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Introduction to Orthogonal Polynomials: Definition and Basic Properties

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Introduction to Orthogonal Polynomials: Definition and Basic Properties Why should we study orthogonal polynomials ? An example of a system of orthogonal polynomials Construction of a system of orthogonal polynomials Definition of orthogonal polynomials Basis properties of orthogonal polynomials Tutorials: Solving assignments Introduction to Orthogonal Polynomials: Definition and basic properties Prof. Dr. Mama Foupouagnigni African Institute for Mathematical Sciences, Limbe, Cameroon and Department of Mathematics, Higher Teachers’ Training College University of Yaounde I, Cameroon Email:[email protected] AIMS-Volkswagen Stiftung Workshop on Introduction to Orthogonal Polynomials and Applications Hotel Prince de Galles, Douala, Cameroon, October 5-12, 2018 Why should we study orthogonal polynomials ? An example of a system of orthogonal polynomials Construction of a system of orthogonal polynomials Definition of orthogonal polynomials Basis properties of orthogonal polynomials Tutorials: Solving assignments Table of Contents 1 Why should we study orthogonal polynomials ? 2 An example of a system of orthogonal polynomials 3 Construction of a system of orthogonal polynomials 4 Definition of orthogonal polynomials 5 Basis properties of orthogonal polynomials 6 Tutorials: Solving assignments Why should we study orthogonal polynomials ? An example of a system of orthogonal polynomials Construction of a system of orthogonal polynomials Definition of orthogonal polynomials Basis properties of orthogonal polynomials Tutorials: Solving assignments Main objectives 1 Give an example of a system of orthogonal polynomials 2 Provide a method for constructing a system orthogonal Polynomials 3 Define the notion of orthogonal polynomials; 4 Provide (with some illustrations on the proof) some basic properties such as: the uniqueness of a family of orthogonal polynomials; the matrix representation; the three-term recurrence relation, the Christoffel-Darboux formula and some of its consequences such as the interlacing properties of the zeros. 5 Finally we discuss and solve, as a short tutorial, some assignments given within the first talk which are mainly proof of some results provided earlier. Why should we study orthogonal polynomials ? An example of a system of orthogonal polynomials Construction of a system of orthogonal polynomials Definition of orthogonal polynomials Basis properties of orthogonal polynomials Tutorials: Solving assignments Why should we study orthogonal polynomials ? Orthogonal polynomials are to be seen as a sequence of polynomials (pn)n with deg(pn) = n with orthogonality property. They are very useful in practice in various domains of mathematics, physics, engineering and so on because of the many properties and relations they satisfy: 1 Orthogonality (all of them) 2 Three term recurrence relation (all of them) 3 Darboux-Christoffel formula (all of them) 4 Matrix representation (all of them); 5 Gauss quadrature (all of them): used for approximation of integrals; 6 second-order holonomic differential, difference or q-difference equation (classical ones); 7 Fourth-order holonomic differential, difference or q-difference equations (Laguerre-Hahn class); 8 Rodrigues formula (classical ones); 9 Partial differential, difference or q-difference equations (OP of several variables); 10 Expansion of continuous function with integrable square derivable in terms of Fourier series of OP (classical OP); 11 .... Why should we study orthogonal polynomials ? An example of a system of orthogonal polynomials Construction of a system of orthogonal polynomials Definition of orthogonal polynomials Basis properties of orthogonal polynomials Tutorials: Solving assignments AMS subject classification for Orthogonal Polynomials Why should we study orthogonal polynomials ? An example of a system of orthogonal polynomials Construction of a system of orthogonal polynomials Definition of orthogonal polynomials Basis properties of orthogonal polynomials Tutorials: Solving assignments The Chebyshev polynomials of the first kind Chebyshev polynomials of the first kind are defined by Tn(x) = cos(nθ); x = cos θ; 0 < θ < π (1) fw0g and fulfil the following properties: 1 Tn is a polynomial of degree n in x: This can be seen from the recurrence relation Tn+1(x) + Tn−1(x) = 2xTn(x); n ≥ 1; T0(x) = 1; T1(x) = x; (2) fw1g 2 (Tn)n satisfies the orthogonality relation Z π Z 1 dx cos(nθ) cos(mθ)dθ = knδn;m = Tn(x) Tm(x)p (3) fw2g 2 0 π −1 1 − x (with k0 = π; kn = 2 ; n ≥ 1), obtained using the change of variable x = cos θ; 0 < θ < π and the