Fourier Transform & Sobolev Spaces

Size: px
Start display at page:

Download "Fourier Transform & Sobolev Spaces"

Transcription

1 Fourier Transform & Sobolev Spaces Michael Reiter, Arthur Schuster Summer Term 2008 Abstract We introduce the concept of weak derivative that allows us to define new interesting Hilbert spaces the Sobolev spaces. Furthermore we discuss the Fourier transform and its relevance for Sobolev spaces. We follow largely the presentation of [Werner, Funktionalanalysis (Springer 2005); V..9 V.2.4]. Test Functions & Weak Derivatives Motivation. (test functions and weak derivatives). In this paragraph we want to extend the concept of derivative to introduce new Hilbert spaces of weakly differentiable functions. Remark.2 (Notation). We are going to use the following notational conventions: (i) Let x R n, we write x for the euclidean norm of x, x := n k= x2 k. (ii) If x, y R n, we denote the standard scalar product simply by xy := n k= x ky k. (iii) We employ the following notation for polynomials: we simply write x f for the function x f (x)x if f : R n R. Definition.3 (test functions). Let Ω R n be open. Then we define the set of test functions (or C - functions with compact support) on Ω as { D(Ω) := ϕ C (Ω, C) : supp(ϕ) := { x : ϕ(x) 0 } } Ω is compact We call supp(ϕ) the support of ϕ. Example.4 (of a test function). The function ϕ(x) = { c exp(( x 2 ) ) for x <, 0 for x, is a test function on R n (c.f. [A, 0.2]). So there actually do exist such functions. Lemma.5 (D(Ω) = L p (Ω)). The space of test functions D(Ω) is dense in L p (Ω) for p <. Proof (sketch). Given f L p we have to find f ε D such that f ε f L p. This can be done using convolution with so called mollifiers: f ε (x) := f (x y)ϕ ε (y)dy where the mollifier is defined by ϕ ε (x) := ε ϕ( x ε ) using ϕ from example.4. Page

2 Then we have f ε C and f ε f in L p by [Evans, Partial Differential Equations; Appendix C, Thm. 6]. The compact support can be achieved by an appropriate cut--off. Motivation.6 (weak derivatives). Many functions that arise in applications are not differentiable in the classical sense but they are still integrable. This allows us to use the following trick called the concept of weak derivatives. Let I R be an open interval, and to begin with let f C (I) and ϕ D(I), it then follows that f (x)ϕ(x)dx = f (x)ϕ (x)dx I I using integration by parts (the boundary values vanish, since the support of ϕ is a compact subset of I). This formula can be rewritten using the scalar product of L 2 (I) as f ϕ = f ϕ. We call a function g that satisfies g ϕ = f ϕ a weak or generalized derivative of f. The n-dimensional analogon can be derived similarly. Let Ω R n open, f C (Ω) and ϕ D(Ω), then f ϕ = f ϕ x i x i now by the theorem of Gauss and the compact support of ϕ. Similarly we obtain the following for m-times continuously differentiable functions f : D α f ϕ = ( ) α f D α ϕ ϕ D(Ω), () where α is a multi--index (i.e. α N n ), and α m (for an introduction to the multi--index notation 0 see [A2, 9.2]). We now drop the assumption f C m and turn equation () into a definition. Definition.7 (weak derivative). Let Ω R n be open, α a multi--index and f L 2 (Ω). Then g L 2 (Ω) is called weak or generalized α-th derivative of f, if g ϕ = ( ) α f D α ϕ ϕ D(Ω). Remark.8 (on weak derivatives). By the above discussion if f C m then the weak derivatives D α f ( α m) coincide with the classical (usual) derivatives. Moreover, such a g is unique: suppose h is another weak derivative of f, then g h ϕ = 0 for all ϕ D(Ω) and by lemma.5 and [ FA, Prop. 2.34: f ϕ = 0 ϕ in a dense set f = 0 ] it follows that g = h. We denote the weak derivative of f by D (α) f, so we can write D (α) f ϕ = ( ) α f D α ϕ ϕ D(Ω). Now we may define spaces of weakly differentiable functions. Definition.9 (Sobolev spaces). Let Ω R n be open. Then we define (i) W m (Ω) := { f L 2 (Ω) : D (α) f L 2 (Ω) exists α m } (ii) f g := W m α m D (α) f D (α) g L 2 Page 2

3 (iii) H m (Ω) := C m (Ω) W m (Ω), where the closure is taken with respect to the norm induced by.. W m (iv) H m 0 (Ω) := D(Ω), again with respect to the induced norm on Wm. These spaces are all called Sobolev spaces. Remark.0 (on Sobolev spaces). (i) W 0 = L 2 by definition and H 0 0 = L2 by lemma.5. (ii) Obviously W m, H m and H m 0 are vector spaces, and.. is a scalar product. Wm (iii) (no proofs) If Ω is a bounded region with sufficiently smooth boundary, then W m (Ω) = H m (Ω) ( i.e. W m (Ω) = C m (Ω) W m (Ω) ). On the other hand, it always holds that W m (Ω) = C m (Ω) W m (Ω). Proposition. (Sobolev spaces are (H)-spaces). W m, H m and H m 0 are Hilbert spaces for all m 0. Proof. Since H m and H m 0 are closed subspaces of Wm is suffices to show the completeness of W m (Ω). Let ( f n ) n be a Cauchy sequence with respect to. W m. Since f 2 W = D (α) f 2 m 2 f W m, α m the sequences (D (α) f n ) n are all. 2 -Cauchy sequences. Therefore there exist f α L 2 (Ω) such that D (α) f n f α 2 0 (n, α m). Set f 0 := f (0,...,0), we ll show that D (α) f 0 = f α, since this implies f 0 W m and f n f 0 with respect to. W m (by definition of that norm). Indeed, for ϕ D(Ω), we have (using the continuity of the L 2 - scalar product in both slots) fα ϕ = lim n D(α) f n ϕ = lim D (α) f n ϕ n.7 = lim ( ) α f n D α ϕ = ( ) α lim n = ( ) α f 0 D α.7 ϕ = D (α) f 0 ϕ And so we obtain f α = D (α) f 0 by lemma.5. n f n D α ϕ Motivation.2 (Sobolev spaces and PDEs). Clearly the Sobolev spaces are nested, i.e., W m (Ω) W m (Ω), and the identity map id : W m (Ω) W m (Ω) is continuous [since the norm on W m can be estimated by. W m]. In applications the following two results are of great importance: Sobolev embedding theorem: almost everywhere. For f W m (Ω) and m > k + n 2 there exists g Ck (Ω) with f = g Rellich-Kondrachov theorem: H m (Ω) is compact. 0 If Ω R n is a bounded open set, then the embedding H m 0 (Ω) These theorems are important tools for proving the existence of solutions of many (especially elliptical) partial differential equations. Page 3

