17.2 Weight Functions and Orthogonal Polynomials 245 en, g 2 0 for all n Z. From the above convergence result we deduce, for all⟨ f ⟩ =1([0, 2π]), ∈ ∈ C N f , g 2 lim cn(f )en, g 2 0. ⟨ ⟩ = N ⟨ ⟩ = →∞ n N !=− Since 1([0, 2π]) is known to be dense in L2([0, 2π], dx) it follows that g 0, by CorollaryC 17.2, hence by Theorem 17.2, this system is an orthonormal basis= of L2([0, 2π], dx). Therefore, every f L2([0, 2π], dx) has a Fourier expansion, which converges (in the sense of the L2-topology).∈ Thus, convergence of the Fourier series in the L2-topology is “natural,” from the point of view of having convergence of this series for the largest class of functions. 17.2 Weight Functions and Orthogonal Polynomials Not only for the interval I [0, 2π] are the Hilbert spaces L2(I,dx) separable, but for any interval I [a, b=], a<b , as the results of this section will show. Furthermore= an orthonormal−∞ ≤ basis will≤ be+∞ constructed explicitly and some interesting properties of the elements of such a basis will be investigated. The starting point is a weight function ρ : I R on the interval I which is assumed to have the following properties: → 1. On the interval I, the function ρ is strictly positive: ρ(x) > 0 for all x I. 2. If the interval I is not bounded, there are two positive constants α and∈ C such that ρ(x)eα x C for all x I. | | ≤ ∈ The strategy to prove that the Hilbert space L2(I,dx) is separable is quite simple. A n first step shows that the countable set of functions ρn(x) x ρ(x), n 0, 1, 2, ... is total in this Hilbert space. The Gram–Schmidt orthonormalization= then= produces easily an orthonormal basis. Lemma 17.1 The system of functions ρn : n 0, 1, 2, ... is total in the Hilbert space L2(I, dx), for any interval I. { = } 2 Proof For the proof we have to show: If an element h L (I,dx) satisfies ρn, h 2 0 for all n, then h 0. ∈ ⟨ ⟩ = = In the case I R we consider h to be be extended by 0 to R I and thus get a 2 ̸= \ function h L (R,dx). On the strip Sα p u iv C : u, v R, v < α , introduce the∈ auxiliary function = { = + ∈ ∈ | | } F (p) ρ(x)h(x)eipx dx. = "R The growth restriction on the weight function implies that F is a well-defined holo- morphic function on Sα (see Exercises). Differentiation of F generates the functions 246 17 Separable Hilbert Spaces ρn in this integral: dnF F (n)(p) (p) in h(x)ρ(x)xneipx dx = dpn = !R (n) n for n 0, 1, 2, ..., and we deduce F (0) i ρn, h 2 0 for all n. Since F is = = ⟨ ⟩ = holomorphic in the strip Sα it follows that F (p) 0 for all p Sα (see Theorem = ∈ 9.5) and thus in particular F (p) 0 for all p R. But F (p) √2π (ρh)(p) where is the inverse Fourier= transform (see∈ Theorem 10.1),= and weL know L 2 f , g 2 f , g 2 for all f , g L (R,dx) (Theorem 10.7). It follows that ⟨L L ⟩ = ⟨ ⟩ ∈ 2 ρh, ρh 2 (ρh), (ρh) 2 0 and thus ρh 0 L (R,dx). Since ρ(x) > 0 for⟨ x ⟩I this= ⟨ impliesL Lh 0⟩ and= we conclude. = ∈ ✷ Technically∈ it is simpler= to do the orthonormalization of the system of functions 2 ρn : n N not in the Hilbert space L (I,dx) directly but in the Hilbert space L{ 2(I, ρd∈x),} which is defined as the space of all equivalence classes of measurable functions f : I K such that f (x) 2ρ(x)dx< equipped with the inner prod- → I | | ∞ uct f , g f (x)g(x)ρ(x)dx. Note that the relation f , g ρf , ρg ρ I " ρ √ √ 2 holds⟨ for⟩ all=f , g L2(I, ρdx). It implies that the Hilbert⟨ spaces⟩ =L⟨2(I, ρdx) and⟩ L2(I,dx) are (isometrically)" ∈ isomorphic under the map L2(I, ρdx) f √ρf L2(I,dx). ∋ (→ ∈ This is shown in the Exercises. Using this isomorphism, Lemma 17.1 can be restated as saying that the system of powers of x, xn : n 0, 1, 2, ... is total in the Hilbert space L2(I, ρdx). { = } We proceed by applying the Gram–Schmidt orthonormalization to the system of powers xn : n 0, 1, 2, ... in the Hilbert space L2(I, ρdx). This gives a sequence { = } of polynomials Pk of degree k such that Pk, Pm ρ δkm. These polynomials are ⟨ ⟩ =0 defined recursively in the following way: Q0(x) x 1, and when for k 1 the = = ≥ polynomials Q0, ..., Qk 1 are