Review and problem list for Applied Math I

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1 Review and problem list for Applied Math I (This is a first version of a serious review sheet; it may contain errors and it certainly omits a number of topic which were covered in the course. Let me know if you find something wrong, or something that you think should be included. It is not meant to be comprehensive, but just to highlight the important points. Drill problems (e.g., solve the following integral equation, etc.) can be found in your text.) Preliminaries The main focus of this first semester is on linear algebra in Hilbert space - that is, linear integral and differential equations which are represented by linear algebraic equations in Hilbert space. The topics of interest to us in finite-dimensional vector spaces are those that have useful generalizations to infinite dimensional inner-product spaces. These include change of basis, orthogonal projections, least squares approximations and pseudoinverses, the maximum principle for the existence of eigenvalues, and self-adjoint operators and their spectral theory. In this section, all vector spaces are assumed finite-dimensional. 1. If W is a subspace of V, and v V, the quantity { v w : w W } is minimized when w = Π W v, where Π W : V W is the orthogonal projection. 2. Show that if A = A is a linear transformation on the complex vector space V, then there exists an orthonormal basis in which the matrix of A is diagonal. 3. If A = A and the quantity <AV,v> <v,v> assumes its maximum value of λ 1 when v = v 1, then λ 1 is an eigenvalue of A and v 1 a corresponding eigenvector. 4. What is a necessary and sufficient condition that there exist a solution to Ax = b? 5. If there is no solution to Ax = b, show that there is always a solution to A Ax = A b, and that this is a least squares solution. When is it unique? 6. How can you use the singular value decomposition to construct a pseudoinverse for an arbitrary matrix A?. Hilbert space The main thing that comes into play that s different from the finite-dimensional case is topology, in particular, whether or not various infinite-dimensional subspaces of H are closed. Suppose, for example, that {e n : n = 1, 2,...} is an infinite orthonormal set of vectors and we define V to be the set of all finite linear combinations of these. Then V is a subspace of H since it s obviously closed under linear combinations. And clearly {e n : n = 1, 2,...} is a basis for V. But V is not a closed subspace. In fact, let W = V ; it should be clear that V W : let {a n } l 2. Then w = n=1 a ne n W, but it s not in V unless {a n } is finite. What we have is that v n w, where v n = n i=1 a ie i. And W is a subspace of H (why?). Since {e n : n = 1, 2,...} is not a basis for W in the usual sense (because not everything in W can be written as a finite linear combination of the 1

2 basis elements), we need to give it a new name in this context, and we say that {e n : n = 1, 2,...} is a complete orhonormal set (c.o.n.s.) in W. A standard example to keep in mind: let L be the linear differential operator D 2. The domain of L is a subspace of H = L 2 [a, b] consisting of those functions f such that f L 2 [a, b]. Finite linear combinations of these functions are also in the domain of L, so the domain is a subspace. Its closure is all of L 2 [a, b], so the domain of L is dense in L 2 [a, b]. Definitions: complete metric space, norm, scalar product, linear functionals and operators, continuity of these, adjoint, Fourier coefficient, orthogonal projection, complete orthonornal set, closed subspace, C[a, b], L 2 [a, b], FFT, wavelets, sampling theorem and optimal filters. In the following problems, H is always assumed to be a complex separable Hilbert space. C[a, b] denotes the space of continuous functions on [a, b] in the uniform (supremum) norm. For the (T/F) questions, either prove the proposition if true or give a counterexample if false. 1. (T/F) Let A H be a subspace. Then ((A ) ) = A. 2. (T/F) The linear operator L : H H is bounded L is continuous. 3. Use the Gram-Schmidt algorithm to find an orthonormal basis for the subspace of L 2 [0, 1] spanned by {1 x, 1 + x, x 2 }. 4. Let A be a finite-dimensional subspace of H and let Π A : H A be the orthogonal projection of H onto A. Let x H be fixed, and let v be any element of A. Show that x v is minimized when v = Π A (x). 5. (T/F) If x n x in C[a, b], then x n x in L 2 [a, b]. 6. (T/F) If x n x in L 2 [a, b], then x n x in C[a, b]. 7. Find the Fourier series for f(x) = x on the interval [ π, π]. Does a 0 + a n cos(nx) + b n sin(nx) n=1 converge pointwise to x on the boundary? 8. State and prove Bessel s inequality. 9. State the Weierstrass approximation theorem and indicate roughly how it is used to prove that the trigonometric functions {1, sin(kπx, cos(kπx), k = 1, 2,...} form a c.o.n.s in L 2 [0, 1]. 10. State and prove the Fredholm alternative for a bounded linear operator L on H. 11. In L 2 [0, 1], what is the best (least squares) approximation to the function x ln x lying in the subspace spanned by the vectors e 1 = 1, e 2 = x? 2

