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Construction of orthogonal functions in Hilbert space

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Construction of orthogonal functions in Hilbert space

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Content UNIVERSITY OF SOUTHERN CALIFORNIA MASTER’S THESIS Construction of Orthogonal Functions in Hilbert Space Author: Jiayu Li Supervisor: Dr. Robert J. Sacker A thesis submitted in fulfillment of the requirements for the degree of M.S. in Applied Mathematics in the Department of Mathematics August 2018 iii v University of Southern California Abstract Dana and David Dornsife College of Letters, Arts and Sciences Department of Mathematics M.S. in Applied Mathematics Construction of Orthogonal Functions in Hilbert Space by Jiayu Li A major problem in mathematical analysis: Whether the sum of a column function can be used to approximate the given function. For example, Taylor expansion, Fourier se- ries, and so on. The orthogonality in Hilbert spaces can promote the concept of Fourier series, thus, a more general orthogonal function system is obtained. This paper studies the origin and theoretical background of orthogonal functions and attempts to construct orthogonal functions. The first part of this paper introduces the Hilbert space, relevant theoretical background ofL 2 spaces and a brief sketch of the Sturm–Liouville problem and its solutions. The second part present the construction of Legendre functions, Her- mite functions and Bessel functions. vii Contents Abstract v 1 Orthonormal Bases in Hilbert Spaces 1 1.1 Orthonormal Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 The Gram-Schmidt Process . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Sturm-Liouville Operators 7 2.1 Sturm-Liouville Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 eigenfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3 Completeness of the eigenfunctions . . . . . . . . . . . . . . . . . . . . . . 10 3 Legendre Functions and Hermite Functions 13 3.1 Legendre polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.1.1 Construction of Legendre polynomials . . . . . . . . . . . . . . . . 13 3.1.2 Legendre polynomials with SL theory . . . . . . . . . . . . . . . . . 15 3.2 Hermite functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4 Bessel Functions 21 4.1 The Gamma Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4.2 Bessel’s equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4.2.1 Bessel functions of the first kind . . . . . . . . . . . . . . . . . . . . 21 4.2.2 Bessel functions of the second kind . . . . . . . . . . . . . . . . . . . 22 4.3 Orthogonality of Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . 22 4.3.1 Fourier-Bessel series . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Bibliography 25 1 Chapter 1 Orthonormal Bases in Hilbert Spaces 1.1 Orthonormal Bases According to the knowledge of linear algebra, we can see that there exists a finite set of orthonormal basis in a finite dimensional inner product space. Now we introduce this concept into Hilbert space. Definition 1.1.1. A set of pointsfx g in an inner product spaceX is said to be orthogonal if x ?x whenever6=. Definition 1.1.2. A set of pointsfx g in an inner product spaceX is said to be orthonormal if hx ;x i = for all and, where is the Kronecker function. A very natural question is: when a set of orthogonal pointsfe 1 ;e 2 ;:::g forms an orthogonal basis? That is to say, for anyf2H, can we findfa i g such that f = 1 X i=1 a i e i : In other words, we can use a set of functions with good properties to approachf. Definition 1.1.3. An orthonormal setfx g in an inner product spaceX is maximal if and only ifx?x for all impliesx = 0. A maximal orthonormal set in a Hilbert spaceH is referred to as an orthonormal basis forH. In the next theorem, we will elaborate on the basic properties of the orthonormal basis. Theorem 1.1.1. Given an orthonormal setfe k g 1 k=1 , the flowing statements are equivalent: (a) The linear combination offe k g 1 k=1 is dense inH. (b) For any integerk 1, ifhf;e k i = 0, thenf = 0. (c) Iff2H, and letS N (f) = P N k=1 a k e k , wherea k =hf;e k i, then lim N!1 jjS N (f) fjj = 0. (d) Leta k =hf;e k i,jjfjj 2 = P 1 k=1 ja k j 2 . 