizzazione {matematico {F}}^{-1}} is so similar to {stile di visualizzazione {matematico {F}}} , this follows very easily from the Fourier inversion theorem (changing variables {displaystyle zeta :=-zeta } ): {stile di visualizzazione {inizio{allineato}f&={matematico {F}}^{-1}({matematico {F}}f)(X)\[6pt]&=int _{mathbb {R} ^{n}}int _{mathbb {R} ^{n}}e^{2pi ixcdot xi },e^{-2pi iycdot xi },f(y),dio,dxi \[6pt]&=int _{mathbb {R} ^{n}}int _{mathbb {R} ^{n}}e^{-2pi ixcdot zeta },e^{2pi iycdot zeta },f(y),dio,bambino \[6pt]&={matematico {F}}({matematico {F}}^{-1}f)(X).fine{allineato}}} In alternativa, this can be seen from the relation between {stile di visualizzazione {matematico {F}}^{-1}f} and the flip operator and the associativity of function composition, da {stile di visualizzazione f={matematico {F}}^{-1}({matematico {F}}f)={matematico {F}}R{matematico {F}}f={matematico {F}}({matematico {F}}^{-1}f).} Conditions on the function When used in physics and engineering, the Fourier inversion theorem is often used under the assumption that everything "behaves nicely". In mathematics such heuristic arguments are not permitted, and the Fourier inversion theorem includes an explicit specification of what class of functions is being allowed. Tuttavia, there is no "best" class of functions to consider so several variants of the Fourier inversion theorem exist, albeit with compatible conclusions.

Schwartz functions The Fourier inversion theorem holds for all Schwartz functions (grosso modo, smooth functions that decay quickly and whose derivatives all decay quickly). This condition has the benefit that it is an elementary direct statement about the function (as opposed to imposing a condition on its Fourier transform), and the integral that defines the Fourier transform and its inverse are absolutely integrable. This version of the theorem is used in the proof of the Fourier inversion theorem for tempered distributions (vedi sotto).

Integrable functions with integrable Fourier transform The Fourier inversion theorem holds for all continuous functions that are absolutely integrable (cioè. {stile di visualizzazione L^{1}(mathbb {R} ^{n})} ) with absolutely integrable Fourier transform. This includes all Schwartz functions, so is a strictly stronger form of the theorem than the previous one mentioned. This condition is the one used above in the statement section.

A slight variant is to drop the condition that the function {stile di visualizzazione f} be continuous but still require that it and its Fourier transform be absolutely integrable. Quindi {displaystyle f=g} almost everywhere where g is a continuous function, e {stile di visualizzazione {matematico {F}}^{-1}({matematico {F}}f)(X)=g(X)} per ogni {displaystyle xin mathbb {R} ^{n}} .

Integrable functions in one dimension Piecewise smooth; one dimension If the function is absolutely integrable in one dimension (cioè. {displaystyle fin L^{1}(mathbb {R} )} ) and is piecewise smooth then a version of the Fourier inversion theorem holds. In this case we define {stile di visualizzazione {matematico {F}}^{-1}g(X):=lim _{Rto infty }int _{-R}^{R}e^{2pi ixxi },g(xi ),dxi .} Then for all {displaystyle xin mathbb {R} } {stile di visualizzazione {matematico {F}}^{-1}({matematico {F}}f)(X)={frac {1}{2}}(f(X_{-})+f(X_{+})),} cioè. {stile di visualizzazione {matematico {F}}^{-1}({matematico {F}}f)(X)} equals the average of the left and right limits of {stile di visualizzazione f} a {stile di visualizzazione x} . At points where {stile di visualizzazione f} is continuous this simply equals {stile di visualizzazione f(X)} .

A higher-dimensional analogue of this form of the theorem also holds, but according to Folland (1992) è "rather delicate and not terribly useful".

