Department Mathematics Stanford University Math 175 Homework 9 This hw will not be graded; solutions will be posted Tue June 2 1. Prove the claim made in lecture that if f 2 L2 .OEa; b/ then there is a sequence {'k }kD1;2;::: of step Rb functions on OEa; b with kf 'k k2 ! 0, where k k2 denotes the L2 .OEa; b/ seminorm . a jf 'k j2 /1=2 . Hint: First check that f 2 L2 .R/ ) jf j 2 L2 .R/ and hence that f D 1 .jf jf / 2 L2 .R/, and show 2 that hence it suffices to check the result claimed in case f 0. Now think about using Lemma 2(ii) of the lecture supplement on f and f 2 . Solution: From lecture jf j 2 L1 and jf j2 D f 2 is in L1 by definition, hence jf j 2 L2 by definition. Then since L2 is a linear space (from lecture), we do indeed have f D 1 .jf j f / 2 L2 .OEa; b/. Thus 2 it is enough to prove the result in case f 0, so we assume that. Then fk D min{f; k }, k D 1; 2; : : : Rb Rb is a bounded non-negative L2 .OEa; b/ function, and kf fk k2 D a .f fk /2 a .f fk / .f Cfk / D 2 Rb 2 2 fk / ! 0 by the monotone convergence theorem. On the other hand, for each k , by Lemma 2 a .f of the lecture supplement there is a step function 'k such that kfk 'k k1 2 k , and (by replacing 'k by the function min{k; max{0; 'k }}) we can assume 0 'k k for each k , so then (since Rb Rb fk C 'k 2k ) a jfk 'k j2 a jfk 'k j.fk C 'k / 2kkfk 'k k1 2k 2 k ! 0. Then by the triangle inequality kf 'k k2 D kf fk C fk 'k k2 kf fk k2 C kfk 'k k2 ! 0. 2. Find explicitly all eigenvalues/eigenfunctions, and show zero is not an eigenvalue, of the SturmLiouville problem (SL) of lecture in the cases (i) OEa; b OE0; , p 1; 1; q 0, D D 1; D i D 0 (so the boundary conditions are f .0/ D f . / D 0 and the equation is f 00 D f ), and (ii) OEa; b OE0; , p 1; 1; q 1, D D 0; D i D 1 (so the boundary conditions are 0.0/ D f 0. / D 0 and the equation is f 00 C f D f ). f Solution (i): As discussed in lecture, for D 0 any C 2 .OEa; b/ solution of the equation can be written in the form u D C1 C C2 x and there are no corresponding non-trivial solutions which satisfy the boundary conditions, so D 0 is not an eigenvalue. If < 0 we can write D ! 2 with ! > 0 and in this case the general solution can be written u.x/ D C1 cosh !x C C2 sinh !x ; the boundary condition u.0/ D 0 implies C1 D 0 and then the boundary condition u. / D 0 cannot be satisfied unless C2 D 0 also. Thus again there are no non-trivial solutions. Finally if > 0 we can write D ! 2 with ! > 0 and in this case the general solution can be written u.x/ D C1 cos !x C C2 sin !x ; the boundary condition u.0/ D 0 implies that C1 D 0 and then the boundary condition u. / D 0 is satisfied with C2 0 if and only if ! D n, n D 1; 2; : : :, so the eigenvalues are n D n2 , n D 1; 2; : : : and the corresponding eigenfunctions (up to non-zero scalar factors) are sin nx . If we use the inner product R .f; g/ D 2 0 f gdx then sin nx has norm 1, and we know from the general Sturm-Liouville theory established in lecture that then {sin nx }nD1;2;::: is a complete orthonormal sequence for L2 .OE0; /. Solution (ii): Similar to (i). We will not give the details here. The conclusion is that the eigenvalues 1 are n D 1 C n2 ; n D 0; 1; 2; : : : with corresponding eigenfunctions p2 ; cos x; cos 2x; : : :, again giving a complete orthonormal system by the same general Sturm-Liouville theory. (Notice that thus for the 00 problem f D f with f 0.0/ D f 0. / D 0 the eigenvalues are 0; 1; 4; : : : ; n2 ; : : : with correspond1 ing complete orthonormal sequence of eigenfunctions given by p2 ; cos x; cos 2x; : : : ; cos nx; : : :.) 3. Using the result of Q.2 give a proof of the completeness of the orthonormal sequence {e i nx }nD0;1;2 in L2 .OE ; / which uses the general theorem on completeness of the non-zero eigenfunctions of K in .ker K/? , for any compact Hermitian operator K on a Hilbert H space . Note: Of course your proof should be independent of the Fejer kernel proof given earlier in the course
Hint: Observe that any function f .x/ on OE ; can be written as the sum of an even and an odd 1 function by writing fP D 1 .f .x/ C f . x// C 2 .f .x/ f . x//, and the real Fourier series (i.e. the .x/ 2 1 Fourier series a0 =2 C nD1 .an cos nx C bn sin nx/ with an ; bn real) for an odd function has all an D 0 and for an even function has all bn D 0. Solution: We know from Q.2 that for any odd function g 2 L2 .OE ; / we can write limN !1 kg PN 2 ; / we can write limN !1 kg nD1 bn sin nxk2 D 0, and likewise any even function h 2 L .OE PN i nx i nx a e i nx Ce i nx p0 and sin nx D e 2e ) and since we can nD1 an cos nxk2 D 0. So (since cos nx D 2 i 2 i nx W n D 0; 1; 2; : : :} are dense write f D g C h with g even and h odd we have shown that span{e in L2 , and hence (by definition) the orthonormal sequence e i nx ; n D 0; 1; 2; : : : is complete.
2 4. For the general Sturm-Liouville problem Lu D u; u 2 CBC .OEa; b/, of lecture, prove that all eigenvalues have multiplicity 1; that is, if is any eigenvalue of the Sturm-Liouville eigenvalue prob2 2 lem problem Lu D u; u 2 CBC .OEa; b/, then span of the set of all eigenfunctions u 2 CBC .OEa; b/ 2 with eigenvalue is a 1-dimensional subspace of CBC .OEa; b/.
Solution: From Q.3 of hw8 we know that if u; v are both eigenfunctions corresponding to one of the eigenvalues n and if u is not a constant multiple of v then we must have that .u.t /; u 0.t //; .v.t /; v 0.t // are l.i. at each point t 2 OEa; b. However .; / .u.a/; u 0.a// D 0 D .; / .v.a/; v 0.a// says that both .u.a/; u 0.a// and .v.a/; v 0.a// are orthogonal to the non-zero vector .; / and hence both of these vectors are multiples of the vector . ; / and hence are l.d., so the latter alternative (that .u.t/; u 0.t//; .v.t /; v 0.t // are l.i. at each point t 2 OEa; b) is impossible and hence u D C v for some constant C . 5. Prove that in fact the eigenvalues n of the general Sturm-Liouville problem Lu D u; u 2 2 CBC .OEa; b/ actually tend to plus infinity as n ! 1 (recall that the general theory of compact selfadjoint operators only ensures that j n j ! 1). Hint: Prove that there is a fixed constant C > 0 (depending only on p; q; ; ; ; i ) such that 2 . Lu; u/L2 C for ...