2 Chapter1. OrthonormalBasesinHilbertSpaces Proof. (a)) (b): By density, given> 0, we can select a linear combination g N = N X i=1 a i e i such that jjfg N jj: If (f;e k ) = 0 for allk 1, then (f;g N ) = 0, thus jjfjj 2 =hf;fi =hfg N ;fijjfg N jjjjfjjjjfjj: We havejjfjj. (b)) (c): ObviouslyS N (f)f;S N (f)) = 0, Thus jjfS N (f)jj 2 +jjS N (f)jj 2 =jjfjj 2 : (1.1) By arbitrariness ofN: jjS N (f)jj 2 = n X i=1 jhf;e i ij 2 jjfjj 2 : This implies 1 X i=1 jhf;e i ij 2 jjfjj 2 : This inequality is called the Bessel’s inequality. Letg = P 1 i=1 hf;e i ie i , for arbitraryi, whenN is sufficiently large, we havehf;e i i = hS N (f);e i i, thus hfg;e i i = lim N!1 hfS N (f);e i i = 0: By (b), we obtainf =g. (c)) (d): By 1.1 and the fact that lim N!1 jjS N (f)fjj = 0: (d)) (a): For arbitraryf, From jjfjj 2 = 1 X k=1 ja k j 2 : and 1.1, we see that for all > 0, there exists integer M, such that when N > M, jjS N (f)fjj<. 1.2 Fourier Series Now we use the above conclusions to discuss the Fourier series. Fourier analysis is undoubtedly one of the core issues of analysis, is a few runs through the entire modern mathematics development problem. For the integrable function f, we make a = 2. Then the Fourier series of a function can be written as follows: 1.3. TheGram-SchmidtProcess 3 In fact, Kolmogorov gave a counterexample in 1926: there is an integrable function f such that its Fourier series does not converge atf at any point. But we can prove that iff2L 2 , then the Fourier series of f convergent almost every where. We define inner product inL 2 [;] hf;gi = Z f(x)g(x)dx; easy to show that 1 2 Z e inx e imx dx = mn : Thus,f 1 p 2 e inx g is an orthogonal set. We will show this set is complete. To do this, we only need to show this functional subspace A 0 :=f M+N X M c n e inx : (c n )2C N+1 g is dense inL 2 (;). Theorem 1.2.1 (Stone–Weierstrass theorem). [2] LetX be a compact metric space, and letA be a subalgebra of the spaceC(X) that possesses the following two properties: (a) (Identity element) The constant functions belong toA. (b) (Reflexive) Iff2A, then f2A. (c) (Separate points) Given any two distinct pointsx;y2 X, then there existsf 2 A such thatf(x)6=f(y): ThenA is dense inC(X). Since A 0 is an algebra with constant function,Reflexive and separate points, Thus dense inC[;], also dense inL 2 [;]. We achieve the flowing theorem Theorem 1.2.2. f2L 2 [;], anda n = 1 2 R f(x)e inx dx, then: (a)jjfjj 2 = 2 P 1 1 ja n j 2 : (b) LetS N (f) = P N N a n e inx , then lim N!1 jjS N (f)fjj = 0: (c) Ifa n = 0 for alln2Z, thenf(x) = 0 almost everywhere. 1.3 The Gram-Schmidt Process The next natural question is: Is there any set of complete orthogonal bases in any Hilbert space? Theorem 1.3.1. Every Hilbert space has an orthonormal basis. Proof. Consider the collection of all orthonormal sets of vectors. It is nonempty, so Haus- dorff’s maximality principle implies that there is a maximal chain of orthonormal sets E . Let E =[ E ; 4 Chapter1. OrthonormalBasesinHilbertSpaces thenE is also an orthonormal set, for pick any two distinct vectorse 2 E ande 2 E E , then e ? e . So E is a maximal set of orthonormal vectors. E ? = 0, otherwise E can be extended further. Note that the set of basis vectors need not be countable, when