4 We are going to prove these theorems in section 3. As a technical tool we introduce the Fourier transform, which is also an interesting topic in itself. 2 Fourier Transform Motivation 2. (decay vs. smoothness). If f L 2 (R n ) this means that f has a certain fall--off property at. In the Sobolev space W m we even ask for such a fall--off property for the (weak) derivatives of f. The Fourier transform allows us to translate derivatives into multiplication with polynomials (see lemma 2.8 below). So we may relate the L 2 -property of derivatives of f into stronger fall--off conditions on f itself. In this way the Sobolev spaces allow us to measures smoothness of functions in terms of their fall--off on the Fourier transform side. Finally the Sobolev embedding theorem links this fall--off back to classical C k -properties. Definition 2.2 (Fourier transform). Let f L (R n ), then we define the Fourier transform F f of f by (F f )(ξ) := f ˆ (ξ) := f (x)e ixξ dx ξ R n. (2) R n [recall the notation xξ for the scalar product of x with ξ] The operator F is called the Fourier transform. Proposition 2.3 (F L(L, C 0 )). If f L (R n ), then F f C 0 (R n ). The Fourier transform F : L (R n ) C 0 (R n ) is a continuous and linear operator with F (2π) n 2. Proof. First we have (F f )(ξ) f (x) e ixξ dx = f R n }{{} (2π) n, 2 = i.e. F f (2π) n 2 f, which implies F (2π) n 2. Therefore F is a continuous operator from L (R n ) L (R n ). To show that F f C 0 (R n ) (i.e. F f is continuous and vanishes at infinity), let (ξ (k) ) be a sequence in R n with ξ (k) ξ. Using the continuity of exp(x) and the scalar product we obtain e ixξ(k) e ixξ 0 x R n. Plugging this into (2) and using dominated convergence we see that (F f )(ξ (k) ) (F f )(ξ) f (x) e ixξ(k) e ixξ dx 0, R n }{{} since the integrand is dominated by the L -function 2 f. This shows that F f is continuous. To obtain im F C 0 (R n ) it suffices to show 2 lim (F ϕ)(ξ) = 0 ϕ ξ D(Rn ) since D(R n ) is dense in L (R n ) by lemma.5. So let ϕ D(R n ), R > 0 and ξ R. Then there exists a coordinate ξ j with ξ j R n [ R ξ = n k= ξ k 2 n max k n ξ k ]. Using integration by parts we finally obtain Page 4

5 (F ϕ)(ξ) = ϕ(x)e ixξ dx (2π) n = ϕ(x) e ixξ dx 2 R n R n x j iξ j (2π) n ϕ(x) 2 R n x j iξ j e ixξ dx n (2π) n ϕ(x) 2 x j }{{}}{{} R 0 n R = (R ), which completes the proof. Remark 2.4 (on proposition 2.3). This result means, that the Fourier transform of an L -function has some decay behaviour, but not enough to secure the L -property. We now hope to find an explicit inverse operator of the Fourier transform starting with a certain subset of L (R n ). It turns out that the rapidly decreasing Schwartz functions do the trick: Definition 2.5 (Schwartz space). Let f : R n C; f is called rapidly decreasing, if [recall x α := x α xα n n ]. The space lim x xα f (x) = 0 α N n 0 (3) S(R n ) := { f C (R n ) : D β f is rapidly decreasing β N n 0 is called Schwartz space, its elements Schwartz functions. Remark 2.6 (on Schwartz functions). (i) Examples for Schwartz functions are γ(x) = e x2 and every testfunction. (ii) The following conditions are equivalent to (3): lim P(x) f (x) = 0 polynomials P : x Rn C (4) lim x x m f (x) = 0 m N 0 (5) (Obviously (3) implies the first condition, which in turn implies (5), because one only has to consider the polynomials P(x) = (x x2 n) m 2 with even m. Finally this again implies (3), because x α x α.) (iii) So by definition, Schwartz functions and all of their derivatives decay faster than the inverse of any polynomial. Using this we obtain S(R n ) L p (R n ) for all p, since for mp (n ) > and f S(R n ) we have: ( ) f (x) p c p dx R n R n + x m dx ( = c p ω n 0 + r m ) p r n dr <, where ω n is the area of the n-sphere { x R n : x = } and c is a properly chosen constant (see [Otto Forster, Analysis 3; 4, Satz 8, Beispiel (4.0)]). (iv) Because D(R n ) is obviously a subspace of S(R n ), the space of Schwartz functions is dense in L p (R n ) for p < as well (lemma.5). (v) A smooth function f C is a Schwartz function if and only if } Page 5

6 sup( + x m ) D β f (x) < m N 0, β N n 0. (6) x R n Clearly every Schwartz function has this property. For the other direction consider x α D β f x m D β f (x) = ( x m + x m+ ) D β f (x) + x (( + x m ) + ( + x m+ )) D α f + x (6) c + x 0. The motivation for the use of the factor ( + x m ) is that it is always possible to divide by it which is not the case for x m. Motivation 2.7 (Schwartz functions and Fourier transform). The importance of the Schwartz space lies in the fact, that the Fourier transform F is a bijection of S(R n ) (we ll prove this in proposition 2.4), which was not the case with L (R n ). Recall the notation x α for the map x x α, then we have for any f S(R n ): x α f S(R n ), D α f S(R n ) α N n 0. (7) This allows us to state the interesting and important interplay between differentiation and multiplication by polynomials using the Fourier transform: Lemma 2.8 (exchange formulas). Let f S(R n ) and α be a multi--index. Then (i) F f C (R n ) and D α (F f ) = ( i) α F(x α f ), (ii) F(D α f ) = i α ξ α F f. Proof. (i) Formally we have: D α (F f )(ξ) = α ξ α = ( i) α f (x)e ixξ dx ( ) = R n Rn f (x) α ξ α e ixξ dx R n f (x)x α e ixξ dx = ( i) α F(x α f )(ξ). We are allowed to pull the differential into the integral in step ( ) by dominated convergence and because x α f S(R n ) L (R n ) (c.f. [Werner, Cor. A.3.3]). (ii) is also a simple calculation using integration by parts (note that since f S(R n ) all boundary terms vanish): F(D α f )(ξ) = Rn (D α f )(x)e ixξ dx = ( ) α = ( ) α ( i) α ξ α (F f )(ξ) = i α ξ α (F f )(ξ). Rn f (x) α x α e ixξ dx Remark 2.9 (on lemma 2.8). This lemma now shows why the Fourier transform is so interesting for PDEs: differentials are turned into multiplications, i.e. analytic operations are turned into algebraic operations and the other way round. Lemma 2.0 (F : S(R n ) S(R n )). If f S(R n ), then also F f S(R n ). Proof. In lemma 2.8(i) we showed that F f C (R n ) holds. So we just have to prove ξ α D β (F f )(ξ) 0 for ξ. This also follows by lemma 2.8: Page 6 ξ α D β (F f )(ξ) = ( i) β ξ α F(x β f )(ξ) = ( i) β ( i) α F(D α x β f )(ξ),

7 since D α (x β f ) S(R n ) L (R n ), proposition 2.3 gives the result. Motivation 2. (e x2 2 ). Next we want to find out what happens to a function f S(R n ) if we apply the Fourier transform twice which is now allowed by lemma 2.0. To do this we first need to calculate the Fourier transform of γ(x) = e x2 2 (x R n ), We also use γ a (x) = γ(ax) for a > 0. Recall [A3-PS, 82]: γ(x)dx =. R n Lemma 2.2 (F of e x2 2 ). We have (F γ)(ξ) = e ξ2 2 = γ(ξ), (F γ a )(ξ) = a n γ ( ξ a ). Proof. The second result is clear once we have proved the first one (just substitute ξ for aξ). First we are going to calculate F γ for n =. In this case γ solves the ordinary differential equation Thus, by lemma 2.8, we have y + xy = 0, y(0) =. ( ) 0 = F(γ + xγ) = iξ F γ + i F γ, which means, that F γ solves the same differential equation; it even has the same initial value: (F γ)(0) = 2π e x2 2 dx =. Because such initial value problems have unique solutions, it follows that γ = F γ. We show the case n > using the result for n = : (F γ)(ξ) = = n ( k= 2π ξ 2 ξ 2 n = e 2 e 2 = e ξ2 2. n n e x2/2 k k= k= e x2 k 2 e ix k ξ k dx k ) e ix kξ k dx dx n Lemma 2.3 (a step closer to F ). Let f S(R n ), then (F F f )(x) = f ( x) x R n, i.e., F 2 is the reflection. Proof. Because we have F f S(R n ) by lemma 2.0, the lefthand side of the equation is defined. If we tried to use Fubini s theorem directly to calculate the double-integral (F F f )(x), we would Page 7

