Chapter 5: Bases in Hilbert Spaces

Transcription

1 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications of Fourier series Chapter 5: Bases in Hilbert Spaces I-Liang Chern Department of Applied Mathematics National Chiao Tung University and Department of Mathematics National Taiwan University Fall, / 94

2 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Characterization of Fourier of orthonormal series basis Existence of bases, Gram Outline 1 Orthogonal bases, general theory Characterization of orthonormal basis Existence of bases, Gram-schmidt orthogonalization 2 The Fourier basis in L 2 (T) Properties of Fourier expansion Convergence Theory 3 Applications of Fourier series Hurwitz s proof for isoperimetric inequality Weyl s ergodic theorem Characterization of Sobolev spaces Solving heat equation on circle Solving Laplace equation on a disk Stability analysis for finite difference methods 2 / 94

3 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Characterization of Fourier of orthonormal series basis Existence of bases, Gram Orthogonal Bases in Hilbert Spaces In this section, we shall discuss how to approximate a point x H in terms of an expansion in an orthogonal set U = {u α α I}. Definition 1 A set U = {u α α I} is called an orthogonal set in a Hilbert space H if any two of them are orthogonal to each other. 2 It is called an orthonormal set if it is orthogonal and each of them is a unit vector. 3 / 94

4 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Characterization of Fourier of orthonormal series basis Existence of bases, Gram For separable Hilbert space (i.e. there exists a countable set A such that A = H), we can choose U to be countable. But in general, U can be uncountable. The index set I may not be ordered, or may even not be countable. Nevertheless, we can still discuss the meaning of the limit of (uncountable) summation. Let U denote the finite linear span of U. 4 / 94

5 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Characterization of Fourier of orthonormal series basis Existence of bases, Gram Definition 1 We say that α I x α converges to x unconditionally, if for any ɛ > 0, there exists a finite set K I such that for any finite set J with K J I, we have x x α < ɛ. α J 2 The summation {x α α I} is called Cauchy if for any ɛ > 0, there exists a finite set K I such that for any finite set J with K J I, we have x α < ɛ. α J K 5 / 94

6 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Characterization of Fourier of orthonormal series basis Existence of bases, Gram Remarks 1 It is clear that if α I x α converges, then it is Cauchy. 2 If α I x α is Cauchy, then for any n N, there exists a finite K n such that for any α K n, x α < 1/n. Then for any α n N K n, x < 1/n for all n, thus, x α = 0. Since n N K n is countable, we conclude that there are at most countable nonzero x α. In this case, we can select J n = j n K j, then S Jn := α J n x α converges to x I x α. 6 / 94

7 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Characterization of Fourier of orthonormal series basis Existence of bases, Gram Example: quasi-periodic function space Important example of non-separable Hilbert space is the space of quasi-periodic functions on R. A function is called quasi-periodic if f(t) = n a k e iω kt k=1 where n N, a k C and ω k R are arbitrary. For quasi-periodic functions f and g, we define 1 (f, g) := lim T 2T T T f(t)g(t) dt. 7 / 94

8 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Characterization of Fourier of orthonormal series basis Existence of bases, Gram H be the completion of all quasi-periodic functions with the above inner product. They are the L 2 -almost periodic periodic functions. The set U = {e iωt ω R} Lemma is a uncountable orthonormal set in H. Let {x α α I} be an orthogonal set. Then α I x α converges if and only if α I x α 2 converges. In this case, 2 x α = x α 2 (1.1) α I α I 8 / 94

9 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Characterization of Fourier of orthonormal series basis Existence of bases, Gram Proof. We check that (a) α I x α is Cauchy if and only if (b) α I x α 2 is Cauchy. This is because if (a) is true, which means that for any ɛ > 0, there exists a finite set K I such that for any J I K, we have i J x α 2 < ɛ. But 2 x α = x α 2. α J α J This is equivalent to say that i I x α 2 is Cauchy. In this case, (1.1) follows from the continuity of the norm. 9 / 94

10 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Characterization of Fourier of orthonormal series basis Existence of bases, Gram Bessel inequality Theorem (Bessel s inequality) Let U := {u α α I} be an orthonormal set in a Hilbert space H. Then for any x H, we have (a) α I (u α, x) 2 x 2. (b) x = α I (u α, x)u α converges unconditionally. (c) (x x) U. (d) x U if and only if x = α I (u α, x)u α. (e) The subspace U has the characterization U = { a α u α a α 2 < } α I α I 10 / 94

11 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Characterization of Fourier of orthonormal series basis Existence of bases, Gram Proof of (a) {x α := (u α, x)u α α I} is orthogonal. From 0 x α J (u α, x)u α 2 = x 2 α J (u α, x) 2, where J I is finite, we have i J x α 2 x 2. Let M = sup{ i J x α 2 J I is a finite set.}. Then for any ɛ > 0, there exists a finite set K I such that M ɛ < i K x α 2. Now for any finite set J I K, we have M ɛ < x α 2 α K x α 2 M. α K J This gives α J x α 2 < ɛ. Thus, α I x α 2 is Cauchy. It converges and has an upper bound x / 94

12 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Characterization of Fourier of orthonormal series basis Existence of bases, Gram (b) From (a) and Lemma 1.3, we get the convergence of the un-order sum x := α I (u α, x)u α. (c) We use continuity of inner product: for any u β U, x (u α, x)u α, u β = (x, u β ) (u α, x)(u α, u β ) = (x, u β ) (u β, x) = 0. α I α I (d) From(b), x U. Thus, we have x U (x x) U. From (c), x x U. Thus, x U x x = 0. Hence any x U can be expressed as x = (u α, x)u α. 12 / 94

13 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Characterization of Fourier of orthonormal series basis Existence of bases, Gram (e) If α I a αu α with α I a α 2 <, then using the same argument of (b), we get the un-order sum α I a αu α converges. And hence it is in U. On the other hand, we have seen from (d) that any element x in U can be expressed as x = α I (u α, x)u α, with α I (u α, x) 2 <. 13 / 94