uncountable, the Hilbert space is not separable. Given any countable linearly independent setfv n g in an inner product space, it is always possible to construct an orthonormal set from it. The construction is called the Gram-Schmidt orthogonalization process. u 0 :=v 0 e 0 := u 0 jju 0 jj u 1 :=v 1 he 0 ;v 1 ie 0 e 1 := u 1 jju 1 jj ::: u n :=v n n1 X i=0 he i ;v n ie i e n := u n jju n jj To check that these formulas yield an orthogonal sequence, first computehu 0 ;u 1 i by substituting the above formula foru 1 : hu 0 ;u 1 i =hu 0 ;v 1 he 0 ;v 1 ie 0 i =hu 0 ;v 1 ihe 0 ;v 1 ihu 0 ;e 0 i =hu 0 ;v 1 ihe 0 ;v 1 iku 0 k =hu 0 ;v 1 ihu 0 ;v 1 i = 0 Then use this to computehu 0 ;u 2 i again by substituting the formula foru 2 : we get zero. The general proof proceeds by mathematical induction. Theorem 1.3.2. Suppose v 1 ;:::;v m is a linearly independent list of vectors in V . Let e 1 = v 1 jjv 1 jj . Forj = 2;:::;m, definee j inductively by e j = v j hv j ;e 1 ie 1 hv j ;e j1 ie j1 jjv j hv j ;e 1 ie 1 hv j ;e j1 ie j1 jj : Thene1;:::;e m is an orthonormal list of vectors in V such that span(v 1 ;:::;v j ) =span(e 1 ;:::e j ) forj = 1;:::;m. Proof. Forj = 1,span(v 1 ) =span(e 1 ). Suppose 1<j <m and we have verified that span(v 1 ;:::;v j1 ) =span(e 1 ;:::e j1 ): 1.3. TheGram-SchmidtProcess 5 Since v 1 ;:::;v m is linearly independent, then v j = 2 span(v1;:::;vj 1). Thus, v j = 2 span(e1;:::;ej 1). Let 1k<j, then he j ;e k i =h v j hv j ;e 1 ie 1 hv j ;e j1 ie j1 jjv j hv j ;e 1 ie 1 hv j ;e j1 ie j1 jj ;e k i = hv j ;e k ihv j ;e k i jjv j hv j ;e 1 ie 1 hv j ;e j1 ie j1 jj = 0 Thus e 1 ;:::;e j is an orthonormal list. From the definition of e j , we see that v j 2 span(e 1 ;:::;e j ). Then we have span(v 1 ;:::;v j )span(e1;:::;e j ): Both lists are linearly independent, thus both subspaces above have dimension j and hence they are equal. 7 Chapter 2 Sturm-Liouville Operators 2.1 Sturm-Liouville Operators Definition 2.1.1 (Sturm–Liouville operator). Letp(x);p 0 (x);q(x); and!(x) be continuous real-valued functions on the finite intervalaxb, and assume thatp(x)> 0 and!(x)> 0 foraxb. LetH be the complex Hilbert space u(x) : Z b a ju(x)j 2 !(x)dx<1 ; where the inner product onH is given by hu;vi = Z b a u(x)v(x)!(x)dx: Now LetD denote the collection of allC 2 -functions inH satisfying the boundary conditions: B a [u] = 1 u(a) + 2 u 0 (a) = 0; B b [u] = 1 u(b) + 2 u 0 (b) = 0; 1 ; 2 ; 1 ; 2 , are real constants with j 1 j +j 2 j> 0; j 1 j +j 2 j> 0: Under these conditions, the operator L = 1 !(x) d dx p(x) d dx +q(x) is said to be a Sturm-Liouville operator ifD =D L . Definition 2.1.2. The eigenvalue equation of Sturm-Liouville operator Lu =u or equivalently, (pu 0 ) 0 + (q +!)u = 0 8 Chapter2. Sturm-LiouvilleOperators is called a Sturm-Liouville differential equation, or SL equation in short. Regular Sturm-Liouville Problem B a [u] = 1 u(a) + 2 u 0 (a) = 0 (j 1 j +j 2 j> 0) B b [u] = 1 u(b) + 2 u 0 (b) = 0 (j 1 j +j 2 j> 0); Singular Sturm-Liouville Problem Ifp(b) = 0: B a [u] = 1 u(a) + 2 u 0 (a) = 0 (j 1 j +j 2 j> 0) or ifp(a) = 0: B b [u] = 1 u(b) + 2 u 0 (b) = 0 (j 1 j +j 2 j> 0); Periodic Sturm-Liouville Problem Ifp(a) =p(b): C[u] =u(a)u(b) = 0 C 0 [u] =u 0 (a)u 0 (b) = 0 Theorem 2.1.1. Every second order linear ODE P (x)u 00 +Q(x)u 0 +R(x)u +u = 0 can be written in SL equation form:[8] 1 !(x) ( d dx p(x) du dx +q(x))u =u: (2.1) Proof. Expanding the left hand-side of 2.1 we obtain p ! u 00 + p 0 ! u 0 + ( q ! +) = 0; Let p ! =P; p 0 ! =Q; 1 ! =R: The first two equations imply thatp 0 =p =Q=P , therefore p(x) = exp Z Q(x) P (x) dx : Then since! =p=P andq =!R we obtain: p(x) = exp Z Q(x) P (x) dx !