8 obtain a non-convergent integral. Therefore we have to be more careful and use the convergenceimproving property of the functions γ a. A first useful observation is the following implication of Fubini s theorem: (F f )(x)g(x)dx = f (ξ)g(x)e iξx dξdx R n R n R n = f (x)g(ξ)e iξx dξdx = f (x)(f g)(x)dx. R n R n R n f, g S(R n ) (8) Here the change in the order of integration is allowed, because (x, ξ) f (ξ)g(x)e iξx is integrable (e iξx is damped in both arguments). Now set g(x) := e ixξ 0γ(ax) with ξ 0 R n and a > 0. Then we have: (F g)(ξ) = R n e ixξ 0 γ(ax)e ixξ dx = (F γ a )(ξ + ξ 0 ). (9) Now comes the actual proof: (F f )(x) e ixξ 0 γ(ax) dx R n }{{} =g(x) (8) = (9) === 2.2 u= x+ξ 0 a ==== 2.2 R n f (x)(f g)(x)dx f (x) R n an (F γ) ( x + ξ0 a R n f (au ξ 0 )γ(u)du. ) dx If we now let a 0 then we see, using dominated convergence, that the first expression tends to (F F f )(ξ 0 ) while the last expression tends to f ( ξ 0 ). Since the first integral is dominated by F f and the last one by f γ, which are both integrable. Proposition 2.4 (F is bijective, Plancherel formula). The Fourier transform is a bijection of S(R n ); the inverse operator is given by (F f )(x) = f (ξ)e ixξ dξ. R n Furthermore we have the so--called Plancherel equation F f F g L 2 = f g L 2 f, g S(R n ). Proof. From the previous lemma 2.3 we immediately obtain F 4 = id S(R n ), which implies that the Fourier transform is bijective and F = F 3. Therefore we have (F f )(x) = (F 2 (F f ))(x) 2.3 === (F f )( x), which implies the first claim. We prove the Plancherel equation using (8): (F f )(ξ)(f g)(ξ)dξ = f (x)(f (F g))(x)dx, R n R n if we write h := F g, we obtain Page 8

9 (F (F g))(x) = (F h)(x) = R n h(ξ)e ixξ dξ = R n h(ξ)e ixξ dξ = (F h)(x) = g(x), which immediately implies F f F g L 2 = R n f (x)g(x)dx = f g L 2. Remark 2.5 (F on L 2 ). (i) The Plancherel equation implies F f 2 = f 2 f S(R n ). (ii) Which means that the Fourier transform F is bijective and isometric with respect to. 2 on the subspace S(R n ) of L 2 (R n ). Since by remark 2.6(iv) S(R n ) is dense in L 2 (R n ), we can extend F to an isometric operator on L 2 (R n ), which is called the Fourier(-Plancherel) transform. Proposition 2.4 implies that F : L 2 (R n ) L 2 (R n ) is also an isometric isomorphism, and that we have: F f F g = f g f, g L 2 (R n ). (0) L 2 L 2 (iii) It is important to understand, that F f for f L 2 (R n ) cannot generally be calculated using the formula for f L (R n ): the integral in (2) need not exist. Lemma 2.6 (exchange formula for W m ). Let f W m (R n ). Then: F(D (α) f ) = i α ξ α F f α m. Proof. For ϕ S(R n ) we have: F(D (α) f ) F ϕ (0) === D (α) f ϕ ( ) = ( ) α f D α ϕ (0) === ( ) α F f F D α ϕ 2.8 = ( ) α F f i α ξ α F ϕ = i α R n ξ α (F f )(ξ)(f ϕ)(ξ)dξ. In step ( ) we used the fact that the integration by parts-trick also works for ϕ S(R n ): More precisely let ϕ S(R n ), (ϕ n ) n D(R n ) and ϕ n ϕ in S(R n ). Then a simple calculation shows D (α) f ϕ = D (α) f lim ϕ n = lim D (α) f ϕ n n n.7 = lim ( ) α f D α ϕ n = ( ) α f lim D α ϕ n n n = ( ) α f D α ϕ. Since the Fourier transform is a bijection of the Schwartz space, we have for h(ξ) := F(D (α) f )(ξ) i α ξ α (F f )(ξ) in particular that Page 9

10 R n h(ξ)ψ(ξ)dξ = 0 ψ D(R n ). We still don t know whether h L 2 (R n ). What we know is, that every restriction of h to an open ball B R := { x R n : x < R } is in L 2 (B R ). Thus the previous equation tells us h ψ L 2 (B R ) = 0 ψ D(B R). And since D(B R ) is dense in L 2 (B R ), we know that h = 0 almost everywhere in B R for all R > 0, so that we have h = 0 almost everywhere in R n the result we were after. Remark 2.7 (the other exchange formula). In lemma 2.6 we only showed the analogon for weak derivatives of one of the exchange formulas of lemma 2.8. Since we only know that F f L 2 (R n ) for f W m (R n ), the weak derivative D (α) (F f ) need not exist. Now we have all the tools we need to prove the promised big theorems: 3 Sobolev embedding theorem & Rellich-Kondrachov theorem Theorem 3. (Sobolev embedding theorem). Let Ω R n be open, and m, k N 0 with m > k + n 2. If f W m (Ω), then there exists a k-times continuously differentiable function on Ω that is equal to f almost everywhere. In other words, the class of equivalent functions f W m (Ω) has a representative in C k (Ω). Remark 3.2 (on theorem 3.). Theorem 3. essentially means that a function that is sufficiently often weakly differentiable is also classically differentiable. Seen differently, decay conditions on a function and its weak derivatives imply smoothness of the function (c.f. also motivation 2.). Proof (of theorem 3.). We will only prove this for Ω = R n, for arbitrary open subsets Ω R n see [Werner; V.2.2]. The idea is to use the proof of lemma 2.8 where we showed that (read step (i) backwards) x α g L (R n ) α k F g C k (R n ). If we use this result for g := F f, we only need to show that f W m (R n ), m > k + n 2 and α k ξα F f L (R n ), () since then we have F F f C k (R n ), which is already nearly the claimed result. To show (), notice that lemma 2.6 already implies ξ α F f L 2 (R n ) for α m. In particular we have ξ 2m j F f (ξ) 2 dξ < j =,..., n and F f (ξ) 2 dξ <. R n R n Using + ξ 2 = (, ξ 2,..., ξ2 n) (,..., ) p (, ξ 2,..., ξ2 n) m = (n + ) p ( + ξ 2m + + ξ 2m n ) m, (Hölder inequality with p + m = ) we also obtain Page 0