Piecewise continuous; one dimension If the function is absolutely integrable in one dimension (cioè. {displaystyle fin L^{1}(mathbb {R} )} ) but merely piecewise continuous then a version of the Fourier inversion theorem still holds. In this case the integral in the inverse Fourier transform is defined with the aid of a smooth rather than a sharp cut off function; specifically we define {stile di visualizzazione {matematico {F}}^{-1}g(X):=lim _{Rto infty }int _{mathbb {R} }varfi (xi /R),e^{2pi ixxi },g(xi ),dxi ,qquad varphi (xi ):=e^{-xi ^{2}}.} The conclusion of the theorem is then the same as for the piecewise smooth case discussed above.

Continuous; any number of dimensions If {stile di visualizzazione f} is continuous and absolutely integrable on {displaystyle mathbb {R} ^{n}} then the Fourier inversion theorem still holds so long as we again define the inverse transform with a smooth cut off function i.e.

{stile di visualizzazione {matematico {F}}^{-1}g(X):=lim _{Rto infty }int _{mathbb {R} ^{n}}varfi (xi /R),e^{2pi ixcdot xi },g(xi ),dxi ,qquad varphi (xi ):=e^{-vert xi vert ^{2}}.} The conclusion is now simply that for all {displaystyle xin mathbb {R} ^{n}} {stile di visualizzazione {matematico {F}}^{-1}({matematico {F}}f)(X)=f(X).} No regularity condition; any number of dimensions If we drop all assumptions about the (piecewise) continuity of {stile di visualizzazione f} and assume merely that it is absolutely integrable, then a version of the theorem still holds. The inverse transform is again defined with the smooth cut off, but with the conclusion that {stile di visualizzazione {matematico {F}}^{-1}({matematico {F}}f)(X)=f(X)} for almost every {displaystyle xin mathbb {R} ^{n}.} [1] Square integrable functions In this case the Fourier transform cannot be defined directly as an integral since it may not be absolutely convergent, so it is instead defined by a density argument (see the Fourier transform article). Per esempio, putting {stile di visualizzazione g_{K}(xi ):=int _{{yin mathbb {R} ^{n}:leftvert yrightvert leq k}}e^{-2pi iycdot xi },f(y),dio,qquad kin mathbb {N} ,} we can set {stile di visualizzazione stile testo {matematico {F}}f:=lim _{kto infty }g_{K}} where the limit is taken in the {stile di visualizzazione L^{2}} -norma. The inverse transform may be defined by density in the same way or by defining it in terms of the Fourier transform and the flip operator. We then have {stile di visualizzazione f(X)={matematico {F}}({matematico {F}}^{-1}f)(X)={matematico {F}}^{-1}({matematico {F}}f)(X)} in the mean squared norm. In one dimension (and one dimension only), it can also be shown that it converges for almost every x∈ℝ- this is Carleson's theorem, but is much harder to prove than convergence in the mean squared norm.

Tempered distributions The Fourier transform may be defined on the space of tempered distributions {stile di visualizzazione {matematico {S}}'(mathbb {R} ^{n})} by duality of the Fourier transform on the space of Schwartz functions. Specifically for {displaystyle fin {matematico {S}}'(mathbb {R} ^{n})} and for all test functions {displaystyle varphi in {matematico {S}}(mathbb {R} ^{n})} prepariamo {angolo dello stile di visualizzazione {matematico {F}}f,varphi rangle := angolo f,{matematico {F}}varphi rangle ,} dove {stile di visualizzazione {matematico {F}}varfi } is defined using the integral formula. Se {displaystyle fin L^{1}(mathbb {R} ^{n})cap L^{2}(mathbb {R} ^{n})} then this agrees with the usual definition. We may define the inverse transform {stile di visualizzazione {matematico {F}}^{-1}colon {matematico {S}}'(mathbb {R} ^{n})a {matematico {S}}'(mathbb {R} ^{n})} , either by duality from the inverse transform on Schwartz functions in the same way, or by defining it in terms of the flip operator (where the flip operator is defined by duality). We then have {stile di visualizzazione {matematico {F}}{matematico {F}}^{-1}={matematico {F}}^{-1}{matematico {F}}=nome operatore {Id} _{{matematico {S}}'(mathbb {R} ^{n})}.} Relation to Fourier series When considering the Fourier series of a function it is conventional to rescale it so that it acts on {stile di visualizzazione [0,2pi ]} (or is {displaystyle 2pi } -periodic). In this section we instead use the somewhat unusual convention taking {stile di visualizzazione f} to act on {stile di visualizzazione [0,1]} , since that matches the convention of the Fourier transform used here.