14 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Characterization of Fourier of orthonormal series basis Existence of bases, Gram Theorem Let U := {u α α I} be an orthonormal set in a Hilbert space H. Then the following conditions are equivalent: (a) (x, u α ) = 0 for all α I implies x = 0; (b) Any x H can be represented as x = α I (u α, x)u α ; (c) U = H; (d) The norm of any x H can be characterized by x 2 = α I (u α, x) 2 ; (e) U is a maximal orthonormal set. 14 / 94

15 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Characterization of Fourier of orthonormal series basis Existence of bases, Gram Proof 1 We see that (a) U = {0} U = 0 H = U. The latter follows from the decomposition theorem H = M M for any closed subspace M. This together with the previous theorem show the equivalent from (a) to (c). 2 The equivalence between (b) and (d) follows from Lemma / 94

16 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Characterization of Fourier of orthonormal series basis Existence of bases, Gram 3 To prove (e) (a), let us suppose there is a v U and v U. That is, U is not maximal since U V := {v} U. Then from (a), v = 0. Conversely, U is maximal means that any orthonormal set V with U V, then U = V. In other word, if there is a v V \U, then v = 0. But this is the statement (a). An orthonormal set in H satisfying one of the statements of this theorem is called an orthonormal basis of H. By (b), any element x H can be represented as x = α I c αu α. By (d), this representation must be unique. And by (b), it is also represented as x = (u α, x)u α. α I 16 / 94

17 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Characterization of Fourier of orthonormal series basis Existence of bases, Gram Existence of bases Theorem Any Hilbert space H has an orthonormal basis. 1 We consider the set S of all orthonormal sets in H and order them by the inclusion relation. 2 S has the property: any totally partial order family has an upper bounded. A total partial order family {U U A} means any two of its elements, say U and V, is either U V or V U. We see that its union V = U A U is an upper bound. 3 With this property, by Zorn s lemma in e set theory, the set S has a maximal element. By the previous theorem, it is an orthonormal basis. 17 / 94

18 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Characterization of Fourier of orthonormal series basis Existence of bases, Gram Separable Hilbert space has countable bases Theorem Any separable Hilbert space H has a countable orthonormal basis. Proof 1 From separability of H, there is a countable set A = {v i i N} such that A = H. 2 We construct a sequence of nested subspaces V n and its basis {w 1, w 2,, w n } by the following procedure. 3 Let w 1 = v 1, V 1 = {v 1 }. If v 2 V 1, then define w 2 = v 2 and V 2 = {v 1, v 2 }. Otherwise, we skip v 2 and continue this process. Either we can find the next one, or we have exhausted all elements in A. 18 / 94

19 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Characterization of Fourier of orthonormal series basis Existence of bases, Gram 4 For the latter case, H is finite dimension. For the former case, we continue this process and select an infinite countable subset B := {w 1, w 2, } from A such that {v 1,, v n } {w 1,, w n } B for all n. Thus, A B From A = H, we get B = H. 5 We can construct an orthonormal set {u 1, u 2, } from the independent set {w 1, w 2, } by the Gram-Schmidt orthonormalization procedure. It is an induction procedure. 19 / 94

20 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Characterization of Fourier of orthonormal series basis Existence of bases, Gram 6 We choose u 1 = w 1 / w 1. Let z 2 = w 2 (w 2, u 1 )u 1 and u 2 = z 2 / z 2. Suppose {u 1,, u n } are found. We define z n+1 = w n+1 and u n+1 = z n+1 / z n+1. 7 By construction, we have n (w n+1, u i )u i i=1 {u 1,, u n } = {w 1,, w n } for all n. Consequently {w 1,, w n } {u 1, u 2, } By taking n and taking closure on the left-hand side, we get H = u 1, u 2,. 20 / 94

21 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Properties of Fourier series expansion Convergence Theory Outline 1 Orthogonal bases, general theory Characterization of orthonormal basis Existence of bases, Gram-schmidt orthogonalization 2 The Fourier basis in L 2 (T) Properties of Fourier expansion Convergence Theory 3 Applications of Fourier series Hurwitz s proof for isoperimetric inequality Weyl s ergodic theorem Characterization of Sobolev spaces Solving heat equation on circle Solving Laplace equation on a disk Stability analysis for finite difference methods 21 / 94

22 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Properties of Fourier series expansion Definition and examples Convergence Theory We study Fourier expansion for 2π-periodic functions f. We will expand f as f(x) k= a k e ikx. By taking the inner product, defined by (f, g) := 1 π f(x)g(x) dx, with 2π π eimx, we find that a m = 1 π f(x)e imx dx. 2π π a m s are called the Fourier coefficients, or Fourier multiples. m is the wave number. We denote a m by ˆf m. 22 / 94

23 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Properties of Fourier series expansion Convergence Theory Examples { 1 for 0 < x < π 1 f(x) = 1 for π < x < 0 2 f(x) = 1 x π 23 / 94

24 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Properties of Fourier series expansion Convergence Theory Basic properties A 2π-periodic function can be identified as a function on circle, which is T = R/(2πZ). Some important properties of Fourier transform are The differentiation becomes a multiplication under Fourier transform. It is also equivalent to say that the differential operator is diagonalized in Fourier basis. The convolution becomes a multiplication under Fourier transform. 24 / 94

25 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Properties of Fourier series expansion Differentiation Convergence Theory Lemma If f C 1 [T], then f k = ik ˆf k. Proof f k = 1 2π 2π 0 f (x)e ikx dx = 1 2π e ikx f(x) x=2π x=0 1 2π = ik ˆf k. 2π 0 ( ik)e ikx f(x) dx Here, we have used the periodicity of f in the last step. 25 / 94