11 R n ( + ξ 2 ) m F f (ξ) 2 dξ <. Plugging this inequality and ξ α ξ α ( + ξ 2 ) α 2 together, it follows that for all α k ξ α F f (ξ) dξ ( + ξ 2 ) α 2 F f (ξ) dξ ( + ξ 2 ) m 2 F f (ξ) ( + ξ 2 ) m k 2 dξ R n R n R n ( ) ( Rn ) ( + ξ 2 ) m F f (ξ) dξ dξ R n ( + ξ 2 ) m k <. To explain the last step in some more detail we denote by ω n the area of the unit sphere in R n (see [Otto Forster, Analysis 3; 4, Satz 8, Beispiel (4.0)]), then we have R n ( + ξ 2 ) dξ = ω m k n 0 ( + r 2 ) m k rn dr <, as long as 2(m k) (n ) >, i.e., m > k+ n 2. Therefore we really have ξα F f L (R n ) which implies, as already mentioned, F F f C k (R n ). Using (σg)(x) := g( x) we now have by lemma 2.3 that the equation σg = F F g holds for all g S(R n ). Since σ and F are continuous and S(R n ) is dense in L 2 (R n ), the equation must also hold for all g L 2 (R n ). This shows that f has to be equal to σ(f F f ) C k (R n ) almost everywhere, which is the claimed result for Ω = R n. Theorem 3.3 (Rellich-Kondrachov theorem). map from H m (Ω) to Hm (Ω) is compact. 0 0 Let Ω R n be bounded and open, then the identity Proof. First we prove the statement for m =. To see that the identity map from H 0 (Ω) to H0 0 (Ω) = L 2 (Ω) (see remark.0) is compact, we need to show that all bounded sequences ( f j ) j in H (Ω) have 0 a subsequence that converges with respect to. 2. Since the space of tesfunctions D(Ω) is dense in H 0 (Ω) (by definition), we can assume that f j D(Ω). By setting f j = 0 outside its compact support we may extend f j to a compactly supported function on all of R n, hence view f j as an element of D(R n ). Since a bounded sequence in D(Ω) is also bounded in L 2 (R n ), we can use [FA2, Theorem 5.26 resp. Corollary 5.28] to obtain the existence of a weakly converging subsequence of ( f j ) j in L 2 (Ω). We again write ( f j ) j for this sequence. The next step is to show its Cauchy property then we re done since L 2 (Ω) is of course complete. Using the Plancherel equality (0) we obtain f k f j 2 2 = F f k F f j 2 2 = R n F f k (ξ) F f j (ξ) 2 dξ = F f k (ξ) F f j (ξ) 2 dξ + ξ R F f k (ξ) F f j (ξ) 2 dξ. (2) ξ >R First we ll show that the second integral can be made small by choosing an appropriate R > 0. Using the exchange formula of lemma 2.6 we have for l =,..., n and f W (R n ) that 2.6 ( ) 2 ξ l F f 2 === F (0) f === x l f x l f W (R n ) (3) 2 (here the derivatives are weak derivatives). This means that the sequence ( ξ F f j ) j is bounded in L 2 (R n ), so for all ε > 0 we may find R > 0 such that Page

12 F f k (ξ) F f j (ξ) 2 dξ ξ >R R 2 ξ 2 F f k (ξ) F f j (ξ) 2 dξ ξ >R }{{} < c, by (3) < ε j, k N. Now we have to deal with the first integral in (2). Set e ξ (x) := e ixξ for x Ω. Since Ω is bounded, we have e ξ L 2 (Ω). Thus f f e ξ = f (x)e ixξ dx = F( f χ Ω )(ξ) Ω defines a continuous functional on L 2 (Ω). Since ( f j ) j converges weakly in L 2 (Ω) we obtain that ((F f j )(ξ)) j is convergent for all ξ R n. Using this we want to conclude that F f k (ξ) F f j (ξ) 2 dξ 0 (j, k ). ξ R This follows using dominated convergence: the integrand is dominated by sup j F f j ( by [FA, Satz.34(iii)] the sequence ( f j ) j is also bounded in L (Ω), which allows us to use proposition 2.3 to obtain that (F f j ) j is bounded with respect to. ). This proves the statement for m =. Now let m > and ( f j ) j a bounded sequence in H m 0 (Ω), then all of the sequences (D(α) f j ) j for 0 α m are bounded in H 0 (Ω). Therefore, by the above, they contain L 2-Cauchy--subsequences. Applying this argument to all multi--indices we obtain a subsequence of ( f j ) j that converges in H m 0 (Ω) since H m 0 (Ω) is complete (by definition). Remark 3.4 (description of W m using F). Lemma 2.6 implies the following characterization: W m (R n ) = { f L 2 (R n ) : ( + ξ 2 ) m 2 F f L 2 (R n ) }. [ f W m D (α) f L 2 α m F(D (α) f ) L 2 α m ξ α F f L 2 α m ( + ξ 2 ) α 2 F f L 2 α m ] This characterization would allow us to extend the definition of the Sobolev spaces W m to positive real exponents m R + [if we also want to allow negative m we have to use tempered distributions]. Page 2

Sobolev spaces. May 18

Sobolev spaces. May 18 Sobolev spaces May 18 2015 1 Weak derivatives The purpose of these notes is to give a very basic introduction to Sobolev spaces. More extensive treatments can e.g. be found in the classical references

More information

Introduction to Microlocal Analysis

Introduction to Microlocal Analysis Introduction to Microlocal Analysis First lecture: Basics Dorothea Bahns (Göttingen) Third Summer School on Dynamical Approaches in Spectral Geometry Microlocal Methods in Global Analysis University of

More information

We denote the space of distributions on Ω by D ( Ω) 2.

We denote the space of distributions on Ω by D ( Ω) 2. Sep. 1 0, 008 Distributions Distributions are generalized functions. Some familiarity with the theory of distributions helps understanding of various function spaces which play important roles in the study

More information

Finite-dimensional spaces. C n is the space of n-tuples x = (x 1,..., x n ) of complex numbers. It is a Hilbert space with the inner product

Finite-dimensional spaces. C n is the space of n-tuples x = (x 1,..., x n ) of complex numbers. It is a Hilbert space with the inner product Chapter 4 Hilbert Spaces 4.1 Inner Product Spaces Inner Product Space. A complex vector space E is called an inner product space (or a pre-hilbert space, or a unitary space) if there is a mapping (, )

More information

Chapter One. The Calderón-Zygmund Theory I: Ellipticity

Chapter One. The Calderón-Zygmund Theory I: Ellipticity Chapter One The Calderón-Zygmund Theory I: Ellipticity Our story begins with a classical situation: convolution with homogeneous, Calderón- Zygmund ( kernels on R n. Let S n 1 R n denote the unit sphere

More information

1 Fourier Integrals on L 2 (R) and L 1 (R).

1 Fourier Integrals on L 2 (R) and L 1 (R). 18.103 Fall 2013 1 Fourier Integrals on L 2 () and L 1 (). The first part of these notes cover 3.5 of AG, without proofs. When we get to things not covered in the book, we will start giving proofs. The

More information

Math 172 Problem Set 8 Solutions

Math 172 Problem Set 8 Solutions Math 72 Problem Set 8 Solutions Problem. (i We have (Fχ [ a,a] (ξ = χ [ a,a] e ixξ dx = a a e ixξ dx = iξ (e iax e iax = 2 sin aξ. ξ (ii We have (Fχ [, e ax (ξ = e ax e ixξ dx = e x(a+iξ dx = a + iξ where

More information

Traces, extensions and co-normal derivatives for elliptic systems on Lipschitz domains

Traces, extensions and co-normal derivatives for elliptic systems on Lipschitz domains Traces, extensions and co-normal derivatives for elliptic systems on Lipschitz domains Sergey E. Mikhailov Brunel University West London, Department of Mathematics, Uxbridge, UB8 3PH, UK J. Math. Analysis

More information

Reminder Notes for the Course on Distribution Theory

Reminder Notes for the Course on Distribution Theory Reminder Notes for the Course on Distribution Theory T. C. Dorlas Dublin Institute for Advanced Studies School of Theoretical Physics 10 Burlington Road, Dublin 4, Ireland. Email: dorlas@stp.dias.ie March

More information

Overview of normed linear spaces

Overview of normed linear spaces 20 Chapter 2 Overview of normed linear spaces Starting from this chapter, we begin examining linear spaces with at least one extra structure (topology or geometry). We assume linearity; this is a natural

More information

Topics in Harmonic Analysis Lecture 1: The Fourier transform

Topics in Harmonic Analysis Lecture 1: The Fourier transform Topics in Harmonic Analysis Lecture 1: The Fourier transform Po-Lam Yung The Chinese University of Hong Kong Outline Fourier series on T: L 2 theory Convolutions The Dirichlet and Fejer kernels Pointwise

More information

1.3.1 Definition and Basic Properties of Convolution

1.3.1 Definition and Basic Properties of Convolution 1.3 Convolution 15 1.3 Convolution Since L 1 (R) is a Banach space, we know that it has many useful properties. In particular the operations of addition and scalar multiplication are continuous. However,

