1 Continuity Classes C m (Ω)

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1 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 + y x, y X (triangle inequality) 1 Continuity Classes C m () Facts. C m () = {f : D α is continuous on α m} C () = {f : D α is continuous on α } (Smooth Functions) Analytic Functions: C ω () = {f : f C } and f equals the Taylor series expansion in a neighborhood around every point in Continuous and Bounded: CB() = {f : f is continuous and bounded on } Absolutely Continuous: AC [a, b] = { f(x) : f(y) f(x) = y x g(s)ds} for some g L 1 (a, b) such that f AC [a, b] implies f = g a.e. where f is the pointwise derivative. C m () C ω () C () 2 Vector Spaces 2.1 Norms A norm on a linear space X is a function : X R with the properties: x 0 x X (nonnegative) λx = λ x x X, λ C(homogeneous) x + y x + y x, y X (triangle inequality) x = 0 x = 0 (stricly positive) 2.2 Vector Norms p ( x p = xi p) 1 p Sum Norm, p = 1 x sum = x 1 = x i 1

2 Euclidean Norm, p = 2 x Euclidean = x 2 = x 2 i Maximum Norm, p = Given a vector x in C N, x max = x = max { x i } x p+a x p p 1 a 0 Examples: x 2 x 1 and x 1 n x 2. x p x r n ( 1 r 1 p) x p p 1 a 0 3 L p Spaces The space of Lebesgue Integrable functions is defined for a given p: } L p () = {f : f L p() < Where the norm on the space is f Lp () = ( ) 1 f p p dx L () is the space of essentially bounded functions. That is, f is bounded on a subset of whose complement has measure zero. The norm on L () is the essential suprenum f = inf {M s.t. f(x) M a.e. } L 2 () is the space of square-integrable functions. This is a Hilbert Space. L 1 () is the space of Lebesgue integrable functions L 0 () is the space of measurable functions L 1 loc () is the space of locally integrable functions. That is, functions integrable on any compact subset of an open set R N L 1 loc() = {f : f L 1 (K) K, K compact} Alternative definition L 1 loc() = { } f : fφ L 1 () < φ C 0 () This definition generalizes to L p loc (). 3.1 Useful Facts Embeddings: u Lp () meas()( 1 p 1 q ) u Lq () Thus if is bounded, then L q () L p () L p loc () for all q > p L () L 2 () L 1 () L 1 loc() 2

3 Continuous functions of compact support are dense in L p () One can discuss weighted L p spaces with 4 l p (N) Space f L p (,wdµ) = ( ) 1 w(x) f p p dµ(x) This is the space of bi-infinite sequences {x n } complete with respect to the norm such that x n p < for 0 < p. l p (N) is and as well for in the case of p = x p = ( There is no norm for 0 < p < 1, but there is a metric l is the space of bounded sequences d ({x n }, {y n }) = x n p ) 1 p, p 1 x = sup x n n x n y n p l 2 is the space of square-summable sequences. This is a Hilbert Space. l 1 is the space of summable sequences 5 Banach Spaces 6 Hilbert Spaces A Hilbert Space is a vector space with a defined inner product u, v which has the following properties Conjugate symmetry: u, v = v, u Linear in the first argument: au + bw, v = a u, v + b w, v Antilinear in the second argument: u, av + bw = ā u, v + b w, v Positive definite when u = v: u, u 0 u, u = 0 if and only if u = 0 A Hilbert Space is complete with respect to its norm: x H = x, x H 3

4 6.1 Convergence 6.1 Convergence x n H converges strongly to x H if x n x H 0. x n H converges weakly to x H if x n, y x, y y H Thus x lim inf n x n which is strict inequality when convergence is not strong. 6.2 Important Hilbert Spaces l 2 is the space of square-summable sequences with x, y = x i y i i= L 2 () is the space of square integrable fuctions with u, v = uvdx W s,2 = H s are the Sobolev Spaces u, v = ugdx + Du Dgdx D s u D s gdx C n m is a Hilbert space which is the space of all m n matrices with the inner product m n A, B = tr(a B) = ā ij b ij i=1 j=1 with the Hilbert-Schmidt norm (aka Frobenius norm) m n A C n m = a ij 2 i=1 j=1 H, the dual of H, is the Hilbert space of continuous linear functionals from H into the base field. For every element u H, there exists a unique element φ u H defined by The norm is 6.3 Hilbert Bases φ u (x) = x, u φ H = sup φ(x) x =1,x H To check if a given sequence If {x n } is a basis for H, we check the following 1. {x n } is closed if the set of all finite linear combinations of them is dense in H 2. {x n } is complete if and only if there is no nonzero vector orthogonal to all x n, that is x, x n = 0 n x = 0 3. If {x n } are orthonormal they are maximal orthonormal if {x n } is not contianed in any strictly larger orthonormal set. 4