(x) = 1 P (x) exp Z Q(x) P (x) dx 2.2. eigenfunctions 9 q(x) = R(x) P (x) exp Z Q(x) P (x) dx 2.2 eigenfunctions Theorem 2.2.1. Sturm-Liouville operator is self-adjoint operator. Proof. LetL be a regular Sturm-Liouville operator and letu;v satisfy the boundary con- ditions. Then we have hLu;vihu;Lvi = Z b a (pu 0 ) 0 v +qu v + (p v 0 ) 0 uqu vdx = Z b a (pu 0 ) 0 v + (p v 0 ) 0 udx Using integration by parts, we obtain hLu;vihu;Lvi =p(b)[u(b)v 0 (b)u 0 (b)v(b)]p(a)[u(a)v 0 (a)u 0 (a)v(a)] =p(b) u(b) u 0 (b) v(b) v 0 (b) p(a) u(a) u 0 (a) v(a) v 0 (a) : ApplyB a tou and v, we have 1 u(a) + 2 u 0 (a) = 0; 1 v(a) + 2 v 0 (b) = 0: Since at least one of 1 ; 2 is nonzero, the determinant of the system must vanish, that is, u(a) u 0 (a) v(a) v 0 (a) = 0: similarly, u(b) u 0 (b) v(b) v 0 (b) = 0: Hence hLu;vi =hu;Lvi SinceL is self adjoint, its eigenvalues are real, and that eigenfunctions corresponding to different eigenvalues are orthogonal. Theorem 2.2.2. If m and n are two distinct eigenvalues of a SL system, with corresponding eigenfunctionsy m andy n , theny m andy n are orthogonal. 10 Chapter2. Sturm-LiouvilleOperators 2.3 Completeness of the eigenfunctions Theorem 2.3.1. LetL be a linear operator on a Hilbert spaceH, If there is a complex number 0 in the resolvent set ofL for which ( 0 IL) 1 is compact and normal, thenL can be expressed as a weighted sum of projections.[7] Proof. Since ( 0 IL) 1 is compact and normal, by spectral theorem, there is a weighted sum of projections P n n P n with ( 0 IL) 1 = X n n P n : Taking the inverse, we get 0 IL = X n 1 n P n ; thus L = X n ( 0 1 n )P n : Theorem 2.3.2. LetL be a Sturm-Liouville operator and assume 0 is not an eigenvalue of L, thenL 1 is compact and self-adjoint. Proof. Letu 1 (x);u 2 (x) be a nontrivial real solution of the boundary value problem: pu 00 +p 0 u 0 qu = 0; B i u i = 0: Since 0 is not an eigenvalue ofL, it follows from the consideration above thatu 1 (x) and u 2 (x) are linearly independent, andB i u j 6= 0 fori6= j. LetW (x) be the Wronskian for these two solutions. Then the general solution ofLu =v or (pu 0 ) 0 +qu =!v becomes u(x) =c 1 u 1 (x) +c 2 u 2 (x) +z(x); where z(x) = Z x a u 1 (x)u 2 (y)u 2 (x)u 1 (y) p(a)W (a) v(y)!(y)dy: We want to choose the coefficients so that when v is continuous then u satisfies the boundary conditions. If we choosec 2 = 0 and c 1 = Z b a u 2 (y) p(a)W (a) v(y)!(y)dy; then the solutionu(x) becomes u(x) = Z b x u 1 (x)u 2 (y) p(a)W (a) v(y)!(y)dy Z x a u 2 (x)u 1 (y) p(a)W (a) v(y)!(y)dy = Z b a g(x;y)v(y)!(y)dy; where 2.3. Completenessoftheeigenfunctions 11 g(x;y) = ( u 1 (x)u 2 (y) p(a)W(a) ; axyb; u 2 (x)u 1 (y) p(a)W(a) ; ayxb Because the kernelg(x;y) is continuous, real-valued andg(x;y) =g(y;x), the oper- atorL 1 is compact and self-adjoint. Theorem 2.3.3. LetL be a Sturm-Liouville operator and assume is not an eigenvalue of L. Then there exits a Green’s functiong(x;y;) defined on [a,b], such that ((IL) 1 v)(x) = Z b a g(x;y;)v(y)!(y)dy Proof. Note thatLI is a Sturm-Liouville operator withD LI =D L Theorem 2.3.4. Eigenfunctions of SL system forms a complete orthogonal basis ofH. Proof. Choose a real number such that is not an eigenvalue ofL. This can be done because the Hilbert space is separable, there are at most a countable number of nonzero mutually orthogonal eigenvectors. By theorem 2.3.3, the operator (IL) 1 is compact and self-adjoint. Apply the spectral theorem to (IL) 1 and letf ~ 1 ; ~ 2 ;:::g be the eigenvalues of (IL) 1 with corresponding eigenfunctionsf 1 ; 2 ;:::g. From Theorem 2.3.1, if n = ~ n , then f 1 ; 2 ;:::g are eigenvalues ofL with corresponding eigenfunctionsf 1 ; 2 ;:::g. 13 Chapter 3 Legendre Functions and Hermite Functions 3.1 Legendre polynomials 3.1.1 Construction of Legendre polynomials Definition 3.1.1. The Legendre polynomials are defined as follows: P n (x) = 1 2 