More information

1 Fourier Integrals of finite measures.

1 Fourier Integrals of finite measures. 18.103 Fall 2013 1 Fourier Integrals of finite measures. Denote the space of finite, positive, measures on by M + () = {µ : µ is a positive measure on ; µ() < } Proposition 1 For µ M + (), we define the

More information

2. Function spaces and approximation

2. Function spaces and approximation 2.1 2. Function spaces and approximation 2.1. The space of test functions. Notation and prerequisites are collected in Appendix A. Let Ω be an open subset of R n. The space C0 (Ω), consisting of the C

More information

1.5 Approximate Identities

1.5 Approximate Identities 38 1 The Fourier Transform on L 1 (R) which are dense subspaces of L p (R). On these domains, P : D P L p (R) and M : D M L p (R). Show, however, that P and M are unbounded even when restricted to these

More information

Tools from Lebesgue integration

Tools from Lebesgue integration Tools from Lebesgue integration E.P. van den Ban Fall 2005 Introduction In these notes we describe some of the basic tools from the theory of Lebesgue integration. Definitions and results will be given

More information

Singular Integrals. 1 Calderon-Zygmund decomposition

Singular Integrals. 1 Calderon-Zygmund decomposition Singular Integrals Analysis III Calderon-Zygmund decomposition Let f be an integrable function f dx 0, f = g + b with g Cα almost everywhere, with b

More information

Topics in Harmonic Analysis Lecture 6: Pseudodifferential calculus and almost orthogonality

Topics in Harmonic Analysis Lecture 6: Pseudodifferential calculus and almost orthogonality Topics in Harmonic Analysis Lecture 6: Pseudodifferential calculus and almost orthogonality Po-Lam Yung The Chinese University of Hong Kong Introduction While multiplier operators are very useful in studying

More information

The Dirichlet s P rinciple. In this lecture we discuss an alternative formulation of the Dirichlet problem for the Laplace equation:

The Dirichlet s P rinciple. In this lecture we discuss an alternative formulation of the Dirichlet problem for the Laplace equation: Oct. 1 The Dirichlet s P rinciple In this lecture we discuss an alternative formulation of the Dirichlet problem for the Laplace equation: 1. Dirichlet s Principle. u = in, u = g on. ( 1 ) If we multiply

More information

S chauder Theory. x 2. = log( x 1 + x 2 ) + 1 ( x 1 + x 2 ) 2. ( 5) x 1 + x 2 x 1 + x 2. 2 = 2 x 1. x 1 x 2. 1 x 1.

S chauder Theory. x 2. = log( x 1 + x 2 ) + 1 ( x 1 + x 2 ) 2. ( 5) x 1 + x 2 x 1 + x 2. 2 = 2 x 1. x 1 x 2. 1 x 1. Sep. 1 9 Intuitively, the solution u to the Poisson equation S chauder Theory u = f 1 should have better regularity than the right hand side f. In particular one expects u to be twice more differentiable

More information

TD 1: Hilbert Spaces and Applications

TD 1: Hilbert Spaces and Applications Université Paris-Dauphine Functional Analysis and PDEs Master MMD-MA 2017/2018 Generalities TD 1: Hilbert Spaces and Applications Exercise 1 (Generalized Parallelogram law). Let (H,, ) be a Hilbert space.

More information

Fourier Series. ,..., e ixn ). Conversely, each 2π-periodic function φ : R n C induces a unique φ : T n C for which φ(e ix 1

Fourier Series. ,..., e ixn ). Conversely, each 2π-periodic function φ : R n C induces a unique φ : T n C for which φ(e ix 1 Fourier Series Let {e j : 1 j n} be the standard basis in R n. We say f : R n C is π-periodic in each variable if f(x + πe j ) = f(x) x R n, 1 j n. We can identify π-periodic functions with functions on

More information

SOLUTIONS TO HOMEWORK ASSIGNMENT 4

SOLUTIONS TO HOMEWORK ASSIGNMENT 4 SOLUTIONS TO HOMEWOK ASSIGNMENT 4 Exercise. A criterion for the image under the Hilbert transform to belong to L Let φ S be given. Show that Hφ L if and only if φx dx = 0. Solution: Suppose first that

More information

Lectures on. Sobolev Spaces. S. Kesavan The Institute of Mathematical Sciences, Chennai.

Lectures on. Sobolev Spaces. S. Kesavan The Institute of Mathematical Sciences, Chennai. Lectures on Sobolev Spaces S. Kesavan The Institute of Mathematical Sciences, Chennai. e-mail: kesh@imsc.res.in 2 1 Distributions In this section we will, very briefly, recall concepts from the theory

More information

THEOREMS, ETC., FOR MATH 515

THEOREMS, ETC., FOR MATH 515 THEOREMS, ETC., FOR MATH 515 Proposition 1 (=comment on page 17). If A is an algebra, then any finite union or finite intersection of sets in A is also in A. Proposition 2 (=Proposition 1.1). For every

More information

NAME: MATH 172: Lebesgue Integration and Fourier Analysis (winter 2012) Final exam. Wednesday, March 21, time: 2.5h

NAME: MATH 172: Lebesgue Integration and Fourier Analysis (winter 2012) Final exam. Wednesday, March 21, time: 2.5h NAME: SOLUTION problem # 1 2 3 4 5 6 7 8 9 points max 15 20 10 15 10 10 10 10 10 110 MATH 172: Lebesgue Integration and Fourier Analysis (winter 2012 Final exam Wednesday, March 21, 2012 time: 2.5h Please

More information

Partial Differential Equations

Partial Differential Equations Part II Partial Differential Equations Year 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2015 Paper 4, Section II 29E Partial Differential Equations 72 (a) Show that the Cauchy problem for u(x,

More information

FRAMES AND TIME-FREQUENCY ANALYSIS

FRAMES AND TIME-FREQUENCY ANALYSIS FRAMES AND TIME-FREQUENCY ANALYSIS LECTURE 5: MODULATION SPACES AND APPLICATIONS Christopher Heil Georgia Tech heil@math.gatech.edu http://www.math.gatech.edu/ heil READING For background on Banach spaces,

More information

RANDOM PROPERTIES BENOIT PAUSADER

RANDOM PROPERTIES BENOIT PAUSADER RANDOM PROPERTIES BENOIT PAUSADER. Quasilinear problems In general, one consider the following trichotomy for nonlinear PDEs: A semilinear problem is a problem where the highest-order terms appears linearly

More information

Math 209B Homework 2

Math 209B Homework 2 Math 29B Homework 2 Edward Burkard Note: All vector spaces are over the field F = R or C 4.6. Two Compactness Theorems. 4. Point Set Topology Exercise 6 The product of countably many sequentally compact

More information

Riemann integral and volume are generalized to unbounded functions and sets. is an admissible set, and its volume is a Riemann integral, 1l E,

Riemann integral and volume are generalized to unbounded functions and sets. is an admissible set, and its volume is a Riemann integral, 1l E, Tel Aviv University, 26 Analysis-III 9 9 Improper integral 9a Introduction....................... 9 9b Positive integrands................... 9c Special functions gamma and beta......... 4 9d Change of

More information

x λ ϕ(x)dx x λ ϕ(x)dx = xλ+1 λ + 1 ϕ(x) u = xλ+1 λ + 1 dv = ϕ (x)dx (x))dx = ϕ(x)

x λ ϕ(x)dx x λ ϕ(x)dx = xλ+1 λ + 1 ϕ(x) u = xλ+1 λ + 1 dv = ϕ (x)dx (x))dx = ϕ(x) 1. Distributions A distribution is a linear functional L : D R (or C) where D = C c (R n ) which is linear and continuous. 1.1. Topology on D. Let K R n be a compact set. Define D K := {f D : f outside

More information

The Calderon-Vaillancourt Theorem

The Calderon-Vaillancourt Theorem The Calderon-Vaillancourt Theorem What follows is a completely self contained proof of the Calderon-Vaillancourt Theorem on the L 2 boundedness of pseudo-differential operators. 1 The result Definition

More information

Functional Analysis. Martin Brokate. 1 Normed Spaces 2. 2 Hilbert Spaces The Principle of Uniform Boundedness 32

Functional Analysis. Martin Brokate. 1 Normed Spaces 2. 2 Hilbert Spaces The Principle of Uniform Boundedness 32 Functional Analysis Martin Brokate Contents 1 Normed Spaces 2 2 Hilbert Spaces 2 3 The Principle of Uniform Boundedness 32 4 Extension, Reflexivity, Separation 37 5 Compact subsets of C and L p 46 6 Weak

More information

2) Let X be a compact space. Prove that the space C(X) of continuous real-valued functions is a complete metric space.