26 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Properties of Fourier series expansion Convergence Theory Convolution If f and g are in L 2 (T), we define the convolution of f and g by (f g)(x) = f(x y)g(y) dy. T T Example: Solutions of differential equations are expressed in convolution forms. Eg. u = f in T, its solution u = g f, where g is the Green s function of d 2 /dx 2 on T. 26 / 94

27 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Properties of Fourier series expansion Convolution and Fourier transform Convergence Theory Lemma If f, g C(T), then ( f g) Proof ( f g) k = = = = = k = 2π ˆf k ĝ k. 1 f g(x)e ikx dx 2π T 1 f(x y)g(y) dye ikx dx 2π T T 1 f(x y)e ik(x y) g(y) dye iky dx 2π T T ( ) 1 f(x y)e ik(x y) dx g(y)e iky dy 2π T T ( ) 1 f(x)e ikx dx g(y)e iky dy = 2π ˆf k ĝ k. 2π T T Here, we have used Fubini theorem. 27 / 94

28 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Properties of Fourier series expansion Regularity and decay Convergence Theory If f is smooth, then its Fourier coefficients decays very fast. Lemma If f C n (T), then ˆf k = o( k n ). Proof. 1 We shall show the case O( k n ) first. 2 By taking integration by part n times, we have ˆf k = 1 π f(x)e ikx dx 2π π 1 1 π = f (n) (x)e ikx dx ( ik) n 2π Thus, if f C n, then π ˆf k 1 1 n f (n) (x) dx = O( k n ). 28 / 94

29 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Properties of Fourier series expansion Remark Convergence Theory ˆf k measures the oscillation of f at scale π/k. We notice that Hence, ˆf k = 1 2π ˆf k = 1 π 2π π = 1 π 2π π := 1 π 2π π π π f(x)e ik(x+π/k) dx f(x)e ikx dx f(x) f(x π/k) e ikx dx 2 D π/k f(x)e ikx dx 29 / 94

30 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Properties of Fourier series expansion Riemann-Lebesgue Lemma Convergence Theory Let s return to study the decay of ˆf k for continuous function f. Lemma If f C(T), then ˆf k = o( k ). Proof. 1 We have seen that ˆf k = 1 2π = 1 2π π π π π f(x)e ikx dx f(x) f(x π/k) e ikx dx 2 30 / 94

31 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Properties of Fourier series expansion Convergence Theory 2 When f is continuous on T, it is uniformly continuous on T. Thus, for any ɛ > 0, we can find K > 0 such that for all k > K we have f(x) f(x π/k) 2 < ɛ. 3 From this, we obtain ˆf k 1 π f(x) f(x π/k) 2π e ikx dx < ɛ. 2 π 31 / 94

32 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Properties of Fourier series expansion Convergence Theory Riemann-Lebesgue Lemma When f is not smooth, say in L 1, we still have ˆf k 0 as k. This is the following Riemann-Lebesgue lemma. Lemma (Riemann-Lebesgue) If f is in L 1 (a, b), then ˆf k := b a f(x)e ikx dx 0, as k. 32 / 94

33 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Properties of Fourier series expansion Convergence Theory Proof. 1 For f L 1 (a, b), we have ˆf k b a f(x)e ikx dx f 1 for all k. 2 Any function f L 1 (a, b) can be approximated by a continuous function g C[a, b] in the L 1 sense. That is, for any ɛ > 0, there exists g C[a, b] such that f g 1 < ɛ. 3 For g C[a, b], we have: for any ɛ > 0, there exists a K > 0 such that for k > K, we have ĝ k < ɛ. 4 Combining these two, we get ˆf k ĝ k + ˆf k ĝ k ĝ k + f g 1 < 2ɛ. Thus, ˆf k 0 as k. 33 / 94

34 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Properties of Fourier series expansion Convergence Theory Remarks. 1 If f is a Dirac delta function, we can also define its Fourier transform ˆf k = 1 π δ(x)e ikx dx = 1 2π π 2π. In this case, δ L 1 and ˆδ k = 1/2π does not converge to 0 as k. 2 If f is a piecewise smooth function with finite many jumps, then it holds that ˆf k = O(1/k). One may consider f has only one jump first. Then f is a superposition of a step function g and a smooth function h. We have seen that ĥk decays fast. For the step function g, we have ĝ k = O(1/k). 34 / 94

35 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Properties of Fourier series expansion Convergence Theory Convergence Theory Let denote the partial sum of the Fourier expansion by f N : f N (x) := N k= N ˆf k e ikx. We shall show that under proper condition, f N will converge to f. Uniform convergence for smooth functions, Convergence in L 2 sense for L 2 functions Convergence in pointwise sense for BV functions. 35 / 94

36 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Properties of Fourier series expansion Convergence Theory Uniform convergence for smooth functions Theorem If f is a 2π-periodic, C -function, then for any n > 0, there exists a constant C n such that sup f N (x) f(x) C n N n. (2.2) 36 / 94

37 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Properties of Fourier series expansion Proof Convergence Theory 1 Express f N = D N f, where D N (x) := k N e ikx = sin(n + 1/2)x sin(x/2) f N (x) := ˆf k e ikx = 1 π f(y)e ik(x y) dy 2π π k N k N = 1 π sin(n + 1 )(x y) 2 2π π sin( 1 f(y) dy (x y)) 2 = 1 π sin(n )t 2π π sin t 2 f(x + t) dt = 1 2π π D N (t)f(x + t) dt π 37 / 94

38 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Properties of Fourier series expansion Convergence Theory 2 Using π 0 D N(x)dx = π, we have f N (x) f(x) = 1 π 2π := 1 2π π π π sin(n + 1)t 2 sin t (f(x + t) f(x)) dt 2 sin((n + 1 )t)g(x, t) dt 2 3 The function g(x, t) := (f(x + t) f(x))/ sin(t/2) = 1 0 f (x + st) ds t/ sin(t/2) is 2π periodic and in C. 4 We can apply integration-by-part n times to arrive f N (x) f(x) = (N+ 1 π ( 1)n/2 ) n t n g(x, t) sin((n+ 1 )t) dt 2 2π π 2 for even n. Similar formula for odd n. 38 / 94

