Sobolev Orthogonal Polynomials in Two Variables and Second Order Partial Differential Equations

We consider polynomials in two variables which satisfy an admissible second order partial differential equation of the form (*) Auxx + 2Buxy + Cuyy +Dux + Euy = u; and are orthogonal relative to a symmetric bilinear form de…ned by '(p; q) = h ; pqi+ h ; pxqxi ; where A; ; E are polynomials in x and y; is an eigenvalue parameter, and are linear functionals on polynomials. We …nd a condition for the partial di¤erential equation ( ) to have polynomial solutions which are orthogonal relative to a symmetric bilinear form '( ; ): Also examples are provided. 1. Introduction In 1967, Krall and She¤er [6] investigated a second order partial di¤erential equation of the form (1.1) A(x; y)uxx + 2B(x; y)uxy + C(x; y)uyy +D(x)ux + E(y)uy = u and classi…ed all weak orthogonal polynomials satisfying the partial di¤erential equation (1.1), where A(x; y), , E(y) are polynomials in x and y; and is an eigenvalue parameter. As a generalization, we consider polynomial solutions to the partial di¤erential equation (1.1) which are orthogonal relative to a symmetric bilinear form ( ; ) on polynomials de…ned by (1.2) (p; q) = h ; pqi+ h ; pxqxi ; where and are moment functionals, and p; q are polynomials in x and y. The case = 0 was investigated by Krall and She¤er. For the partial di¤erential equation (1.1) considered by Krall and She¤er, we know that (i) Cx = 0 (up to a linear change of independent variables) and (ii) partial derivatives with respect to x satisfy the partial di¤erential equation of the same type as the partial di¤erential equation (1.1) (see [4]). These facts remind us of the Hahn-Sonnine characterization theorem for classical orthogonal polynomials ([2, 5, 10]) which states that : The only polynomial sequences fPn(x)gn=0 (up to a complex change of variable) which are simultaneously orthogonal with respect to bilinear forms of the form (p; q)0 = R R p(x)q(x) d 0, (p; q)1 = R R p(x)q(x) d 0 + R R p 0(x)q0(x) d 1; 1991 Mathematics Subject Classi…cation. 33C50,35P99. Key words and phrases. Orthogonal polynomials in two variables, bilinear symmetric form, Sobolev orthogonal polynomials in two variables, second order partial di¤erential equations. 1 2 JEONGKEUN LEE AND L. L. LITTLEJOHN with i(i = 0; 1) real-valued signed Borel measures, are the classical orthogonal polynomials of Jacobi, Laguerre, Hermite and Bessel polynomials. Naturally they lead us to the problem of investigating polynomials orthogonal relative to ( ; ) in (1.2). But contrary to the classical orthogonal polynomials in one variable, orthogonal polynomials in two variables whose partial derivatives with respect to x or y are orthogonal dose not satisfy the partial di¤erential equation of the form (1.1) (See [3] for these materials). Instead, we consider polynomials in two variables which are orthogonal relative to a symmetric bilinear form ( ; ) in (1.2) and satisfy the partial di¤erential equation (1.1). In this paper, we give some basic facts on Sobolev orthogonal polynomials and the relationship between Sobolev orthogonal polynomials relative to a symmetric bilinear form ( ; ) in (1.2) and the partial di¤erential equation (1.1). Also, we give some examples of the partial di¤erential equation having Sobolev orthogonal polynomials as solutions. 2. Preliminaries: Basic Theory of Orthogonal Polynomials in Two Variables Let Pn be the space of all polynomials in x and y of degree n: The set of all polynomials in two variables is denoted by P. By a polynomial system (in short, PS), we mean a sequence f mn(x; y)gm;n=0 of polynomials such that deg mn = m+ n for each m;n 0 and f n j;jgj=0 is linearly independent modulo Pn 1:We denote f n j;j(x; y)gj=0 by an (n+1)-dimensional column vector n and a PS f mn(x; y)gm;n=0 by f ngn=0. We say that a PS f ngn=0 is monic if mn(x; y) = xy modulo Pm+n 1 for each m;n 0:. To a given PS f ngn=0; there corresponds a unique monic PS fPngn=0 which is de…ned by Pn = A 1 n n; where An = (ajk) n j;k=0 and n j;j(x; y) = Pn k=0 a n j;kx n y modulo Pn 1: It will be called the normalization of f ngn=0: A linear functional on P is called a moment functional. We denote the action of a moment functional on polynomial by h ; i instead of the customary ( ): Similarly, for a matrix Q = (Qi;j) with Qi;j being a polynomial, h ;Qi is de…ned to be the matrix (h ;Qi;ji). We see that ;AB = ;BA T for any column vectors A and B of polynomials. For a moment functional and any polynomial , we de…ne the partial derivatives of by the formulas (2.1) h@x ; i = h ; @x i; h@y ; i = h ; @y i for 2 P; and de…ne the multiplication on by a polynomial through the formula (2.2) h ; i = h ; i for 2 P: De…nition 2.1. A PS f ngn=0 is called an orthogonal basis (OB) relative to if there is a nonzero moment functional such