2) Let X be a compact space. Prove that the space C(X) of continuous real-valued functions is a complete metric space. University of Bergen General Functional Analysis Problems with solutions 6 ) Prove that is unique in any normed space. Solution of ) Let us suppose that there are 2 zeros and 2. Then = + 2 = 2 + = 2. 2)

More information

i=1 α i. Given an m-times continuously

i=1 α i. Given an m-times continuously 1 Fundamentals 1.1 Classification and characteristics Let Ω R d, d N, d 2, be an open set and α = (α 1,, α d ) T N d 0, N 0 := N {0}, a multiindex with α := d i=1 α i. Given an m-times continuously differentiable

More information

Sobolev Spaces. Chapter 10

Sobolev Spaces. Chapter 10 Chapter 1 Sobolev Spaces We now define spaces H 1,p (R n ), known as Sobolev spaces. For u to belong to H 1,p (R n ), we require that u L p (R n ) and that u have weak derivatives of first order in L p

More information

1 Continuity Classes C m (Ω)

1 Continuity Classes C m (Ω) 0.1 Norms 0.1 Norms A norm on a linear space X is a function : X R with the properties: Positive Definite: x 0 x X (nonnegative) x = 0 x = 0 (strictly positive) λx = λ x x X, λ C(homogeneous) x + y x +

More information

TOOLS FROM HARMONIC ANALYSIS

TOOLS FROM HARMONIC ANALYSIS TOOLS FROM HARMONIC ANALYSIS BRADLY STADIE Abstract. The Fourier transform can be thought of as a map that decomposes a function into oscillatory functions. In this paper, we will apply this decomposition

More information

MATH MEASURE THEORY AND FOURIER ANALYSIS. Contents

MATH MEASURE THEORY AND FOURIER ANALYSIS. Contents MATH 3969 - MEASURE THEORY AND FOURIER ANALYSIS ANDREW TULLOCH Contents 1. Measure Theory 2 1.1. Properties of Measures 3 1.2. Constructing σ-algebras and measures 3 1.3. Properties of the Lebesgue measure

More information

Lecture 5: Hodge theorem

Lecture 5: Hodge theorem Lecture 5: Hodge theorem Jonathan Evans 4th October 2010 Jonathan Evans () Lecture 5: Hodge theorem 4th October 2010 1 / 15 Jonathan Evans () Lecture 5: Hodge theorem 4th October 2010 2 / 15 The aim of

More information

Measurable functions are approximately nice, even if look terrible.

Measurable functions are approximately nice, even if look terrible. Tel Aviv University, 2015 Functions of real variables 74 7 Approximation 7a A terrible integrable function........... 74 7b Approximation of sets................ 76 7c Approximation of functions............

More information

i=1 β i,i.e. = β 1 x β x β 1 1 xβ d

i=1 β i,i.e. = β 1 x β x β 1 1 xβ d 66 2. Every family of seminorms on a vector space containing a norm induces ahausdorff locally convex topology. 3. Given an open subset Ω of R d with the euclidean topology, the space C(Ω) of real valued

More information

A review: The Laplacian and the d Alembertian. j=1

A review: The Laplacian and the d Alembertian. j=1 Chapter One A review: The Laplacian and the d Alembertian 1.1 THE LAPLACIAN One of the main goals of this course is to understand well the solution of wave equation both in Euclidean space and on manifolds

More information

Regularity for Poisson Equation

Regularity for Poisson Equation Regularity for Poisson Equation OcMountain Daylight Time. 4, 20 Intuitively, the solution u to the Poisson equation u= f () should have better regularity than the right hand side f. In particular one expects

More information

Quintic deficient spline wavelets

Quintic deficient spline wavelets Quintic deficient spline wavelets F. Bastin and P. Laubin January 19, 4 Abstract We show explicitely how to construct scaling functions and wavelets which are quintic deficient splines with compact support

More information

LECTURE 5: THE METHOD OF STATIONARY PHASE

LECTURE 5: THE METHOD OF STATIONARY PHASE LECTURE 5: THE METHOD OF STATIONARY PHASE Some notions.. A crash course on Fourier transform For j =,, n, j = x j. D j = i j. For any multi-index α = (α,, α n ) N n. α = α + + α n. α! = α! α n!. x α =

More information

Sobolev spaces, Trace theorems and Green s functions.

Sobolev spaces, Trace theorems and Green s functions. Sobolev spaces, Trace theorems and Green s functions. Boundary Element Methods for Waves Scattering Numerical Analysis Seminar. Orane Jecker October 21, 2010 Plan Introduction 1 Useful definitions 2 Distributions

More information

THE HODGE DECOMPOSITION

THE HODGE DECOMPOSITION THE HODGE DECOMPOSITION KELLER VANDEBOGERT 1. The Musical Isomorphisms and induced metrics Given a smooth Riemannian manifold (X, g), T X will denote the tangent bundle; T X the cotangent bundle. The additional

More information

A Walking Tour of Microlocal Analysis

A Walking Tour of Microlocal Analysis A Walking Tour of Microlocal Analysis Jeff Schonert August 10, 2006 Abstract We summarize some of the basic principles of microlocal analysis and their applications. After reviewing distributions, we then

More information

An introduction to some aspects of functional analysis

An introduction to some aspects of functional analysis An introduction to some aspects of functional analysis Stephen Semmes Rice University Abstract These informal notes deal with some very basic objects in functional analysis, including norms and seminorms

More information

ON THE FRACTIONAL CAUCHY PROBLEM ASSOCIATED WITH A FELLER SEMIGROUP

ON THE FRACTIONAL CAUCHY PROBLEM ASSOCIATED WITH A FELLER SEMIGROUP Dedicated to Professor Gheorghe Bucur on the occasion of his 7th birthday ON THE FRACTIONAL CAUCHY PROBLEM ASSOCIATED WITH A FELLER SEMIGROUP EMIL POPESCU Starting from the usual Cauchy problem, we give

More information

Some SDEs with distributional drift Part I : General calculus. Flandoli, Franco; Russo, Francesco; Wolf, Jochen

Some SDEs with distributional drift Part I : General calculus. Flandoli, Franco; Russo, Francesco; Wolf, Jochen Title Author(s) Some SDEs with distributional drift Part I : General calculus Flandoli, Franco; Russo, Francesco; Wolf, Jochen Citation Osaka Journal of Mathematics. 4() P.493-P.54 Issue Date 3-6 Text

More information

Here we used the multiindex notation:

Here we used the multiindex notation: Mathematics Department Stanford University Math 51H Distributions Distributions first arose in solving partial differential equations by duality arguments; a later related benefit was that one could always

More information

Commutative Banach algebras 79

Commutative Banach algebras 79 8. Commutative Banach algebras In this chapter, we analyze commutative Banach algebras in greater detail. So we always assume that xy = yx for all x, y A here. Definition 8.1. Let A be a (commutative)

More information

Notions such as convergent sequence and Cauchy sequence make sense for any metric space. Convergent Sequences are Cauchy

Notions such as convergent sequence and Cauchy sequence make sense for any metric space. Convergent Sequences are Cauchy Banach Spaces These notes provide an introduction to Banach spaces, which are complete normed vector spaces. For the purposes of these notes, all vector spaces are assumed to be over the real numbers.