39 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Properties of Fourier series expansion Convergence Theory 5 Thus, we get sup f N (x) f(x) CN n x = O(N n ). n t g(x, t) sin((tn )t) dt Remark. The constant C n, which depends on g (n) dt, is in general not big, as compared with the term N n. Hence, the approximation (2.2) is highly efficient for smooth functions. For example, N = 20 is sufficient in many applications. The accuracy property (2.2) is called spectral accuracy. 39 / 94

40 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Properties of Fourier series expansion Convergence Theory L 2 Convergence Theory Theorem If f L 2 (T), then the Fourier expansion f N (f) f in L 2 (T). That is, {e ikx k Z} is an orthonormal basis in L 2 (T). Proof. In the proof below, I shall use the fact that C (T) is dense in L 2 (T). Therefore I shall prove the case for smooth function, then use approximation argument to show it is also true for general L 2 functions. 40 / 94

41 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Properties of Fourier series expansion Convergence Theory Method 1 1 We have seen that C can be approximated by trigonometric polynomials. That is, C (T) U, where U = {e ikx k Z}. 2 From C (T) = L 2 (T), we get L 2 (T) = U. 41 / 94

42 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Properties of Fourier series expansion Method 2 Convergence Theory 1 We show that if f e ikx for all k Z, then f = 0. 2 Case 1: f is smooth. From its finite Fourier expansion f N, which is zero, converges to f, we get that f 0. 3 Case 2: f L 2 (T). We regularize f by f ɛ := ρ ɛ f C. Their Fourier coefficients f ɛ are ( ρ ɛ f) = ( ρ ɛ ) k ( ˆf) k = 0. Thus, f ɛ 0. k 4 From f ɛ f in L 2 (T), we get f 0 also. 42 / 94

43 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Properties of Fourier series expansion Convergence Theory Remark By the general theorem of orthogonal basis in Hilbert space, we have seen the equivalence of each statement below. (a) U = {0} (b) U = L 2 (T) (c) Parvesal equality: f 2 = k ˆf k 2. Notice that the expansion is unordered. Nevertheless, we shall still discuss the L 2 convergence in more detail below. 43 / 94

44 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Properties of Fourier series expansion Fourier transform Convergence Theory We may view the Fourier transform maps L 2 (T) space into l 2 (Z) space. L 2 (T) is endowed with the inner product (f, g) := 1 2π π π f(x)g(x) dx l 2 (Z) is endowed with inner product (a, b) := k a kb k. The Bessel s inequality k= ˆf k 2 f 2 says that the Fourier transform maps continuously from L 2 (T) to l 2 (Z). 44 / 94

45 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Properties of Fourier series expansion Convergence Theory Theorem (Isometry property) The Fourier transform is an isometry from L 2 (T) to l 2 (Z): (f, g) = k ˆf k ĝ k. 1 We show the case when f is smooth and g L 2 (T). From the convergence theorem for smooth f, we get (f, g) = 1 2π = 1 2π = 1 2π π π π f(x)g(x) dx π k π k π ˆf k e ikx g(x) dx ˆf k e ikx g(x) dx = k ˆf k ĝ k. 45 / 94

46 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Properties of Fourier series expansion Convergence Theory The interchange of k and π π is due to that k converges fast and is independent of x (from smoothness of f). 2 We show the case when f, g L 2 (T). we approximate f by f ɛ := ρ ɛ f, which are in C and converge to f in L 2. The isometry property is valid for f ɛ and g: (f ɛ, g) = ( f ɛ, ĝ). As ɛ 0, (f ɛ f, g) f ɛ f g 0, ( f ɛ ˆf, ĝ) f ɛ ˆf ĝ f ɛ f g 0. The last inequality is from the Bessel inequality. Thus, we obtain (f, g) = ( ˆf, ĝ). 46 / 94

47 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Properties of Fourier series expansion Convergence Theory The isometry property says that the Fourier transformation preserves the inner product. When g = f in the above isometry property, we obtain the following Parseval identity. Corollary (Parseval identity) For f L 2, we have f 2 = k ˆf k 2. Theorem (L 2 -convergence theorem) If f L 2, then f N = N k= N ˆf k e ikx f in L / 94

48 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Properties of Fourier series expansion Convergence Theory Proof. 1 First, the sequence {f N } is a Cauchy sequence in L 2. This follows from f N f M 2 = N k <M ˆf k 2 and the Bessel inequality. 2 Suppose f N converges to g. Then it is easy to check that the Fourier coefficients of f g are all zeros. From the Parvesal identity, we have f = g. 48 / 94

49 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Properties of Fourier series expansion Convergence Theory BV functions A function is called a BV function (or a function of finite total variation) on an interval (a, b), if for any partition π = {a = x 0 < x 1 < < x n = b}, f BV := sup π f(x i ) f(x i 1 ) <. i Property of BV function: its singularity can only be jump discontinuities, i.e., at a discontinuity, say, x 0, f has both left limit f(x 0 ) and right limit f(x 0 +). 49 / 94

50 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Properties of Fourier series expansion Convergence Theory A BV function f can be decomposed into f = f 0 + f 1, where f 0 is a piecewise constant function andf 1 is absolutely continuous (i.e. f 1 is differentiable and f 1 is integrable). The jump points of f 0 are countable. The BV-norm of f is exactly equal to f BV = [f(x i )] + f 1(x) dx. i where x i are the jump points of f (also f 0 ) and [f(x i )] := f(x i +) f(x i ) is the jump of f at x i. 50 / 94

51 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Properties of Fourier series expansion Convergence Theory Fourier expansion for BV function Theorem (Fourier inversion theorem for BV functions) If f is in BV (T), then for any x, f N (x) := N k= N ˆf k e ikx 1 (f(x+) + f(x )) / 94