defined, we define the polynomial Qk by − k 1 k − Qn, x ρ Qk(x) xk ⟨ ⟩ Qn. = − Q , Q n 0 n n ρ #= ⟨ ⟩ Finally, the polynomials Qk are normalized and we arrive at an orthonormal system of polynomials Pk: 1 Pk Qk, k 0, 1, 2, ... = Qk ρ = ∥ ∥ Note that according to this construction, Pk is a polynomial of degree k with positive coefficient for the power xk. Theorem 17.1 and Lemma 17.1 imply that the system of polynomials Pk : k 0, 1, 2, ... is an orthonormal basis of the Hilbert space L2(I, ρdx). If we{ now introduce= the} functions ek(x) Pk(x) ρ(x), x I = ∈ we obtain an orthonormal basis of the Hilbert$ space L2(I,dx). This shows Theorem 17.3. 17.2 Weight Functions and Orthogonal Polynomials 247 Theorem 17.3 For any interval I (a, b), a<b the Hilbert space 2 = −∞ ≤ ≤ +∞ L (I, dx) is separable, and the above system ek : k 0, 1, 2, ... is an orthonormal basis. { = } Proof Only the existence of a weight function for the interval I has to be shown. Then by the preceding discussion we conclude. A simple choice of a weight function for any of these intervals is for instance the exponential function ρ(x) e αx2 , = − x R, for some α > 0. ✷ ∈ Naturally, the orthonormal polynomials Pk depend on the interval and the weight function. After some general properties of these polynomials have been studied we will determine the orthonormal polynomials for some intervals and weight functions explicitly. Lemma 17.2 If Qm is a polynomial of degree m, then Qm, Pk ρ 0 for all k>m. ⟨ ⟩ = 2 Proof Since Pk : k 0, 1, 2, ... is an ONB of the Hilbert space L (I, ρdx) the { = } polynomial Qm has a Fourier expansion with respect to this ONB: Qm n∞ 0 cnPn, k = = cn Pn, Qm ρ . Since the powers x , k 0, 1, 2, ... are linearly independent = ⟨ ⟩ = ! functions on the interval I and since the degree of Qm is m and that of Pn is n, the m coefficients cn in this expansion must vanish for n>m, i.e., Qm n 0 cnPn and = = thus Pk, Qm ρ 0 for all k>m. ✷ ⟨ ⟩ = ! Since, the orthonormal system Pk : k 0, 1, 2, ... is obtained by the Gram– Schmidt orthonormalization from the{ system= of powers}xk for k 0, 1, 2, ... with = respect to the inner product , ρ , the polynomial Pn 1 is generated by multiplying ⟨· ·⟩ + the polynomial Pn with x and adding some lower order polynomial as correction. Indeed one has Proposition 17.1 Let ρ be a weight for the interval I (a, b) and denote the complete system of orthonormal polynomials for this weight= and this interval by Pk : k 0, 1, 2, ... Then, for every n 1, there are constants An, Bn, Cn such that{ = } ≥ Pn 1(x) (Anx Bn)Pn(x) CnPn 1(x) x I. + = + + − ∀ ∈ k Proof We know Pk(x) akx Qk 1(x) with some constant ak > 0 and some = + − polynomial Qk 1 of degree smaller than or equal to k 1. Thus, if we define An an 1 − − = + , it follows that Pn 1 AnxPn is a polynomial of degree smaller than or equal an + − to n, hence there are constants cn,k such that n Pn 1 AnxPn cn,kPk. + − = k 0 "= Now calculate the inner product with Pj , j n: ≤ n Pj , Pn 1 AnxPn ρ cn,k Pj , Pk ρ cn,j . ⟨ + − ⟩ = ⟨ ⟩ = k 0 "= 248 17 Separable Hilbert Spaces Since the polynomial Pk is orthogonal to all polynomials Qj of degree j k 1 ≤ − we deduce that cn,j 0 for all j<n 1, cn,n 1 An xPn 1, Pn ρ , and cn,n = − − = − ⟨ − ⟩ = An xPn, Pn ρ . The statement follows by choosing Bn cn,n and Cn cn,n 1. ✷ − ⟨ ⟩ = = − Proposition 17.2 For any weight function ρ on the interval I, the kth orthonormal polynomial Pk has exactly k simple real zeroes. Proof Per construction the orthonormal polynomials Pk have real coefficients, have the degree k, and the coefficient ck is positive. The fundamental theorem of algebra (Theorem 9.4) implies: The polynomial Pk has a certain number m k of simple ≤ real roots x1, ..., xm and the roots which are not real occur in pairs of complex conjugate numbers, (zj , zj ), j m 1, ..., M with the same multiplicity nj , m M = + + 2 j m 1 nj k. Therefore the polynomial Pk can be written as = + = ! m M nj nj Pk(x) ck (x xj ) (x zj ) (x zj ) . = − − − j 1 j m 1 "= ="+ m Consider the polynomial Qm(x) ck j 1 (x xj ).