5 6.3 Hilbert Bases Orthonormal Basis Let {e n } n=1 be orthonormal, the following are equivalent 1. {e n } is maximal orthonormal 2. {e n } is closed 3. {e n } is complete 4. {e n } is a basis of H 5. Bessel s Inequality holds for all x H 6. x, e n e n, y = x, y x, y H n=1 So we know a complete orthonormal set forms a basis, with x, e n known as the n th Generalized Fourier Coefficient of x with respect to {e n } Gram Schmidt If M = L {e 1, e 2,..., e n } is linearly independent but not orthonormal, then create a new orthonormal basis M = L {f 1, f 2,..., f n } with f n = f n n f n f n = e n e n, f i f i Reisz Fischer Critertion For an infinite dimensional basis, c n e n converges in H if and only if c n 2 < n=1 n=1 This implies c n e n converges if and only if {c n } i=1 l2 n= Riemann Lebesgue Lemma Version 1 If given {e n } n=1 is ON and x H then Bessel s Inequality n=1 i=1 x, e n 2 < implies lim n x, e n = 0 For an infinite dimensional basis, S N = x, e n e n = P MN x where M N = L {e 1, e 2,..., e n }, which implies x, e n 2 x 2 n=1 5

6 7 Sobolev Spaces Intuitively, a Sobolev space is a space of functions with sufficiently many derivatives for some application domain, such as partial differential equations, and equipped with a norm that measures both the size and regularity of a function. It is a vector space of functions equipped with a norm that is a combination of L p norms and the function itself as well as derivatives up to a given order. The derivatives are understood in a suitable weak sense to make the space complete, thus a Banach space. Their importance comes from the fact that solutions of partial differential equations are naturally found in Sobolev spaces, rather than in spaces of continuous functions and with the derivatives understood in the classical sense. 7.1 Weak Derivatives Given f L p (), D α f is defined in the D sense for any α if g L q () s.t. D α f = g in the D sense then we say g is the weak α derivative of f in L q (). That is, f(x)d α φ(x)dx = ( 1) α g(x)φ(x)dx φ C0 () Weak derivatives need not be uniformly convergent Weak derivatives are defined almost everywhere Weak and classical derivatives coincide when u C k () and α = k Example: If f(x) = x on ( 1, 1), then f = [f ] + f 0 δ = sign (x). So since f; L q () for 1 q, f has a weak first derivative in L q () Example: If f(x) = H(x) then f = δ / L q () for any q. So f has does not have a weak first derivative in L q () for any q 7.2 Definition The Sobolev space W k,p () of order k is defined to be the set of all functions u L p () such that for every multi-index α with α k, the weak partial derivative D α u belongs to L p (), i.e. for an open set and for 1 p W k,p () = {u L p () : D α u L p () α k} The first (default) of norm on W k,p () is ( u W k,p () = D α u p L p () α k max α k Dα u L () ) 1 p 1 p < p = An equivalent norm is ( u W k,p () = u p L p () + D k u p L p () max α k Dα u L () ) 1 p 1 p < p = Another equivalent norm on W k,p () is u W k,p () = D α u L p () α k 6

7 7.3 Embeddings W k,p () is a Banach space Alternative definiton: We say f has a strong α derivative in L p () if f n, g C () s.t. f n f in L p () and D α f n g in L p () It turns out that the space of f L p () that have a strong α derivative for α k is the same space as W k,p (). That is for α k f has a strong α derivative if and only if it has a weak α derivative 7.3 Embeddings W k,2 () is the Hilbert space H k () with norm W k,2 () that can be equivalently defined as { H k () = f L 2 ( () : 1 + n 2 + n n 2k) } 2 ˆf(n) < with norm and inner product f 2 H k () = f, g Hk () = (1 + n 2 ) k ˆf(n) 2 k D i f, D i g L 2 () i=0 H0() 1 is subspace of H 1 () where functions vanish on the bondary, complete with respect to its norm ( f H1 () ( f = 2 + f 2)) 1 2 f H 1 (T) if c n = 1 2π 2π f(x)e inx dx satisfies 0 H 1 (T) L 2 (T) nc n 2 < Sobolev Embedding Theorem If k > l, 1 p, q, (k l)p < n, and 1 p 1 q = k l n then W k,p (R N ) W l,q (R N ) When k = 1 and l = 0, let p 1 be the Sobolev conjugate of p: p = 1 p 1 n W 1,p (R N ) L p (R N ) Existence of sufficiently many weak derivatives implies some continuity of the classical derivatives. H s (R N ) C k 0 (R N ) s > k + N 2 Notably, H 1 (R) C 0 (R) 7