52 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Properties of Fourier series expansion Proof Convergence Theory 1 Recall that Here, π f N (x) = 1 2π ( 0 = π D N (x y)f(y) dy π ) + D N (t)f(x + t) dt π = f + N (x) + f N (x). D N (x) = k N 0 e ikx = sin(n + 1/2)x sin(x/2) 2 Using π 0 sin(n + 1/2)x sin(x/2) dx = π, 52 / 94

53 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Properties of Fourier series expansion Convergence Theory we have f + N (x) 1 2 f(x+) = 1 2π := 1 2π π 0 π 0 sin(n + 1)t 2 sin t (f(x + t) f(x)) dt 2 sin((n + 1 )t)g(t) dt 2 3 From f being in BV, the function g(t) is in L 1 (0, π). By the Riemann-Lebesgue lemma, f + N (x) 1 f(x+) 0 as 2 N. Similarly, we have f N (x) f(x ) 0 as N. 53 / 94

54 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Properties of Fourier series expansion Gibbs phenomenon Convergence Theory In applications, we encounter piecewise smooth functions frequently. In this case, the approximation is not uniform. An overshoot and undershoot always appear across discontinuities. Such a phenomenon is called Gibbs phenomenon. Since a BV function can be decomposed into a piecewise constant function and a smooth function, we concentrate to the case when there is only one discontinuity. The typical example is the function { 1 for 0 < x < π f(x) = 1 for π < x < 0 54 / 94

55 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Properties of Fourier series expansion Convergence Theory The corresponding f N is f N (x) = 1 x sin((n )t) dt 1 x+π sin((n )t) dt 2π x π sin(t/2) 2π x sin(t/2) First, we show that we may replace 1 2 sin(t/2) by 1 t with possible error o(1/n). This is because the function 1 1 is in t 2 sin(t/2) C1 on [ π, π] and the Riemann-Lebesgue lemma. Thus, we have f N (x) = 1 x sin((n )t) dt 1 x+π sin((n )t) dt + o(1/n) π x π t π x t = 1 x(n+1/2) sinc(t) dt 1 (x+π)(n+1/2) sinc(t) dt + o(1/n). π (x π)(n+1/2) π x(n+1/2) 55 / 94

56 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Properties of Fourier series expansion Convergence Theory Here, the function sinc(t) := sin(t)/t. It has the following properties: sinc(t) dt = π/2. 0 For any z > 0, sinc(t) dt = O ( ) 1 z z The formal can be obtained by considering F (x) := tx sin t dt. 0 e t To see the latter inequality, we rewrite ( nπ sinc(t) dt = + z z k n (n+1)π nπ ) sinc(t) dt where n = [z/π] + 1. Notice that the series is an alternating series. Thus, the series is bounded by its leading term, which is of O(1/z). Let us denote the integral z 0 sinc(t) dt by Si(z). 56 / 94

57 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Properties of Fourier series expansion Convergence Theory To show that the sequence f N does not converge uniformly, we pick up x = z/(n + 1/2) with z > 0. After changing variable, we arrive z f N ( (N + 1/2) ) = 1 z sinc(t) dt 1 z+(n+1/2)π sinc(t) dt + o(1/n) π z (N+1/2)π π z = 1 z sinc(t) dt 1 sinc(t) dt + O(1/(z + N)) + O(1/(z N)) π π z = 2 z sinc(t) dt + (1/(z + N)) + O(1/(z N)) π 0 = 1 2 sinc(t) dt + (1/(z + N)) + O(1/(z N)) π z In general, for function f with arbitrary jump at 0, we have z [f] f N ( ) = f(0+) (N + 1/2) π sinc(t) dt + (1/(z + N)) + O(1/(z N)) z = f(0+) + O(1/z) + O(1/(z N)). where, the jump [f] := f(0+) f(0 ). 57 / 94

58 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Properties of Fourier series expansion Convergence Theory We see that the rate of convergence is slow if z = N α with 0 < α < 1. This means that if the distance of x and the nearest discontinuity is N 1+α, then the convergent rate at x is only O(N α ). If the distance is O(1), then the convergent rate is O(N 1 ). This shows that the convergence is not uniform. The maximum of Si(z) indeed occurs at z = π where 1 Si(π) π This yields π f N ( ) = f(0+) (f(0+) f(0 )). N + 1/2 Hence, there is about 9% overshoot. This is called Gibbs phenomenon. 58 / 94

59 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Hurwitz s proof of Fourier for isoperimetric series inequality Weyl s ergodic the Outline 1 Orthogonal bases, general theory Characterization of orthonormal basis Existence of bases, Gram-schmidt orthogonalization 2 The Fourier basis in L 2 (T) Properties of Fourier expansion Convergence Theory 3 Applications of Fourier series Hurwitz s proof for isoperimetric inequality Weyl s ergodic theorem Characterization of Sobolev spaces Solving heat equation on circle Solving Laplace equation on a disk Stability analysis for finite difference methods 59 / 94

60 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Hurwitz s proof of Fourier for isoperimetric series inequality Weyl s ergodic the Hurwitz s proof for isoperimetric inequality (see Hunter s book) The isoperimetric inequality involves to find the maximal area enclosed by a closed curve with given perimeter. If the perimeter is L, the area is A, then the isoperimeter inequality is 4πA L 2. The equality holds when the closed curve is a circle. There are many proofs of this inequality. In 1902, Hurwitz provided a proof using Fourier expansion. Let us show his proof here as an application of Fourier expansion. 60 / 94

61 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Hurwitz s proof of Fourier for isoperimetric series inequality Weyl s ergodic the Let the closed curve is given by (x, y) = (f(s), y(s)), where s is the arc length. We may assume the length of the curve is 2π, otherwise we rescale it by (x, y) by (2πx/L, 2πy/L). Since s is the arc length, we have f(s) 2 + ġ(s) 2 = 1. The area of the enclosed region is given by A = 1 f(s)ġ(s) g(s) f(s) 2 ds. T Our goal is to maximize A subject to the perimeter constraint T f(s) 2 + ġ(s) 2 ds = 2π. 61 / 94