3 Compact operators In operator form, linear integral and differential equations look like Lu = f, to be solved for u. There are the usual finite-dimensional issues: Is L one-to-one? If not, what s a basis for N(L), the null space of L? And in this case, what s the condition on f that a solution exist? The infinitedimensional issue is the same as above: are the domain and range of L closed, and if not what can be said? Compact operators have properties that are quite similar to those of matrices, largely because they are limits of finite-rank operators. But the operators that come closest to having the properties of matrices are those of the form I λk, where K is compact (These are examples of Fredholm operators.). Definitions: relatively compact set, compact linear operator, Hilbert-Schmidt operator, resolvent operator, Neumann series, contraction map 1. Show that if K is compact, then K is bounded. 2. Show that if K is compact, and {e n } H is an infinite orthonormal set, lim n Ke n = Show that if K is compact and its range is infinite-dimensional, then K 1 is unbounded. 4. If K is compact and 0 is not an eigenvalue of K, then R K, the range of K, is a proper subset of H, but R K = H. A standard example of this phenomenon: let L = D 2 on L 2 [0, 1], with boundary conditions u(0) = u(1) = 0, and let Kf(x) = 1 0 g(x, y)f(y) dy, where g is the Green s function. Then K is compact, of course, and its range is contained in {u L 2 [0, 1] u L 2 [0, 1]}. And the eigenvectors of K are the same as those of L, namely {e n (x) = 2 sin(nπx) : n = 1, 2,...} which form a c.o.n.s. for L 2 [0, 1]. 5. Show that if K is a degenerate integral kernel on L 2 [a, b], n k(x, y) = φ i (x)ψ i (y), then the integral equation u(x) = λ reduces to a (finite) matrix equation. b a i=1 k(x, y)u(y) dy + f(x) 6. If K = K is compact, and has infinitely many distinct eigenvalues λ n, then lim n λ n = Show that if K is bounded and self-adjoint, and KA A for some subspace A, then K(A ) A. 8. The adjoint of the Hilbert-Schmidt operator with integral kernel k(x, y) is Hilbert-Schmidt with kernel k (x, y) = k(y, x). 9. Let µ i be real, µ i µ i+1, and µ i 0. Let {e i } be any orthonormal set in H, and define Kx = µ i < x, e i > e i. Show that K is compact and self-adjoint. i=0 3

4 10. Let K be compact, and let L = I λk. (a) Show that L is bounded away from 0 on N(L) ; i.e., c > 0 u N(L) Lu c u. (b) Use this fact to show that the range of L is closed. 11. If K = K is compact, with eigenvalues {µ i } and corresponding orthonormal eigenvectors {φ i }, and λ 1 is not an eigenvalue of K, solve the equation u = f + λku for u as a function of f. 12. Same question when λ 1 is an eigenvalue of K with multiplicity j. 13. Show that the boundary value problem u + λu = 0; u(0) = u(r) = 0 can be converted into the integral equation u = λku, with K corresponding to the integral kernel k(x, y) = { y r x r (r x) } : 0 y x r (r y) : 0 x y r 14. When using Neumann iterates to approximate the solution to the equation (I λk)u = f estimate the error made if only the first n terms of the series are used. 15. Let K = K be compact with eigenvalues {µ i } and o.n. eigenvectors {φ i }. (a) Show that (b) If λk < 1, then sum the series Kf = i µ i < f, φ i > φ i. λ n K n f, n=0 and show that it s equal to (I + λr)f, where R is the resolvent. Can λ 1 be an eigenvalue of K? Why or why not? 4

5 Differential operators: A nice linear differential equation Lu = f with separated homogeneous boundary conditions has a unique solution of the form u = Gf, where G is the compact integral operator Gf(x) = b a g(x, y)f(y) dy. Since G = L 1, L must be an unbounded operator whose domain is a proper subspace of L 2 [a, b]. However, if the Green s function exists, then its eigenvectors are the same as those of L, and we can use our knowledge of compact operators to give a complete description of the solutions to the differential equation. Definitions: linear boundary conditions of the form Bu = Φ, formal adjoint of a linear differential operator, adjoint boundary conditions, Green s functions, distributions, test functions, weak solutions, generalized functions and their derivatives. 1. If g(x, y) is a Green s function for the differential operator L, what is the meaning of the statement Lg(x, y) = δ(x y)? 2. The operator L = D 2 + 4I; u(0) = u(π); u (0) = u (π) has no Green s function. Why? 3. For the boundary value problem (1 + x 2 )u (x) + xu (x) + u(x) = f(x), u(0) + 3u (0) = 0; u(1) 2u (1) = 0, find the adjoint operator L, and the adjoint boundary conditions L v = Show that L = a 2 D 2 + a 1 D + a 0 I is formally self-adjoint a 1 = a Show that the unique solution to the IVP Lu = f; u(0) = u (0) = 0, where L is the second order linear differential operator a 2 (x)d 2 + a 1 (x)d + a 0 (x), f, a 2, a 1, and a 0 are continuous and a 2 0 in [0, r] is given by u(x) = x 0 u 1 (y)u 2 (x) u 1 (x)u 2 (y) f(y) dy a 2 (y)w (y) where u 1 and u 2 are any two linearly independent solutions to the homeogenous equation Lu = 0, and W (y) is the Wronskian of u 1 and u Find the condition(s) for a 3rd order linear differential operator to be formally self-adjoint. 7. Give an example of a 2nd order differential operator (and BCs) such that L is formally selfadjoint, but not truly self adjoint. 8. Convert the initial value problem u (x) + xu(x) = f(x), u(0) = 0 to an integral equation of the form (I + K)u = F, where F (x) = x 0 f(s) ds. Argue that, at least in L2 [0, 1], we have K < 1, so the Neumann series converges. Try to sum the Neumann series to get u. (Hint: Solve the differential equation, so you know what you re looking for....) 9. Let L be a formally self-adjoint second order linear differential operator, and suppose that, for some separated, homogenous boundary conditions Bu on [a, b], the homogeneous problem Lu = 0, Bu = 0. has only the trivial solution u = 0. Show that the eigenfunctions of L satisfying Bu = 0 form a complete orthonornal set in L 2 [a, b]. 5

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