8 7.4 Facts 7.4 Facts For finite p, W k,p () is a separable space (contains a countable dense subset). If E W k,p (), E, {u n } n N W k,p (), then n s.t. u n E W 0,p () is simply L p () W 1,1 () is the space of absolutely continuous functions AC() W 1, () is the space of Lipschitz continous functions on Formally if f(x) = Bessel s equality says f L 2 (R N ) = 8 Linear Operators c n e inx then the weak derivative of f is f (x) = nc n 2 < The space of Linear Operators LX, Y The norm on this space is the Operator norm c n ine inx T = T x sup x D(T ),x 0 x = sup T x = sup T x x D(T ), x =1 x D(T ),0< x 1 which has the following properties T u T u T + S T + S T S T S T = 0 if and only if it is the Zero map 0u = Convergence The space of Bounded linear operators B(H) is a Banach space complete with respect to the operator norm with the property that T < If T n T 0, we say T n T in B(H) uniformly If T n T pointwise or strongly if T n x T x x H Uniform convergence implies strong convergence. Converse is false. The Identity Map I X X is bounded on any normed space and has I = 1. A series n=1 T n converges uniformly or strongly if the corresponding sequence of partial sums does. We know T n converges uniformly if T n < 8

9 8.2 Compact Operators Example T B, S = T n n! converges uniformly since T n n! T n n! < Limit is deifned to be e T Let E B(H), E < 1. Then S = If S N = n E n, j=0 as N, E N+1 E N+1 0 so or In particular, (I E) 1 E n is convergent since E j E j 1 = 1 E < j=0 S N (I E) = (I + E + E E N )(I E) = I E N+1 j=0 8.2 Compact Operators S(I E) = I = S = (I E) 1 E j = 1 1 E (I E) 1 = E n This implies 1 ρ(e) The space of compact operators is the linear operators L(X, Y ) such that Y is precompact (the closure of Y is compact). Compact operators are bounded and contiuous. Any operator with finite rank is a compact operator. Any compact operator is a limit of finite rank operators. If T n K(H) and T n T L(X,Y ) 0, then T is compact 9

10 9 Distribution Theory 9.1 Test Functions For open R n, C 0 () = D() is the vector space of test functions D() = {f C () : supp f R} (f with compact support) C 0 is not a metric space, but rather a Topological Vector Space (TVS) Convergence in C 0 (R N ) Let φ n C 0 We say φ n 0 in C 0 1. K s.t. supp φ n K n 2. D α φ n C() 0 as n α if And so we say φ n φ in C 0 if 1. φ n φ 0 (uniformly) in C Properties of C 0 C0 () is closed under the following operations: Given φ 1 (x) C0 (), C0 is closed under the following operations: Scaling φ(x) = αφ 1 (x) Translation φ(x) = φ 1 (x c) Dilation φ(x) = φ 1 (αx) for any α 0 Differentiation φ(x) = D α φ 1 (x) for any multiindex α Sums of such terms Products of such terms If φ 1 (x) C 0 () then φ 1 ( x ) C 0 () Example: The Common Mollifier φ(x) = {e 1 x 2 1 x < 1 0 x > Distributions T D () Distributions T are continuous linear functionals on C0 () that satisfy 1. T : C0 () C 2. T (c 1 φ 1 + c 2 φ 2 ) = c 1 T (φ 1 ) + c 2 T (φ 2 ) c 1, c 2 C and φ 1, φ 2 C0 () 3. If φ n φ in C0 () then T (φ n ) T (φ) That is, D () is the set of distributions over, and it is the dual space of C0 10

11 9.2 Distributions T D () Convergence in D () T n T in D () if T n (φ) T (φ) for all φ C 0 () T n, φ T, φ φ = T n T Regular Distributions T is a regular distribution if T = T f : φ T f (φ) for some f L 1 loc The set of T f s.t. f L 1 loc () may be regarded as a subset of D () and is defined T f (φ) = f(x)φ(x)dx T f1 = T f2 if and only if f 1 = f 2 a.e. If f n f in the L 1 loc sense, then T f n T f since T fn (φ) T f (φ) f n f L 1 (K) φ L (K) 0 We abuse notation and say f n f in the D sense Every distribution is either regular or singular: δ is singular Properties of D D is closed under linear combinations. Multiplication for distributions is not defined in a general way. For a(x) C, we define (at )(φ) = T (aφ) since aφ C0 and linearity is preserved. Similiarly, at f = T f (aφ). Example: T = δ, (aδ)(φ) = δ(aφ) = a(0)φ(0) = a(0)δ(φ) so aδ = a(0)δ Distributional Derivatives are defined as T, φ = T, φ. For T = T f for f C 1 (R) we define T f (φ) = T f (φ ) In R N, for a multiindex α, (D α T )(φ) = ( 1) α T (D α φ). Rather, D α T, φ = ( 1) α T, D α φ Any locally integrable function (f L 1 loc ()) has a distributional derivative T (φ) D since T : φ C and is linear and if φ n 0 then T (φ n) 0 All distributions are infinitely differentiable in the sense of distribution, and you may suffle mixed partial derivatives. ( ( )) Product Rule For a C, ( x i (at ))(φ) = a x i T (φ) + a x i T (φ), or (at ), φ = a T, φ + at, φ D α : D () D (). That is, if T n T in D () then D α T n D α T in D () for all α Example: If f n δ then f n δ Classically function convergence does not imply derivative convergence 11