62 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Hurwitz s proof of Fourier for isoperimetric series inequality Weyl s ergodic the We expand f and g in Fourier series: f(s) = ˆf n e ins, g(s) = n= n= From the Parvesal equality: 1 f(s) 2π 2 ds = n 2 ˆf n 2. n Z Thus, for arc length, we have T 1 = n Z n 2 ( ˆf n 2 + ĝ n 2 ). ĝ n e ins For the area functional, we get A π = 1 fġ gf 2π ds = T n Z ˆf n inĝ n ĝ n in ˆf n 62 / 94

63 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Hurwitz s proof of Fourier for isoperimetric series inequality Weyl s ergodic the Subtracting these two series, we get 1 A π = n 0 ( n ˆf n iĝ n 2 + nĝ n + i ˆf n 2 + (n 2 1)( ˆf ) n 2 + ĝ n 2 ). We then get 1 A π 0. The equality holds only when ˆf n = ĝ n = 0 for all n 2 and ˆf 1 = iĝ 1. Plug this into the arc length constraint, we get Thus, ˆf 1 = 1 2 e iδ, ĝ 1 = i 2 e iδ. f(s) = x 0 + cos(s + δ), g(s) = y 0 + sin(s + δ). 63 / 94

64 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Hurwitz s proof of Fourier for isoperimetric series inequality Weyl s ergodic the Weyl s ergodic theorem In discrete dynamical systems, the dynamics of x n is characterized by an iterative map x n+1 = F (x n ). The simplest example is x n T and F (x n ) = x n + 2πγ. It is arisen from the Poincare section map of the flow on T 2 : (x, y) (x(t), y(t)) := (x+2πω x t, y+2πω y t), ω x, ω y R. The cross section is taken to be y 0( mod 2π). Then the trajectory with y 0 = 0 will revisit y = 0 at time with 2πω y t = 2πn, that is, t n = n/ω y. The corresponding x(t n ) = x + 2πnω x /ω y. If we call γ := ω x /ω y, then the map: x(t n ) x(t n+1 ) is the above linear discrete map. 64 / 94

65 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Hurwitz s proof of Fourier for isoperimetric series inequality Weyl s ergodic the For a continuous function f : T C, we are interested in two kinds of averages: phase space average: f s := 1 2π Tf(x) dx, time average: f t := lim N 1 N+1 N n=0 f(xn ). Theorem (Weyl ergodic 1916) If γ is irrational, then for all f C(T) and for all x 0 T. f t (x 0 ) = f s (3.3) 65 / 94

66 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Hurwitz s proof of Fourier for isoperimetric series inequality proof Weyl s ergodic the 1 It is easy to check that (3.3) holds for all f = e imx : lim N 1 N + 1 f(xn ) = lim N 1 N + 1 e imx0 = lim N N + 1 N n=0 e im(x0 +2πnγ) ( ) 1 e 2πimγ(N+1) = 0. 1 e 2πimγ Here, we have used e 2πimγ 1 for irrational γ. On the other hand, e imx s = 1 e imx dx = 0. 2π T Thus, e imx t = e imx s. With this, (3.3) also holds for all trigonometric polynomials. 66 / 94

67 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Hurwitz s proof of Fourier for isoperimetric series inequality Weyl s ergodic the 2 The trigonometric polynomials are dense in C(T). 3 Given any f C(T), for any ɛ > 0, there exists a trig. polynomial p such that f p < ɛ, 1 N f(x n 1 N ) f s 2ɛ+ p(x n ) p N + 1 s N + 1 n=0 n=0 Taking limit sup, we get 1 lim sup N N + 1 N f(x n ) f s 2ɛ. n=0 Thus, (3.3) also holds for any f C(T). 67 / 94

68 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Hurwitz s proof of Fourier for isoperimetric series inequality Weyl s ergodic the Characterization of Sobolev spaces Let H m (T) be the completion of C (T) under the norm u 2 H := m u 2 + u u (m) 2. From û k = ikû k, we get û(m) k = (ik) m û k. From Parseval equality, we obtain u 2 = û k 2,, u (m) 2 = k 2m û k 2. k Z k Z Thus, we have u 2 H m = k Z(1 + k k 2m ) û k 2. The regularity of u is characterized by the decay of û k. 68 / 94

69 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Hurwitz s proof of Fourier for isoperimetric series inequality Weyl s ergodic the Sobolev space with general exponents Notice that for a fixed m 0, (1 + k ) 2m (1 + k 2 ) m (1 + k 2m ) for all k Z. This means that there are positive constants C i such that (1+ k ) 2m C 1 (1+ k 2 ) m C 2 (1+ k 2m ) C 3 (1+ k ) 2m k We can define the Sobolev norm u 2 H m (1 + k ) 2m û k 2. k Z with m R by 69 / 94

70 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Hurwitz s proof of Fourier for isoperimetric series inequality Sobolev embedding Weyl s ergodic the Theorem For m > 1/2, we have H m (T) C(T). Proof 1 For any smooth function f, we have f(x) = ˆf k e ikx k (1 + k ) m (1 + k ) m ˆfk k ( ) 1/2 ( ) 1/2 (1 + k ) 2m (1 + k ) 2m ˆf k 2 k k When m > 1/2, k Z (1 + k ) 2m < thus, we obtain f C f H m 70 / 94

71 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Hurwitz s proof of Fourier for isoperimetric series inequality Weyl s ergodic the 2 For any f H m (T), we can approximate f by f N in H m. (Check by yourself) From f N f M C f N f M H m we get that f N is Cauchy in uniform norm. Thus it converges to f in. 71 / 94