The Fourier inversion theorem is analogous to the convergence of Fourier series. In the Fourier transform case we have {displaystyle fcolon mathbb {R} ^{n}a matematicabb {C} ,quad {cappello {f}}colon mathbb {R} ^{n}a matematicabb {C} ,} {stile di visualizzazione {cappello {f}}(xi ):=int _{mathbb {R} ^{n}}e^{-2pi iycdot xi },f(y),dio,} {stile di visualizzazione f(X)=int _{mathbb {R} ^{n}}e^{2pi ixcdot xi },{cappello {f}}(xi ),dxi .} In the Fourier series case we instead have {displaystyle fcolon [0,1]^{n}a matematicabb {C} ,quad {cappello {f}}colon mathbb {Z} ^{n}a matematicabb {C} ,} {stile di visualizzazione {cappello {f}}(K):=int _{[0,1]^{n}}e^{-2pi iycdot k},f(y),dio,} {stile di visualizzazione f(X)=somma _{kin mathbb {Z} ^{n}}e^{2pi ixcdot k},{cappello {f}}(K).} In particolare, in one dimension {displaystyle kin mathbb {Z} } and the sum runs from {displaystyle -infty } a {displaystyle infty } .

Applications Some problems, such as certain differential equations, become easier to solve when the Fourier transform is applied. In that case the solution to the original problem is recovered using the inverse Fourier transform.

In applications of the Fourier transform the Fourier inversion theorem often plays a critical role. In many situations the basic strategy is to apply the Fourier transform, perform some operation or simplification, and then apply the inverse Fourier transform.

More abstractly, the Fourier inversion theorem is a statement about the Fourier transform as an operator (see Fourier transform on function spaces). Per esempio, the Fourier inversion theorem on {displaystyle fin L^{2}(mathbb {R} ^{n})} shows that the Fourier transform is a unitary operator on {stile di visualizzazione L^{2}(mathbb {R} ^{n})} .

Properties of inverse transform The inverse Fourier transform is extremely similar to the original Fourier transform: as discussed above, it differs only in the application of a flip operator. For this reason the properties of the Fourier transform hold for the inverse Fourier transform, such as the Convolution theorem and the Riemann–Lebesgue lemma.

Tables of Fourier transforms may easily be used for the inverse Fourier transform by composing the looked-up function with the flip operator. Per esempio, looking up the Fourier transform of the rect function we see that {stile di visualizzazione f(X)=nome operatore {rect} (ascia)quad Rightarrow quad ({matematico {F}}f)(xi )={frac {1}{|un|}}nome operatore {sinc} sinistra({frac {xi }{un}}Giusto),} so the corresponding fact for the inverse transform is {stile di visualizzazione g(xi )=nome operatore {rect} (axi )quad Rightarrow quad ({matematico {F}}^{-1}g)(X)={frac {1}{|un|}}nome operatore {sinc} sinistra(-{frac {X}{un}}Giusto).} Proof The proof uses a few facts, dato {stile di visualizzazione f(y)} e {stile di visualizzazione {matematico {F}}f(xi )=int _{mathbb {R} ^{n}}e^{-2pi iycdot xi }f(y),dio} .