n n! ( d dx ) n (x 2 1) n forx2 [1; 1] andn = 0; 1; 2;::: . Theorem 3.1.1. LetP n be a Legendre polynomial of ordern, then hf;P n i = (1) n 2 n n! Z 1 1 (x 2 1) n d n f(x) dx n dx: Proof. hf;P n i = Z 1 1 f(x)P n (x)dx = 1 2 n n! Z 1 1 f(x) d n dx n (x 2 1) n dx = 1 2 n n! f(x) d n1 dx n1 (x 2 1) n 1 1 Z 1 1 d n1 dx 1 (x 2 1) n d dx f(x)dx : = (1) 1 2 n n! Z 1 1 d n1 dx 1 (x 2 1) n d dx f(x)dx Integrate by parts forn times. Notice that the firstn1 derivatives of (x 2 1) n are equal to 0 atx =1. Therefore we have hf;P n i = (1) n 2 n n! Z 1 1 (x 2 1) n d n f(x) dx n dx . In particular, form = 0; 1;:::n 1, 14 Chapter3. LegendreFunctionsandHermiteFunctions Z 1 1 x m P n (x)dx = 0: Theorem 3.1.2. Ifp(x) is a polynomial of degreen, andp(x) is orthogonal to 1;x;:::x n1 on interval [1; 1], thenp(x) =aP n (x). Wherea is a constant.[3] Proof. Sincep(x) andP n (x) are both polynomial of degreen, then there exists a constant a such that the coefficient ofx n inaP n (x) andp(x) are the same. Consider the polyno- mial f(x) = aP n (x)p(x). On the one hand, it is a linear combination of P n (x) and p(x), then it is orthogonal to all polynomials of order less thann. On the other hand, f(x) itself is a polynomial of order less thann. This implies Z 1 1 (aP n (x)p(x)) 2 dx = 0; henceaP n (x) =p(x). Theorem 3.1.3. Apply Gram-Schmidt process to the sequence 1;x;x 2 ;:::;x n ;::: , the result- ing orthonormal sequence is 1 p 2 ; r 3 2 x; 3 2 r 5 2 (x 2 1 3 );:::; p n + 1=2 2 n n! ( d dx ) n (x 2 1) n : The resulting polynomials are called the normalized Legendre polynomials. Proof. We first need to prove that R 1 1 P n (x) 2 dx = 2 2n+1 : d n P n dx n = d 2n dx 2n (x 2n +::: ) = (2n)! 2 n n! Takef(x) =P n (x): Z 1 1 P n (x) 2 dx = (1) n 2 n n! (2n)! 2 n n! Z 1 1 (x 2 1) n dx (3.1) To estimate R 1 1 (x 2 1) n dx, 3.1. Legendrepolynomials 15 Z 1 1 (x 2 1) n dx = Z 0 (cos 2 (x) 1) n d(cos(x)) = Z 0 (1) n sin 2n+1 (x)dx = 2n 2n + 1 Z 0 (1) n sin 2n1 (x)dx = 2n 2n + 1 2n 2 2n 1 Z 0 (1) n sin 2n3 (x)dx =::: = (1) n 2 2n+1 (n!) 2 (2n + 1)! (3.2) Combine 3.1 and 3.2 we have Z 1 1 P n (x) 2 dx = 2 2n + 1 : (3.3) From theorem 3.1.2 we know that applying Gram-Schmidt process to the sequence 1;x;x 2 ;:::;x n ;::: , the resulting orthonormal sequence is a scalar multiple of the Leg- endre polynomials. And by 3.3, the constant should be q 2n+1 2 . 3.1.2 Legendre polynomials with SL theory We have proved that the set of polynomials is dense in L 2 [a;b]. Now, consider the simplest polynomials: 1;x;x 2 ;x 3 ::: they are not orthogonal. By applying Gram–Schmidt process on the setf1;x;x 2 ;x 3 :::g, we obtain the Legendre polynomials. 1 p 2 ; r 3 2 x; 3 2 r 5 2 (x 2 1 3 ) the general formula is: p n (x) = p n + 1=2 2 n n! ( d dx ) n (x 2 1) n These polynomials satisfy the differential equation Lp n =n(n + 1)p n whereL =D(1x 2 )D = (1x 2 )D 2 2xD. This equation has regular singular points at x =1 and x = 1 while x = 0 is an ordinary point. We can find power series solutions centered atx = 0. Now we construct such series solutions. Assume that y = 1 X i=1 c i x i 16 Chapter3. LegendreFunctionsandHermiteFunctions be a solution of the Legendre equation. By substituting such a series in (1), we get (1x 2 ) 1 X i=2 i(i 1)c i x i2 2x 1 X i=1 ic i x i1 + 1 X i=1 c i x i = 0 1 X i=2 i(i 1)c i x i2 1 X i=2 i(i 1)c i x i 1 X i=1 2ic i x i + 1 X i=0 c i x i = 0: Re-index the series we get 1 X j=0 [(i + 2)(i + 1)c i+2 (i 2 +i)c i ]x i = 0: By equating each coefficient to 0, we can obtain the recurrence relations c i+2 = (i + 2)i (i + 2)(i + 1) c i ; i = 0; 1; 2;::: If we assumec 0 6= 0 andc 1 = 0, the recurrence relation gives c 3 = 2 6 c 1 = 0; c 5 = 12 20 c 3 = 0;:::c 2i+1 = 0: The coefficients with even index can be written in term ofc 0 : c 2 = 2 c 