More information

On Fréchet algebras with the dominating norm property

On Fréchet algebras with the dominating norm property On Fréchet algebras with the dominating norm property Tomasz Ciaś Faculty of Mathematics and Computer Science Adam Mickiewicz University in Poznań Poland Banach Algebras and Applications Oulu, July 3 11,

More information

L p -boundedness of the Hilbert transform

L p -boundedness of the Hilbert transform L p -boundedness of the Hilbert transform Kunal Narayan Chaudhury Abstract The Hilbert transform is essentially the only singular operator in one dimension. This undoubtedly makes it one of the the most

More information

EXPOSITORY NOTES ON DISTRIBUTION THEORY, FALL 2018

EXPOSITORY NOTES ON DISTRIBUTION THEORY, FALL 2018 EXPOSITORY NOTES ON DISTRIBUTION THEORY, FALL 2018 While these notes are under construction, I expect there will be many typos. The main reference for this is volume 1 of Hörmander, The analysis of liner

More information

Analysis Comprehensive Exam Questions Fall F(x) = 1 x. f(t)dt. t 1 2. tf 2 (t)dt. and g(t, x) = 2 t. 2 t

Analysis Comprehensive Exam Questions Fall F(x) = 1 x. f(t)dt. t 1 2. tf 2 (t)dt. and g(t, x) = 2 t. 2 t Analysis Comprehensive Exam Questions Fall 2. Let f L 2 (, ) be given. (a) Prove that ( x 2 f(t) dt) 2 x x t f(t) 2 dt. (b) Given part (a), prove that F L 2 (, ) 2 f L 2 (, ), where F(x) = x (a) Using

More information

CHAPTER 6 bis. Distributions

CHAPTER 6 bis. Distributions CHAPTER 6 bis Distributions The Dirac function has proved extremely useful and convenient to physicists, even though many a mathematician was truly horrified when the Dirac function was described to him:

More information

1 Fourier transform as unitary equivalence

1 Fourier transform as unitary equivalence Tel Aviv University, 009 Intro to functional analysis 1 1 Fourier transform as unitary equivalence 1a Introduction..................... 1 1b Exponential map................... 1c Exponential map as an

More information

Functional Analysis Exercise Class

Functional Analysis Exercise Class Functional Analysis Exercise Class Week 9 November 13 November Deadline to hand in the homeworks: your exercise class on week 16 November 20 November Exercises (1) Show that if T B(X, Y ) and S B(Y, Z)

More information

Heisenberg s inequality

Heisenberg s inequality Heisenberg s inequality Michael Walter 1 Introduction 1 Definition. Let P be a probability measure on R. Then its mean or expectation is defined as E(P) := x dp(x) Its variance is the expected square deviation

More information

g(x) = P (y) Proof. This is true for n = 0. Assume by the inductive hypothesis that g (n) (0) = 0 for some n. Compute g (n) (h) g (n) (0)

g(x) = P (y) Proof. This is true for n = 0. Assume by the inductive hypothesis that g (n) (0) = 0 for some n. Compute g (n) (h) g (n) (0) Mollifiers and Smooth Functions We say a function f from C is C (or simply smooth) if all its derivatives to every order exist at every point of. For f : C, we say f is C if all partial derivatives to

More information

It follows from the above inequalities that for c C 1

It follows from the above inequalities that for c C 1 3 Spaces L p 1. In this part we fix a measure space (, A, µ) (we may assume that µ is complete), and consider the A -measurable functions on it. 2. For f L 1 (, µ) set f 1 = f L 1 = f L 1 (,µ) = f dµ.

More information

MATH 51H Section 4. October 16, Recall what it means for a function between metric spaces to be continuous:

MATH 51H Section 4. October 16, Recall what it means for a function between metric spaces to be continuous: MATH 51H Section 4 October 16, 2015 1 Continuity Recall what it means for a function between metric spaces to be continuous: Definition. Let (X, d X ), (Y, d Y ) be metric spaces. A function f : X Y is

More information

Bernstein s inequality and Nikolsky s inequality for R d

Bernstein s inequality and Nikolsky s inequality for R d Bernstein s inequality and Nikolsky s inequality for d Jordan Bell jordan.bell@gmail.com Department of athematics University of Toronto February 6 25 Complex Borel measures and the Fourier transform Let

More information

MATH 220: THE FOURIER TRANSFORM TEMPERED DISTRIBUTIONS. Beforehand we constructed distributions by taking the set Cc

MATH 220: THE FOURIER TRANSFORM TEMPERED DISTRIBUTIONS. Beforehand we constructed distributions by taking the set Cc MATH 220: THE FOURIER TRANSFORM TEMPERED DISTRIBUTIONS Beforehand we constructed distributions by taking the set Cc ( ) as the set of very nice functions, and defined distributions as continuous linear

More information

Introduction to Index Theory. Elmar Schrohe Institut für Analysis

Introduction to Index Theory. Elmar Schrohe Institut für Analysis Introduction to Index Theory Elmar Schrohe Institut für Analysis Basics Background In analysis and pde, you want to solve equations. In good cases: Linearize, end up with Au = f, where A L(E, F ) is a

More information

Locally convex spaces, the hyperplane separation theorem, and the Krein-Milman theorem

Locally convex spaces, the hyperplane separation theorem, and the Krein-Milman theorem 56 Chapter 7 Locally convex spaces, the hyperplane separation theorem, and the Krein-Milman theorem Recall that C(X) is not a normed linear space when X is not compact. On the other hand we could use semi

More information

LECTURE 3 Functional spaces on manifolds

LECTURE 3 Functional spaces on manifolds LECTURE 3 Functional spaces on manifolds The aim of this section is to introduce Sobolev spaces on manifolds (or on vector bundles over manifolds). These will be the Banach spaces of sections we were after

More information

13. Fourier transforms

13. Fourier transforms (December 16, 2017) 13. Fourier transforms Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ [This document is http://www.math.umn.edu/ garrett/m/real/notes 2017-18/13 Fourier transforms.pdf]

More information

Eigenvalues and Eigenfunctions of the Laplacian

Eigenvalues and Eigenfunctions of the Laplacian The Waterloo Mathematics Review 23 Eigenvalues and Eigenfunctions of the Laplacian Mihai Nica University of Waterloo mcnica@uwaterloo.ca Abstract: The problem of determining the eigenvalues and eigenvectors

More information

Chapter 4. The dominated convergence theorem and applications

Chapter 4. The dominated convergence theorem and applications Chapter 4. The dominated convergence theorem and applications The Monotone Covergence theorem is one of a number of key theorems alllowing one to exchange limits and [Lebesgue] integrals (or derivatives

More information

TOPICS IN FOURIER ANALYSIS-IV. Contents

TOPICS IN FOURIER ANALYSIS-IV. Contents TOPICS IN FOURIER ANALYSIS-IV M.T. NAIR Contents 1. Test functions and distributions 2 2. Convolution revisited 8 3. Proof of uniqueness theorem 13 4. A characterization of distributions 14 5. Distributions

More information

8 Singular Integral Operators and L p -Regularity Theory

8 Singular Integral Operators and L p -Regularity Theory 8 Singular Integral Operators and L p -Regularity Theory 8. Motivation See hand-written notes! 8.2 Mikhlin Multiplier Theorem Recall that the Fourier transformation F and the inverse Fourier transformation