72 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Hurwitz s proof of Fourier for isoperimetric series inequality Heat equation on a circle Weyl s ergodic the We can solve the heat flow on circle exactly. This problem indeeds motivated Fourier invent the Fourier expansion. Let us consider u t = u xx, x T, t > 0, with initial data u(x, 0) = f(x). If we expand u(x, t) = n Z u n(t)e inx, then, formally, u n e inx = n 2 u n e inx n Z n Z and f n e inx. n Z u n (0)e inx = n Z 72 / 94

73 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Hurwitz s proof of Fourier for isoperimetric series inequality Weyl s ergodic the Since {e inx n Z} are independent, we get u n = n 2 u n, u n (0) = f n. Thus, u n (t) = f n e n2t. Thus, we define the function u(x, t) = n Z ˆf n e n2t e inx. In the following, we need to check: 1 u, u t and u xx exist and u t = u xx for t > 0 and x T; 2 u(, t) f in L 2 (T) as t / 94

74 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Hurwitz s proof of Fourier for isoperimetric series inequality Proof of (1) Weyl s ergodic the We show that u x exists here. It is clearly that n Z ˆf n e n2t e inx converges absolute and uniformly w.r.t. x for t > 0, as long as ˆf n grows at most algebraically in n. Since n Z ine n2 t ˆfn e inx converges absolute and uniformly w.r.t. x for t > 0. This implies u is differentiable in x and the differentiation can be interchange with the infinite summation: x u = x ˆf n e n2t = ˆf n e n2t ine inx. n Z n Z Similar proof for the existence of u xx and u t for t > 0. Since the Fourier coefficients of u t and u xx are identical on t > 0, we thus get u t = u xx. 74 / 94

75 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Hurwitz s proof of Fourier for isoperimetric series inequality Proof of (2) Weyl s ergodic the Let us denote u(, t) by T (t)f and itself Fourier transform û k (t) by ˆT (t) f. That is, T (t) f n = e n2 t ˆfn. Goal: prove T (t)f f in L 2 (T), or T (t) f f in l 2 (Z) lim T (t) f f 2 2 = lim t 0+ t 0+ = n Z n Z (e n2t 1) 2 f n 2 lim t 0+ (e n2t 1) 2 f n 2 = 0. The interchange of and lim here is due to the dominant convergence theorem and (e n2t 1) 2 f n f n 2 <. n Z n Z 75 / 94

76 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Hurwitz s proof of Fourier for isoperimetric series inequality Weyl s ergodic the Solving Laplace equation on a disk We consider the Laplace equation u xx + u yy = 0 on the domain Ω : x 2 + y 2 < 1, with the Dirichlet boundary condition: u = f on Ω. In the polar coordinate, the equation has the form: u rr + 1 r u r + 1 r u 2 θθ = 0. The boundary condition is u(1, θ) = f(θ), θ T. 76 / 94

77 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Hurwitz s proof of Fourier for isoperimetric series inequality Weyl s ergodic the The solution can be expanded as u(r, θ) = n Z u n (r)e inθ. Plug this into the Laplace equation, we get (u n + 1r ) u n n2 e inθ = 0. This leads to n Z u n + 1 r u n n2 = 0 for all n Z. r2 The two independent solutions are u n = r n or u n = r n. But only r n is finite at r = 0. Thus, we obtain r 2 u(r, θ) = n Z a n r n e inθ. 77 / 94

78 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Hurwitz s proof of Fourier for isoperimetric series inequality Weyl s ergodic the At r = 1, we get f(θ) = n Z a n e inθ. Thus, a n = 1 f(θ)e inθ dθ. 2π T For r < 1, the L 2 norm of n Z a nr n e inθ is bounded by n Z a n 2, uniformly in r < 1. Thus, from dominant convergence theorem, we have lim u(r, ) = f( ) in r 1 L2 (T). 78 / 94

79 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Hurwitz s proof of Fourier for isoperimetric series inequality Weyl s ergodic the If we differentiate the infinite series in r term-by-term, we get a n n r n 1 e inθ. n Z This infinite series converges absolutely and uniformly for r r 0 for any fixed r 0 < 1. This implies that u is differentiable in r and the differentiation can be performed term-by-term in the infinite series: r u = n Z a n n r n 1 e inθ. By the same argument, we get rr u and θθ u exist and u satisfies the Laplace equation in polar coordinate form. 79 / 94

80 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Hurwitz s proof of Fourier for isoperimetric series inequality Weyl s ergodic the Alternatively, we can write the above summation in convolution form: u(r, θ) = where g(r, θ) = 1 2π = 1 2π T g(r, θ φ)f(φ) dφ, r n e inθ n Z ( ) 1 e iθ + 1 reiθ 1 e iθ = 1 1 r 2 2π 1 2r cos θ + r 2 The function g is called the Poisson kernel. It is infinitely differentiable for r < 1. This implies g f C (Ω). 80 / 94

81 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Hurwitz s proof of Fourier for isoperimetric series inequality Weyl s ergodic the Von Neumann stability analysis for finite difference methods In numerical PDEs, the stability analysis is a crucial step to the convergence theory of a numerical scheme. Below, I shall demonstrate the von Neumann stability analysis for heat equation in one dimension. It is a L 2 stability analysis suitable for for (the interior part of) numerical PDEs with constant coefficients. Let us consider the heat equation: u t = u xx in one dimension with initial data u(x, 0) = f(x). 81 / 94

82 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Hurwitz s proof of Fourier for isoperimetric series inequality Weyl s ergodic the Meshes Let h = x, k = t be the spatial and temporal mesh sizes. Define x j = jh, j Z and t n = nk, n 0. Let us abbreviate u(x j, t n ) by u n j. We shall approximate u n j by Uj n, where Uj n satisfies some finite difference equations. 82 / 94

83 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Hurwitz s proof of Fourier for isoperimetric series inequality Weyl s ergodic the Spatial discretization The simplest one is to use the centered finite difference approximation for u xx : u xx = u j+1 2u j + u j 1 h 2 + O(h 2 ) This results in the following systems of ODEs U j (t) = U j+1(t) 2U j (t) + U j 1 (t) h 2 In vector form U = 1 h 2 AU where U = (U 0, U 1,...) t, A = diag (1, 2, 1). 83 / 94