Se {displaystyle xin mathbb {R} ^{n}} e {stile di visualizzazione g(xi )=e^{2pi mathrm {io} xcdot xi }psi (xi )} , poi {stile di visualizzazione ({matematico {F}}g)(y)=({matematico {F}}psi )(y-x)} . Se {displaystyle varepsilon in mathbb {R} } e {stile di visualizzazione psi (xi )=varphi (varepsilon xi )} , poi {stile di visualizzazione ({matematico {F}}psi )(y)=({matematico {F}}varfi )(y/varepsilon )/|varepsilon |} . Per {stile di visualizzazione f,gin L^{1}(mathbb {R} ^{n})} , Fubini's theorem implies that {displaystyle textstyle int g(xi )cdot ({matematico {F}}f)(xi ),dxi =int ({matematico {F}}g)(y)cdot f(y),dio} . Definire {stile di visualizzazione varphi (xi )=e^{-pi vert xi vert ^{2}}} ; poi {stile di visualizzazione ({matematico {F}}varfi )(y)=varphi (y)} . Definire {displaystyle varphi _{varepsilon }(y)=varphi (y/varepsilon )/varepsilon ^{n}} . Then with {displaystyle ast } denoting convolution, {displaystyle varphi _{varepsilon }} is an approximation to the identity: for any continuous {displaystyle fin L^{1}(mathbb {R} ^{n})} e punto {displaystyle xin mathbb {R} ^{n}} , {displaystyle lim _{varepsilon a 0}(varfi _{varepsilon }ast f)(X)=f(X)} (where the convergence is pointwise).

Da, per assunzione, {stile di visualizzazione {matematico {F}}fine L^{1}(mathbb {R} ^{n})} , then it follows by the dominated convergence theorem that {displaystyle int _{mathbb {R} ^{n}}e^{2pi ixcdot xi }({matematico {F}}f)(xi ),dxi =lim _{varepsilon a 0}int _{mathbb {R} ^{n}}e^{-pi varepsilon ^{2}|xi |^{2}+2pi ixcdot xi }({matematico {F}}f)(xi ),dxi .} Definire {stile di visualizzazione g_{X}(xi )=e^{-pi varepsilon ^{2}vert xi vert ^{2}+2pi mathrm {io} xcdot xi }} . Applying facts 1, 2 e 4, repeatedly for multiple integrals if necessary, otteniamo {stile di visualizzazione ({matematico {F}}g_{X})(y)={frac {1}{varepsilon ^{n}}}e^{-{frac {pi }{varepsilon ^{2}}}|x-y|^{2}}=varphi _{varepsilon }(x-y).} Using fact 3 Su {stile di visualizzazione f} e {stile di visualizzazione g_{X}} , per ciascuno {displaystyle xin mathbb {R} ^{n}} , noi abbiamo {displaystyle int _{mathbb {R} ^{n}}e^{-pi varepsilon ^{2}|xi |^{2}+2pi ixcdot xi }({matematico {F}}f)(xi ),dxi =int _{mathbb {R} ^{n}}{frac {1}{varepsilon ^{n}}}e^{-{frac {pi }{varepsilon ^{2}}}|x-y|^{2}}f(y),dy=(varfi _{varepsilon }*f)(X),} the convolution of {stile di visualizzazione f} with an approximate identity. But since {displaystyle fin L^{1}(mathbb {R} ^{n})} , fact 5 Dillo {displaystyle lim _{varepsilon a 0}(varfi _{varepsilon }*f)(X)=f(X).} Putting together the above we have shown that {displaystyle int _{mathbb {R} ^{n}}e^{2pi ixcdot xi }({matematico {F}}f)(xi ),dxi =f(X).qquad square } Notes ^ An operator is a transformation that maps functions to functions. The flip operator, the Fourier transform, the inverse Fourier transform and the identity transform are all examples of operators. References This article includes a list of references, letture correlate o collegamenti esterni, ma le sue fonti rimangono poco chiare perché mancano di citazioni inline. Aiutaci a migliorare questo articolo introducendo citazioni più precise. (Gennaio 2013) (Scopri come e quando rimuovere questo messaggio modello) Folland, G. B. (1992). Fourier Analysis and its Applications. Belmont, circa, Stati Uniti d'America: Wadsworth. ISBN 0-534-17094-3. Folland, G. B. (1995). Introduction to Partial Differential Equations (2nd ed.). Princeton, Stati Uniti d'America: Princeton Univ. Premere. ISBN 978-0-691-04361-6. ^ "DMat0101, Appunti 3: The Fourier transform on L^1". I Woke Up In A Strange Place. 2011-03-10. Recuperato 2018-02-12. Categorie: Generalized functionsTheorems in Fourier analysis