More information

The Hilbert transform

The Hilbert transform The Hilbert transform Definition and properties ecall the distribution pv(, defined by pv(/(ϕ := lim ɛ ɛ ϕ( d. The Hilbert transform is defined via the convolution with pv(/, namely (Hf( := π lim f( t

More information

Laplace s Equation. Chapter Mean Value Formulas

Laplace s Equation. Chapter Mean Value Formulas Chapter 1 Laplace s Equation Let be an open set in R n. A function u C 2 () is called harmonic in if it satisfies Laplace s equation n (1.1) u := D ii u = 0 in. i=1 A function u C 2 () is called subharmonic

More information

Analysis IV : Assignment 3 Solutions John Toth, Winter ,...). In particular for every fixed m N the sequence (u (n)

Analysis IV : Assignment 3 Solutions John Toth, Winter ,...). In particular for every fixed m N the sequence (u (n) Analysis IV : Assignment 3 Solutions John Toth, Winter 203 Exercise (l 2 (Z), 2 ) is a complete and separable Hilbert space. Proof Let {u (n) } n N be a Cauchy sequence. Say u (n) = (..., u 2, (n) u (n),

More information

Fourier Series. 1. Review of Linear Algebra

Fourier Series. 1. Review of Linear Algebra Fourier Series In this section we give a short introduction to Fourier Analysis. If you are interested in Fourier analysis and would like to know more detail, I highly recommend the following book: Fourier

More information

Friedrich symmetric systems

Friedrich symmetric systems viii CHAPTER 8 Friedrich symmetric systems In this chapter, we describe a theory due to Friedrich [13] for positive symmetric systems, which gives the existence and uniqueness of weak solutions of boundary

More information

Distribution Theory and Fundamental Solutions of Differential Operators

Distribution Theory and Fundamental Solutions of Differential Operators Final Degree Dissertation Degree in Mathematics Distribution Theory and Fundamental Solutions of Differential Operators Author: Daniel Eceizabarrena Pérez Supervisor: Javier Duoandikoetxea Zuazo Leioa,

More information

Bochner s Theorem on the Fourier Transform on R

Bochner s Theorem on the Fourier Transform on R Bochner s heorem on the Fourier ransform on Yitao Lei October 203 Introduction ypically, the Fourier transformation sends suitable functions on to functions on. his can be defined on the space L ) + L

More information

MATH 205C: STATIONARY PHASE LEMMA

MATH 205C: STATIONARY PHASE LEMMA MATH 205C: STATIONARY PHASE LEMMA For ω, consider an integral of the form I(ω) = e iωf(x) u(x) dx, where u Cc (R n ) complex valued, with support in a compact set K, and f C (R n ) real valued. Thus, I(ω)

More information

Notes for Elliptic operators

Notes for Elliptic operators Notes for 18.117 Elliptic operators 1 Differential operators on R n Let U be an open subset of R n and let D k be the differential operator, 1 1 x k. For every multi-index, α = α 1,...,α n, we define A

More information

ON MATRIX VALUED SQUARE INTEGRABLE POSITIVE DEFINITE FUNCTIONS

ON MATRIX VALUED SQUARE INTEGRABLE POSITIVE DEFINITE FUNCTIONS 1 2 3 ON MATRIX VALUED SQUARE INTERABLE POSITIVE DEFINITE FUNCTIONS HONYU HE Abstract. In this paper, we study matrix valued positive definite functions on a unimodular group. We generalize two important

More information

Elliptic Regularity. Throughout we assume all vector bundles are smooth bundles with metrics over a Riemannian manifold X n.

Elliptic Regularity. Throughout we assume all vector bundles are smooth bundles with metrics over a Riemannian manifold X n. Elliptic Regularity Throughout we assume all vector bundles are smooth bundles with metrics over a Riemannian manifold X n. 1 Review of Hodge Theory In this note I outline the proof of the following Fundamental

More information

Metric spaces and metrizability

Metric spaces and metrizability 1 Motivation Metric spaces and metrizability By this point in the course, this section should not need much in the way of motivation. From the very beginning, we have talked about R n usual and how relatively

More information

Math The Laplacian. 1 Green s Identities, Fundamental Solution

Math The Laplacian. 1 Green s Identities, Fundamental Solution Math. 209 The Laplacian Green s Identities, Fundamental Solution Let be a bounded open set in R n, n 2, with smooth boundary. The fact that the boundary is smooth means that at each point x the external

More information

3. Fourier decomposition of functions

3. Fourier decomposition of functions 22 C. MOUHOT 3.1. The Fourier transform. 3. Fourier decomposition of functions Definition 3.1 (Fourier Transform on L 1 (R d )). Given f 2 L 1 (R d ) define its Fourier transform F(f)( ) := R d e 2i x

More information

We have to prove now that (3.38) defines an orthonormal wavelet. It belongs to W 0 by Lemma and (3.55) with j = 1. We can write any f W 1 as

We have to prove now that (3.38) defines an orthonormal wavelet. It belongs to W 0 by Lemma and (3.55) with j = 1. We can write any f W 1 as 88 CHAPTER 3. WAVELETS AND APPLICATIONS We have to prove now that (3.38) defines an orthonormal wavelet. It belongs to W 0 by Lemma 3..7 and (3.55) with j =. We can write any f W as (3.58) f(ξ) = p(2ξ)ν(2ξ)

More information

GENERALIZED FUNCTIONS LECTURES. Contents

GENERALIZED FUNCTIONS LECTURES. Contents GENERALIZED FUNCTIONS LECTURES Contents 1. The space of generalized functions on R n 2 1.1. Motivation 2 1.2. Basic definitions 3 1.3. Derivatives of generalized functions 5 1.4. The support of generalized

More information

Unbounded operators on Hilbert spaces

Unbounded operators on Hilbert spaces Chapter 1 Unbounded operators on Hilbert spaces Definition 1.1. Let H 1, H 2 be Hilbert spaces and T : dom(t ) H 2 be a densely defined linear operator, i.e. dom(t ) is a dense linear subspace of H 1.

More information

EECS 598: Statistical Learning Theory, Winter 2014 Topic 11. Kernels

EECS 598: Statistical Learning Theory, Winter 2014 Topic 11. Kernels EECS 598: Statistical Learning Theory, Winter 2014 Topic 11 Kernels Lecturer: Clayton Scott Scribe: Jun Guo, Soumik Chatterjee Disclaimer: These notes have not been subjected to the usual scrutiny reserved

More information

Chapter 3: Baire category and open mapping theorems

Chapter 3: Baire category and open mapping theorems MA3421 2016 17 Chapter 3: Baire category and open mapping theorems A number of the major results rely on completeness via the Baire category theorem. 3.1 The Baire category theorem 3.1.1 Definition. A

More information

Sobolev Spaces. Chapter Hölder spaces

Sobolev Spaces. Chapter Hölder spaces Chapter 2 Sobolev Spaces Sobolev spaces turn out often to be the proper setting in which to apply ideas of functional analysis to get information concerning partial differential equations. Here, we collect

More information

P-adic Functions - Part 1

P-adic Functions - Part 1 P-adic Functions - Part 1 Nicolae Ciocan 22.11.2011 1 Locally constant functions Motivation: Another big difference between p-adic analysis and real analysis is the existence of nontrivial locally constant

More information

II. FOURIER TRANSFORM ON L 1 (R)

II. FOURIER TRANSFORM ON L 1 (R) II. FOURIER TRANSFORM ON L 1 (R) In this chapter we will discuss the Fourier transform of Lebesgue integrable functions defined on R. To fix the notation, we denote L 1 (R) = {f : R C f(t) dt < }. The

More information

B. Appendix B. Topological vector spaces

B. Appendix B. Topological vector spaces B.1 B. Appendix B. Topological vector spaces B.1. Fréchet spaces. In this appendix we go through the definition of Fréchet spaces and their inductive limits, such as they are used for definitions of function

More information