0 ; c 4 = 2 3 4! c 0 ;::: Prove by induction we get c 2k = c 0 (2k)! ( k1 Y i=1 [(2i + 1)(2i)]): If we assumec 0 = 0 andc 1 6= 0, the recurrence relation gives c 2k+1 = c 1 (2k + 1)! ( k Y i=1 [2i(2i 1)]) The corresponding solutions are (respectively): y 1 (x) =c 0 1 X k=0 ( k1 Y i=1 [(2i + 1)(2i)]) x 2k (2k)! y 2 (x) =c 1 1 X k=0 ( k Y i=1 [2i(2i 1)]) x 2k+1 (2k + 1)! The only case when Legendre equation has a bounded solution on [1; 1] is when the eigenvalues has the form =n(n + 1) wheren = 0; 1; 2;::: 3.1. Legendrepolynomials 17 Theorem 3.1.4. Legendre’s equation has solutions that are bounded in the interval [1; 1] if and only if =n(n + 1) for somen = 0; 1; 2;::: Proof. we have a nontrivial function f that satisfies (21) for some lambda, and is or- thogonal to everyP n . Since it is orthogonal to everyP n , and every polynomial can be written as a linear combination of Legendre polynomials, it must also be orthogonal to every polynomial. We have Z 1 1 f(x)p n (x)dx = 0 for every polynomial p n (x). Since f is bounded and continuous, there exists a se- quence of polynomials converging uniformly to f. Taking this sequence to be p n (x), and passing to the limit inside the integral (24), which is justified since the sequence is uniformly convergent, we obtain Z 1 1 f(x) 2 dx = 0 Sincef is continuous, this impliesf(x) = 0. 18 Chapter3. LegendreFunctionsandHermiteFunctions 3.2 Hermite functions Definition 3.2.1. For eachn2N, the Hermite polynomialsH n :R!R is defined by H n (x) = (1) n e x 2 d n dx n e x 2 : The Hermite functionsh n (x) is defined by h n (x) = ( p 2 n n!) 1=2 H n (x) exp( x 2 2 ): The first five Hermite polynomials are: H 0 (x) = 1: H 1 (x) = 2x: H 2 (x) = 4x 2 2: H 3 (x) = 8x 3 12x: H 4 (x) = 16x 4 48x 2 + 12: Theorem 3.2.1. H n is a polynomial of degreen. Proof. By induction, we have d n dx n e x 2 = (2x) n e x 2 +p(x)e x 2 ; wherep(x) is a polynomial of degree less thann. Therefore H n (x) = (1) n e x 2 [(2x) n +p(x)]e x 2 = (2x) n + (1) n p(x): which is a polynomial of degreen. Theorem 3.2.2.fH n :n2Ng is orthogonal inL 2 e x 2 (R). Proof. Assumem<n, hH m ;H n i e x 2 = Z 1 1 H m (x)H n (x)e x 2 dx = (1) n Z 1 1 H m (x) d n dx n e x 2 dx Integrating by partsn times, notice that,p(x)e x 2 ! 0 asjxj!1 for any polynomial p(x), we obtain hH m ;H n i e x 2 = (1) 2n Z 1 1 d n dx n H m (x) e x 2 dx = 0: 3.2. Hermitefunctions 19 Theorem 3.2.3. LetG(x;t) = exp(2txt 2 ). Then G(x;t) = 1 X n=0 1 n! H n (x)t n : G(x;t) is the generating function of the Hermite polynomials. Proof. For everyx2R the functionG(x;t) is analytic int overR, and is therefore repre- sented by the power series G(x;t) = 1 X n=0 1 n! @ n G @t n t=0 t n : Because @ n G @t n t=0 = @ n @t n e x 2 (xt) 2 t=0 =e x 2 @ n @t n e (xt) 2 t=0 = (1) n e x 2 d n dy n e y 2 y=x = (1) n e x 2 d n dx n e x 2 ; It follows that @ n G @t n t=0 = (1) n e x 2 @ n @x n e x 2 =H n (x) Theorem 3.2.4. H n satisfies the second-order differential equation u 00 2xu 0 + 2nu = 0; x2R: Proof. DifferentiatingG(x;t) with respect tox, we have 2te 2xtt 2 = 1 X n=1 1 n! H 0 n (x)t n ; on the other hand 2te 2xtt 2 = 2 1 X n=0 1 n! H n (x)t n+1 = 2 1 X n=1 n n! H n1 (x)t n : This implies H 0 n (x) = 2nH n1 (x); n = 1; 2;::: (3.4) 20 Chapter3. LegendreFunctionsandHermiteFunctions DifferentiatingG(x;t) with respect tot gives 2(xt)e 2xtt 2 = 1 X n=0 1 n! H n+1 (x)t n ; 2xG(x;t) = 2tG(x;t) + 1 X n=0 1 n! H n+1 (x)t n from which 2x 1 X n=0 1 n! H n (x)t n = 2 1 X n=0 1 n! H n (x)t n+1 + 1 X n=0 H n+1 (x)t n = 2 1 X n=0 n + 1 (n + 1)! H n (x)t n+1 + 1 X n=0 1 n! H n+1 (x)t n = 2 1 X n=1 n n! H n1 (x)t n + 1 X n=0 1 n! H n+1 (x)t n : Comparing the coefficients, we obtain that 2xH n (x) = 2nH n1 (x) +H n+1 (x) =H 0 n (x) +H n+1 (x): (3.5) Differentiating 3.5 we have H 00 n (x) = 2xH 0 n (x) 2nH n (x): Theorem 3.2.5. Show that the Hermite functions form an orthonormal basis forL 2 (1;1).