84 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Hurwitz s proof of Fourier for isoperimetric series inequality Temporal discretization Weyl s ergodic the We can apply numerical ODE solvers Forward Euler method: Backward Euler method: 2nd order Runge-Kutta (RK2): U n+1 = U n + k h 2 AU n (3.4) U n+1 = U n + k h 2 AU n+1 (3.5) U n+1 U n = k h 2 AU n+1/2, U n+1/2 = U n + Crank-Nicolson: U n+1 U n = k 2h 2 AU n (3.6) k 2h 2 (AU n+1 + AU n ). (3.7) 84 / 94

85 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Hurwitz s proof of Fourier for isoperimetric series inequality Weyl s ergodic the These linear finite difference equations can be solved formally as where U n+1 = GU n Forward Euler: G = 1 + k h 2 A, Backward Euler: G = (1 k A) 1, ( h 2 RK2: G = 1 + k A + 1 k ) 2 h 2 2 h A 2 2 Crank-Nicolson: G = 1+ k 2h 2 A 1 k 2h 2 A For the Forward Euler, We may abbreviate it as where U n+1 j = G(U n j 1, U n j, U n j+1), (3.8) G(U j 1, U j, U j+1 ) = U j + k h 2 (U j 1 2U j + U j+1 ) 85 / 94

86 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Hurwitz s proof of Fourier for isoperimetric series inequality Stability and Convergence Weyl s ergodic the Convergence theory: Uj n mesh sizes h, k 0. converges to u(x j, t n ) as the Stability theory: U n remain bounded in some norm n. We first see the truncation error produced by a true solution by the finite difference equation. Plug a true solution u(x, t) into (3.4): where u n+1 j u n j = k h 2 ( u n j+1 2u n j + u n j 1) + kτ n j (3.9) τ n j = D t,+ u n j (u t ) n j (D + D u n j (u xx ) n j ) = O(k) + O(h 2 ). 86 / 94

87 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Hurwitz s proof of Fourier for isoperimetric series inequality Weyl s ergodic the Let e n j denote for u n j U n j. Then subtract (3.4) from (3.9), we get e n+1 j e n j = k h 2 ( e n j+1 2e n j + e n j 1) + kτ n j. (3.10) This can be expressed in operator form: e n+1 = Ge n + kτ n. (3.11) e n Ge n 1 + k τ n 1 G 2 e n 2 + k( Gτ n 2 + τ n 1 ) G n e 0 + k( G n 1 τ Gτ n 2 + τ n 1 ) 87 / 94

88 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Hurwitz s proof of Fourier for isoperimetric series inequality Weyl s ergodic the Suppose G satisfies the stability condition G n U C U for some C independent of n. Then e n C e 0 + C max m τ m. If the local truncation error has the estimate and the initial error e 0 satisfies max τ m = O(h 2 ) + O(k) m e 0 = O(h 2 ), then so does the global true error satisfies e n = O(h 2 ) + O(k) for all n. 88 / 94

89 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Hurwitz s proof of Fourier for isoperimetric series inequality Weyl s ergodic the Definition 1 A finite difference method U n+1 = G h,k (U n ) is called consistent if its local truncation error τ satisfies τ h,k 0 as h, k 0. 2 It is called stable under the norm in a region (h, k) R if G n h,ku C U for all n with nk fixed. Here, C is a constant independent of n. 3 It is called convergence if the true error e h,k 0 as h, k / 94

90 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Hurwitz s proof of Fourier for isoperimetric series inequality Weyl s ergodic the For forward Euler method for the heat equation, stability + consistency convergence. We have seen ( ). The other way is simple for you to figure out. 90 / 94

91 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Hurwitz s proof of Fourier for isoperimetric series inequality L 2 Stability von Neumann Analysis Weyl s ergodic the Given {U j } j Z, we define and its Fourier transform The advantages: U 2 = j Z Û(ξ) = 1 2π U j 2 U j e ijξ. j Z the shift operator is transformed to a multiplier: T U(ξ) = e iξ Û(ξ), where (T U) j := U j+1 ; the Parseval equility: U 2 = Û / 94

92 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Hurwitz s proof of Fourier for isoperimetric series inequality Weyl s ergodic the If a finite difference scheme is expressed as U n+1 j = (GU n ) j = m a i (T i U n ) j, i= l then Û n+1 (ξ) = Ĝ(ξ)Û n (ξ). From the Parseval equality, U n+1 2 = Û n+1 2 = π π Ĝ(ξ) 2 Û n (ξ) 2 dξ π max ξ Ĝ(ξ) 2 Û n (ξ) 2 dξ = Ĝ 2 U 2 π 92 / 94

93 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Hurwitz s proof of Fourier for isoperimetric series inequality Necessary and sufficient conditions for stability Weyl s ergodic the Thus a sufficient condition for stability is Ĝ 1. (3.12) Conversely, suppose Ĝ(ξ 0) > 1, fromĝ being smooth, we can find ɛ and δ such that Ĝ(ξ) 1 + ɛ for all ξ ξ 0 < δ. Let us choose U 0 in l 2 (Z) such that Û 0 (ξ) = 1 for ξ ξ 0 δ. Then Û n 2 = Ĝ 2n (ξ) Û 0 2 Ĝ 2n (ξ) Û 0 2 ξ ξ 0 δ (1 + ɛ) 2n δ as n Thus, the scheme can not be stable. 93 / 94

94 Orthogonal bases, general theory The Fourier basis in L 2 (T) Applications Hurwitz s proof of Fourier for isoperimetric series inequality Weyl s ergodic the Theorem A finite difference scheme U n+1 j = m a k Uj+k n k= l with constant coefficients is stable if and only if m Ĝ(ξ) := a k e ikξ k= l satisfies max Ĝ(ξ) 1. (3.13) π ξ π 94 / 94