that for all n 0 h ; n k i = 0; 2 Pn 1; 0 k n: And f ngn=0 is called a weak orthogonal polynomial set (WOPS) relative to if there is a nonzero moment functional such that h ; m;n k;li = Kmn mk nl if m+ n 6= k + l: If Kmn 6= 0 (respectively, Kmn > 0) for each m;n 0; we say that f ngn=0 is an orthogonal polynomial set (in short, OPS) (respectively,a positive-de…nite OPS) relative to : It is obvious that there is an OB relative to if and only if there is a WOPS relative to : De…nition 2.2. A moment functional is quasi-de…nite (respectively, weakly quasi-de…nite) if there is an OPS (respectively, a WOPS) relative to : SOBOLEV ORTHOGONAL POLYNOMIALS AND PARTIAL DIFFERENTIAL EQUATIONS 3 From De…nition 2.1 and 2.2, we see that a PS f ngn=0 is an OPS (respectively, a positive-de…nite OPS) relative to if and only if h ; m n i = Hn mn and Hn := h ; n n i is a nonsingular (respectively, a positive-de…nite) diagonal matrix. For any PSf ngn=0, there is a unique moment functional , which is called the canonical moment functional of f ngn=0, de…ned by the conditions h ; 1i = 1; h ; mni = 0; m+ n 1: Note that if a PS f ngn=0 is an OB relative to , then is a constant multiple of canonical moment functional of f ngn=0: Although f ngn=0 is an OPS relative to ; its normalization fPngn=0 need not be an OPS relative to but fPngn=0 is an OB relative to . It is not easy to produce an OPS relative to , if any, from an OB relative to : Theorem 2.1. ([4, 6]) For any moment functional , the following statements are equivalent. (i) is quasi-de…nite. (ii) There is a unique monic OB fPngn=0 relative to . (iii) There is a monic OB fPngn=0 such that Hn := h ;PnPn i is nonsingular for all n 0. Theorem 2.2 (Favard’s Theorem). [11] Let f ngn=0 be a PS. Then the following statements are equivalent. (i) f ngn=0 is a WOPS relative to a quasi-de…nite moment functional : (ii) For n 0 and i = 1; 2, there are matrices Ani of order (n+1) (n+2), Bni of order (n+1) (n+1), and Cni of order (n+ 1) n such that (a) xi n = Ani n+1 +Bni n + Cni n 1 (here x1 = x; x2 = y) (b) rankCn = n+ 1, where Cn = (Cn1; Cn2): Lemma 2.3. Let be a moment functional and be a polynomial. Then we have (i) = 0 if and only if x = 0 or y = 0 (ii) ( )x = x + x and ( )y = y + y: Proof. (i): The proof is obvious. (ii) A computation shows that for any p 2 P; we have h( )x; pi = h ; pxi = h ; pxi = h ; ( p)x + xpi = h x; pi+ h ; xpi = h x; pi+ h x ; xpi = h x + x; pi ; which means ( )x = x + x: By a similar calculation, we have ( )y = y + y: If the partial di¤erential equation (1.1) has a PS f ngn=0 as solutions, then it must be of the form (2.3) Auxx + 2Buxy + Cuyy +Dux + Euy = (ax + d1x+ e1y + f1)uxx + (2axy + d2x+ e2y + f2)uxy + ay + d3x+ e3y + f3 uyy +(gx+ h1)ux + (gy + h2)uy = nu; where n = an(n 1) + dn: We say that the partial di¤erential equation (2.3) is admissible if m 6= n for m 6= n: (2.3) has a unique monic PS as solutions if and only if it is admissible. Theorem 2.4. [4] Let be the canonical moment functional of a PS f ngn=0: If f ngn=0 satis…es the partial di¤erential equation (2:3), then satis…es the equation (2.4) L [ ] = (A )xx + 2(B )xy + (C )yy (D )x (E )y = 0; where L [u] := (Au)xx +2(Bu)xy + (Cu)yy (Du)x (Eu)y is the formal Lagrange adjoint operator of L[ ]: 4 JEONGKEUN LEE AND L. L. LITTLEJOHN Furthermore, (2:4) has a unique solution up to a multiplication constant if the partial di¤erential equation (2:3) is admissible. Theorem 2.5. ([4, 6]) Let f ngn=0 be an OB relative to : Then the following statements are all equivalent. (i) f ngn=0 satis…es the partial di¤erential equation (2.3). (ii) satis…es the moment equations (2.5) M1[ ] := (A )x + (B )y D = 0; M2[ ] := (B )x + (C )y E = 0: Remark 2.1. For any moment functional ; L [ ] is written in the following form: L [ ] = (M1[ ])x + (M2[ ])y : This formula will be used in section 4: Theorem 2.6. [4] Let f ngn=0 be a PS satisfying the admissible partial di¤erential equation (2:3) and the canonical moment functional of f ngn=0: Then the following statements are equivalent. (i) f ngn=0 is a WOPS relative to : (ii) M1[ ] = 0: (iii) M2[ ] = 0: 3. Theory of Sobolev Orthogonal Polynomials in Two Variables We know that any moment functional de…nes a symmetric bilinear form '( ; ) on P P through the formula '(p; q) = h ; pqi : Conversely, a symmetric bilinear form can be generated by a moment functional provided some conditions are ful…lled. Theorem 3.1. Let '( ; ) be a symmetric bilinear form on P P. Then the following statements are equivalent. (i) there is a moment functional such that '(p; q) = h ; pqi for any p; q 2 P: (ii) '(xp; q) = '(p; xq) and '(yp; q) = '(p; yq) for any p; q 2 P: Proof. ()) It is obvious. (() De…ne a moment functional by h ; pi = '(p; 1); p 2 P: Then we have for any p; q 2 P '(p; q) = '(p; deg q X i+j=0 aijx y) = deg q X i+j=0 aij'(p; x y) = deg q X