Lecture. Fourier transform Definition and main resuts Let f L 2 (R). The Fourier transform of a function f is a function f(α) = f(x)t iαx dx () The normaized Fourier transform of f is a function R ˆf = f (2) It is not obvious that the Fourier transform exists for any function f L 2 R. This is proved beow. The main resuts of this ecture are: Theorem (Panchere equaity) f L 2 (R) ˆf L 2 (R), and f = ˆf. Theorem 2 [Inversion Formua] f L 2 (R), f(x) = f(α)e iαx dα R We wi prove these theorems for the case when f is a C 2 function with a finite support, and then extend it to the whoe of L 2 (R). The Fourier transform is a continuous anaog of the Fourier series. The Fourier series are defined on a circe, and the Fourier transform is defined on the rea ine. The atter is obtained from the first one by passing to the imit when the circe ength tends to infinity.
2 Decreasing of the Fourier transform for a smooth function Theorem 3 Fourier transform of a C m sower than x m. function with a compact support decreases no The proof is quite the same as for the Fourier series. Proof The Fourier transform brings the derivation to the mutipication by iα: f (α) = iα f(α). (3) This formua is proved by the integration by parts. Denoting the Fourier transform by F, we have: F(f )(α) = f (x)e iαx dx = e iαx df(x) = f(x)de iαx dx = iα f(x)e iαx dx = iαff(α). This impies By induction in M we get: Ff(α) = iα F(f )(α). Ff(α) = (iα) m F(f (m) )(α). But the function F(f (m) ) is bounded because f (m) is continuous with the finite support. Coroary If f C 2,0, then there exists a C such that: f(α) < C + α 2. (4) 3 Fourier series on a ong circe The circe may be represented by a segment with the endpoints identified. Another representation: the functions on a circe of ength T are represented by T -periodic functions on the rea ine. We wi use this atter representation. Consider a -periodic function f : R R and decompose it to the Fourier series with respect to a system that we wi now construct. Consider a -periodic function g(x) = f(x) and decompose it into a cassica Fourier series: g(x) = Σ k Z g k e ikx. (5) 2
The factors ik in the exponentia ar caed the wave numbers. For the cassica Fourier series the set of wave numbers coincides with Z. By definition of g, ( x ) f(x) = g = Σg k e ik x = Σ α Z c a e iαx (6) The wave numbers in the atter series range over the set Z. For the arge this set is much thicker than Z. Somewhat oosey we may say that as this set tends to R. Equaity (6) impies that the vectors e iαx, a Z form a basis in L 2 ([ π, π]). Their norms equa. Hence, for α Z, c α = π f(x)e iαx dx π Note that for α Z, c α = F(fχ [ π,π])(α) (7) The Panchere equaity for f has the form The Fourier decomposition of f is given by (6), (7). f 2 = Σ α Z c α 2. (8) 4 Passing to the imit: heuristic proofs of the Panchere equaity and Inversion Formua Fix a function f C 2,0 with the support supp f [ π 0, π 0 ], and for any > 0 consider the restriction of f to the segment [ π, π].formay, these restrictions are different functions, but we wi denote them a by f. Let us prove the Panchere equaity: f = ˆf. It is sufficient to prove that f 2 = f 2 (9) Let us extend the function f -periodicay, end decompose the resuting function to the Fourier series. By (8) and (7), f 2 = Σ α Z f(α) 2 := Σ. (0) 3
The expression Σ is an integra sum for the integra f 2 = f(α) 2 dα := I This sum corresponds to the partition of the ine by the segments of ength with the endpoints in the set Z. On one hand, this sum tends to the integra (this shoud be proved!); on the other hand, the sequence Σ is stationary (does not depend on ). This proves (9). In the same way, the Inversion Formua may be proved. By (6), for x <, S is an integra sum for the integra f(x) = Σ α Z I(x) = f(α)e iαx := S. f(α)e iαx dα. Passing to the imit as above (this taking of the imit shoud be justified), we obtain the Inversion Formua: f(x) = f(α)e iαx dα () 5 Forma proof Lemma S R f(α) 2 dα provided that: f C 2,0. Lemma 2 If C 2,0, then Σ (x) I(x) for. R The previous arguments and Lemma impy the Panchere equaity, and Lemma 2 impies the Inversion Formua. Proof [of Lemma ]. If the integra I have been proper, the integra sums woud tend to the integra by the cassica theorem about Riemann integras. We need to overcome an improper integra. This is done by the estimates beow. By Coroary, there exists C > 0 : f(α) < C( + α 2 ). Take ε > 0 and such N that Σ α Z \[ N,N] f(α) 2 < Σ C α Z \[ N,N] ( + α 2 ) 2 < C dα ( + α 2 ) 2 ε 3. α N Then α N f(α) 2 dα < ε 3. 4
Take so arge that Then S f 2 < ε. Σ α Z, α N f(α) 2 α N f(a) 2 dx < ε 3. Lemma 2 is proved in the same way. Concusion. We have proved Theorems and 2: the Panchere equaity and Inversion Formua for smooth functions f with the compact support. The goa: prove the same for f L 2. 6 Operators and their extensions Definition A map A : H H is a inear operator, provided that A(αξ +βη) = αa(ξ)+ βa(η). Definition 2 A map A is an isometry provided that Aξ = ξ ξ H; by defaut, A is inear. Theorem 4 Let E be a dense subset of H, A : E E H be an isometry. Then A may be extended to H up to an isometry A : H H. Proof Let x H, (x r ) E, x n x for n. Lety n = Ax n. Then (x n ) is a Cauchy sequence (y n ) is a Cauchy sequence as we. Let y = im n yn. Set A(x) = y. Exercise The map A is we defined: y depends on x ony, and not on x n x. The map A is an isometry because y n = x n y = x. Therefore, the Fourier transform may be extended to the whoe of L 2 (R) as an isometry; that is, on the whoe os this space the Panchere equaity hods. The Inversion Formua is extended to the space L 2 (R) in the same way. Namey, et, S : f(x) f( x) be an operator of the reversion of the indeterminate. The Inversion Formua is equivaent to the foowing one: F 2 = S. The operators in both sides of this equation are the isometries. Their coincidence on a dense set C 2,0 impies the same on the whoe space L 2 (R). 5
7 Case of many variabes In case of many variabes the formua for the Fourier transform has the same form () as for one variabe, with the use of the mutiindex notations. The formua for the normaized Fourier transform is: ˆf = f (2) () n The Panchere equaity for f L 2 (R n ) is quite the same as for n =. The inversion formua obtains a different factor f(x) = f(α)e iαx dα. () n A this is proved in the very same way as for n =. We wi use these facts, but wi not repeat the proofs. R n 6