[5] Proof. Supposef? k for allk. Letg(x) =f(x)e x 2 =2 . Consider the Fourier transform of g: ^ g() = Z R g(x)e 2ix dx = Z R f(x)e x 2 =2 e 2ix dx = Z R f(x)e x 2 =2 1 X n=0 (2ix) n n! dx The linear span ofH 0 ;:::;H m is the set of all polynomials of degreem. Thus ^ g() = Z R f(x)e x 2 =2 1 X n=0 a n H n (x)dx = Z R f(x)e x 2 =2 1 X n=0 b n n (x)e x 2 =2 dx Wherefa n g;fb n g are the coefficients ofH n (x) and n (x). Then we have ^ g() = 1 X n=0 b n Z R f(x) n (x)dx = 0 It follows thatf(x) = 0: 21 Chapter 4 Bessel Functions 4.1 The Gamma Function 4.2 Bessel’s equation The differential equation x 2 y 00 +xy 0 + (x 2 )y = 0 (4.1) is called Bessel’s equation. We will show that this is another example of Strum-Liouville eigenvalue equation. 4.2.1 Bessel functions of the first kind Equation 4.1 is singular at pointx = 0, thus, we cannot create a power series solution to such a differential equation. Instead, we will use the Frobenius method. Theorem 4.2.1 (Frobenius method). Every equation of the form y 00 + q(x) x y 0 + r(x) x 2 y = 0; whereq andr are analytic atx = 0, has a solution of the form y(x) =x t 1 X k=0 c k x k ; (4.2) wheret is a real or complex number. Substituting 4.2 into 4.1 , we obtain: 1 X k=0 (k +t) 2 c k x t+k + 1 X k=0 c k x t+k+2 2 1 X k=0 c k x t+k : Compare the coefficients ofx t ;x t+1 ;::: we obtain the following recurrence formula: (t +j) 2 x j 2 c j +c j2 = 0; j = 0; 1;::: By choosing c 0 = 1 2 ( + 1) ; 22 Chapter4. BesselFunctions the coefficients can be determined by c 2k = (1) k 2 +2k k!( +m + 1) : And the Corresponding function J (x) =x 1 X k=0 (1) k 2 +2k k!( +m + 1) x 2k is called Bessel’s function of the first kind. 4.2.2 Bessel functions of the second kind Because Bessel’s equation is second-order differential equation, there is a second solu- tion that is linearly independent fromJ . There are several ways we can define a second solution to Bessel’s equation.[6] Y (x) = ( 1 sin [J cosJ (x)]; 6= 0; 1; 2;::: lim !n Y ; n = 0; 1; 2;::: (4.3) 4.3 Orthogonality of Bessel functions In this section, we will prove the orthogonality of Bessel functions and introduce the Fourier-Bessel series. First of all, divide the Bessel’s equation byx: xy 00 +y 0 + (x 2 x )y = 0 (4.4) we introduce a parameter through change of variables x =x;y =y(x) =ux: Under this transformation, equation 4.4 takes the form (xu 0 ) 0 + ( 2 x + 2 x)u = 0 (4.5) which is a SL equation with eigenvalue parameter = 2 . With boundary conditions: 1 u(a) + 2 u 0 (a) = 0; 1 u(b) + 2 u 0 (b) = 0: To simplify the problem, we assume a = 0 , 2 = 0, and restrict = n to the nonnegative integers. The general solution of 4.5 is given by u(x) =c n J n (x) +d n Y n (x): 4.3. OrthogonalityofBesselfunctions 23 Sinceu(x) has a limit atx = 0, the coefficient ofY n is zero. 2 = 0 gives J n (b) = 0; n = 0; 1;::: (4.6) Now, we need to find the roots ofJ n , we will prove the flowing theorem: Theorem 4.3.1. For anyn2N 0 ,J n (x) has infinitely many positive roots n1 < n2 <::: and nk !1 ask!1. Proof. consider equation y 00 + 1 x y 0 + (1 2 x 2 )y = 0; x> 0 defineu = p xy we obtain u 00 + (1 + 1 4 2 4x 2 )u = 0 (4.7) According to Sturm comparison theorem[4], we compare 4.7 with u 00 +u = 0 we see that if 0 1 2 then, in every subinterval of (0;1) of length, any solution of Bessel’s equation has at least one zero. So we have proved the statement forJ 0 . On the other hand d dx [x J (x)] = d dx 1 X m=0 (1) m 2 2m+ m!(m + + 1) x 2m = 1 X m=1 (1) m 2m 2 2m+ m!(m + + 1) x 2m1 =x 1 X m=0 (1) m 2 2m++1 m!