12 9.2 Distributions T D () Properties of Distributions T (φ) = φ(x 0 ) for some x 0 and φ C 0 is a distribution but not a function The Dirac Delta Function is defined as such: pick x 0, let T (φ) = φ(x 0 ), denoted 1. δ x0 (x x 0 ) = 0 if x 0 2. δ(x)dx = 1 The Dipole Distributionis defined δ x0 : δ x0 (φ) = φ(x 0 ) δ : δ(φ) = φ(0) T (φ) = δ = φ (x 0 ) D H (φ) = φ(0) = δ(φ) and H (φ) = δ (φ) = δ(φ ) = φ (0) (the dipole distributon) Convolutions in D is defined for f C 0 : f T f T, φ = T, ˇfφ, (f T )(x) = T (τ x ˇf) and for S T where T is a distribution of compact support: S (T φ) = (S T ) φ Aproximations to the Identity f is said to be an approximate identity if f L 1 (R n ) such that 1. R n f n (x)dx = 1 n 2. c s.t. f n L 1 (R n ) c 3. lim n x >ɛ f n(x) dx = 0 ɛ > 0 Then f n δ in D sense Principle Value Distributions We see that T (φ) = φ(x) x dx looks like T f for f(x) = 1 x but f(x) = 1 x / L1 loc, so instead we consider the Principle Value of the integral T (φ) = lim ɛ 0 x >ɛ φ(x) dx = pv x φ(x) dx = pv x ( ) 1 x Note that the principle value distribution is not the same as the improper integral. 12

13 9.3 Tempered Distributions Pseudofunctions For asymptotic singularities, { we cannot use f(c) log(x) x > 0 Example: f(x) = 0 x < 0, note that f / L1 loc T f (φ) = lim ɛ 0 + ɛ ( log(x)φ (x)dx = lim [log(x)φ(x)] ɛ 0 + ɛ + ɛ ) φ(x) x dx Neither limit exists but their sum does due to cancellation. Using φ(ɛ) = φ(0) + φ (θ)ɛ by MVT, ( ) T f φ(x) (φ) = lim log(ɛ)φ(0) + ɛ 0 + ɛ x dx = pf H(x) x Where pf means pseudofunction or finite part, and is itself a distribution 9.3 Tempered Distributions Norm on the Schwartz Space p α,β (φ) = φ α,β = sup x R N x α β φ(x) Definition of the Schwartz Space S The Schwartz Space S(R N ) consists of all functions for which all derivatives are bounded and all derivatives are bonded by multiplication by x α for any α. That is, S = { φ C (R N ) : p α,β (φ) < α, β Z n } + If φ S, then there exists a constant C d,α such that C d,α α φ(x) (1 + x ) d 2 x R N Thus, and element of S is a smooth function such that the function and all of its derivatives decay faster than the reciprocal of any polynomial (that is it decays faster than any polynomial grows) as x. Elements of S are called Schwartz functions and known as rapidly decreasing functions. Schwartz Functions Facts C0 (R N ) S Gaussians multiplied by any finite degree polynomial are Schwartz functions q(x)e c x x0 2 S c > 0, x 0 R N, q(x) = c α x α S L 1 (R N ) α d S is dense in L p (R N ) for 1 p < since C 0 is dense in R N 13

14 9.4 Distributions of Compact Support 9.4 Distributions of Compact Support E is the subspace of distributions with compact support. Define E = C (R N ) with φ n φ E if D α (φ n φ) L (K) 0 for all K RN That is, T E if and only if T D and supp T R N So C0 (R N ) S E are toplogical inclusions and so E S D Support of Distributions: x supp T if and only if ɛ > 0 φ C 0 (B ɛ (x)) s.t. T (φ) 0 T = δ If x 0 let ɛ = x 2 then φ C 0 (B ɛ (x)) then δ(φ) = φ(0) = 0 and so supp δ = {0} ˆT is defined for all T E since E D 14