(m + + 1) x 2m++1 =x J +1 : If 1 ; 2 are two consecutive zeros ofJ m (x), then by Rolle’s theorem that there is at least one point between 1 and 2 where the derivative of x m J m (x) is equal to zero. henceJ m+1 (x) has at least one zero between 1 and 2 . Consequently, the statement is true for allJ n (x). We shall denote the k-th positive root ofJ n (x) by nk We have already proved that for alln2 N, the roots ofJ n is an infinite increasing sequence tends to1: n1 < n2 <::: the solutions of 4.6 are given by k b = nk ; 24 Chapter4. BesselFunctions and the eigenvalues are k = 2 k = ( nk b ) 2 : and the corresponding sequence of eigenfunctions is J n ( 1 x);J n ( 2 x);::: Thus, we have proved the flowing theorem. Theorem 4.3.2. For alln2N 0 , the sequenceJ n ( 1 x);J n ( 2 x);::: is orthogonal and complete inL 2 x (0;b): 4.3.1 Fourier-Bessel series Fourier–Bessel series is a particular kind of generalized Fourier series based on Bessel functions [1]. f(x) = 1 X k=1 hf(x);J n ( k x)i kJ n ( k x)k 2 J n ( k x) (4.8) Fourier–Bessel series are used in the solution to partial differential equations, particu- larly in cylindrical coordinate systems. 25 Bibliography [1] F. Bowman. Introduction to Bessel Functions. Dover Books on Mathematics. Dover Publications. ISBN: 9780486604626. [2] Philippe G. Ciarlet. Linear and Nonlinear Functional Analysis with Applications. Soci- ety for Industrial and Applied Mathematics, 2013. [3] H.F. Davis. Fourier Series and Orthogonal Functions. Dover Books on Mathematics. Dover Publications, 2012. ISBN: 9780486140735. [4] JB Diaz and Joyce R McLaughlin. “Sturm comparison theorems for ordinary and partial differential equations”. In: Bulletin of the American Mathematical Society 75.2 (1969), pp. 335–339. [5] G.B. Folland. Real analysis: modern techniques and their applications. Pure and applied mathematics. Wiley, 1999. ISBN: 9780471317166. [6] M. A Al-Gwaiz. Sturm-Liouville Theory and its Applications. Springer, 2008, pp. 14333– 14333. [7] Arch W Naylor and George R Sell. Linear Operator Theory in Engineering and Science. Springer-Verlag, 1982. [8] P .V . O’Neil. Advanced Engineering Mathematics. Cengage Learning, 2011. ISBN: 9781111427412.

Abstract (if available)

Abstract A major problem in mathematical analysis: Whether the sum of a column function can be used to approximate the given function. For example, Taylor expansion, Fourier series, and so on. The orthogonality in Hilbert spaces can promote the concept of Fourier series, thus, a more general orthogonal function system is obtained. This paper studies the origin and theoretical background of orthogonal functions and attempts to construct orthogonal functions. The first part of this paper introduces the Hilbert space, relevant theoretical background of L² spaces and a brief sketch of the Sturm–Liouville problem and its solutions. The second part present the construction of Legendre functions, Hermite functions and Bessel functions.

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Creator Li, Jiayu (author)

Core Title Construction of orthogonal functions in Hilbert space

School College of Letters, Arts and Sciences

Degree Master of Science

Degree Program Applied Mathematics

Publication Date 06/26/2018

Defense Date 06/26/2018

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag Bessel function,Hermite function,Hilbert space,Legendre function,OAI-PMH Harvest,orthogonal function,Sturm-Liouville operator

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Haydn, Nicolai (committee chair), Lototsky, Sergey (committee member), Tiruviluamala, Neelesh (committee member)

Creator Email jiayul@usc.edu,zzlijiayu@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c40-506053

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Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

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