linearization formula 2 cos nθ cos mθ = cos(n + m)θ + cos(n − m)θ: 3 Monic Chebyshev polynomial of degree n is the polynomial deviating less from zero on [−1; 1] among monic polynomials of degree n: n 1−n 1−n min max jqn(x)j; qn 2 R[x]; qn(x) = x + ::: = max j2 Tn(x)j = 2 : −1≤x≤1 −1≤x≤1 (4) fw3g 4 Second-order holonomic differential equation: 2 00 0 2 (1 − x ) Tn (x) − x Tn(x) + n Tn(x) = 0; n ≥ 0: (5) fw4g Why should we study orthogonal polynomials ? An example of a system of orthogonal polynomials Construction of a system of orthogonal polynomials Definition of orthogonal polynomials Basis properties of orthogonal polynomials Tutorials: Solving assignments First 10 Chebyshev I Polynomials From the three-term recurrence relation, one can generate any Tn: Tn+1(x) = 2xTn(x) − Tn−1(x); n ≥ 1; T0(x) = 1; T1(x) = x; 2 T2(x) = 2 x − 1; 3 T3(x) = 4 x − 3 x; 4 2 T4(x) = 8 x − 8 x + 1; 5 3 T5(x) = 16 x − 20 x + 5 x; (6) fw5g 6 4 2 T6(x) = 32 x − 48 x + 18 x − 1; 7 5 3 T7(x) = 64 x − 112 x + 56 x − 7 x; 8 6 4 2 T8(x) = 128 x − 256 x + 160 x − 32 x + 1; 9 7 5 3 T9(x) = 256 x − 576 x + 432 x − 120 x + 9 x: The zeros xn;k of Tn ranked in increasing order are: (2(n − k) + 1 x = cos π ; k = 1::n: (7) fw6g n;k 2n Why should we study orthogonal polynomials ? An example of a system of orthogonal polynomials Construction of a system of orthogonal polynomials Definition of orthogonal polynomials Basis properties of orthogonal polynomials Tutorials: Solving assignments Graphic of the first 10 Chebyshev I Polynomials Why should we study orthogonal polynomials ? An example of a system of orthogonal polynomials Construction of a system of orthogonal polynomials Definition of orthogonal polynomials Basis properties of orthogonal polynomials Tutorials: Solving assignments Orthogonality relations for the Chebyshev I Polynomials Summing up, we have seen that the Chebyshev polynomials Tn satisfy: deg(Tn) = n ≥ 0; and the orthogonality condition: Z 1 dx Z 1 dx Tn(x) Tm(x)p = 0; n 6= m; Tn(x) Tn(x)p 6= 0; n ≥ 0: 2 2 −1 1 − x −1 1 − x The polynomial sequence (Tn)n is said to be orthogonal with respect to the weight function ρ(x) = p 1 defined over the interval ] − 1; 1[: It is an 1−x2 orthogonal polynomial sequence. Assignment 1: Establish relations (1)-(7). Why should we study orthogonal polynomials ? An example of a system of orthogonal polynomials Construction of a system of orthogonal polynomials Definition of orthogonal polynomials Basis properties of orthogonal polynomials Tutorials: Solving assignments Construction of a system of orthogonal polynomials:Part 1 Let us consider a scalar product ( ; ) defined on R[x] × R[x] where R[x] is the ring of polynomials with real variable. As scalar product, it fulfills the following properties: (p; p) ≥ 0; 8p 2 R[x]; and (p; p) = 0 =) p = 0; (8) fw7g (p; q) = (q; p); 8p; q 2 R[x]; (9) (λ p; q) = λ (p; q); 8λ 2 R; 8p; q 2 R[x]; (10) (p + q; r) = (p; r) + (q; r); 8p; q; r 2 R[x]: (11) As examples of scalar products on R[x] with connections to known systems of orthogonal polynomials, we mention: Z 1 dx (p; q) = p(x) q(x)p ; (12) fw8g 2 −1 1 − x connected to Chebyshev polynomials; N X (p; q) = wk p(k) q(k); N 2 N [ f1g; (13) fw9g k=0 leading to orthogonal polynomials of a discrete variable. Why should we study orthogonal polynomials ? An example of a system of orthogonal polynomials Construction of a system of orthogonal polynomials Definition of orthogonal polynomials Basis properties of orthogonal polynomials Tutorials: Solving assignments Construction of a system of orthogonal polynomials: Part 2 Theorem (Gram-Schmidt orthogonalisation process) The polynomial systems (qn)n and (pn)n defined recurrently by the relations n−1 n n X (qk; x ) qk q0 = 1; qn = x − qk; n ≥ 1; pk = p ; k ≥ 0; (14) fw10g (qk; qk) k=0 (qk; qk) satisfy the relations deg(qn) = deg(pn) = n; 8n ≥ 0; (qn; qm) = 0; n 6= m; (qn; qn); 6= 0 8n ≥ n; (pn; pm) = 0; n 6= m; (pn; pn) = 1; 8n ≥ n: The proof is done by induction on n: Assignment 2. The polynomial systems (qn)n and (pn)n are said to be orthogonal with respect to the scalar product ( ; ). They represent the same orthogonal polynomial system with different normalisation: (qn)n is monic (to say the coefficient of the leading monomial is equal to 1) while (pn)n is orthonormal ( (pn; pn) = 1). Why should we study orthogonal polynomials ? An example of a system of orthogonal polynomials Construction of a system of orthogonal polynomials Definition of orthogonal polynomials Basis properties of orthogonal polynomials Tutorials: Solving assignments Definition of orthogonal polynomials: Part 1 Orthogonality with respect to a scalar product A system (pn)n of polynomials is said to be orthogonal with respect to the scalar product ( ; ) if it satisfies the following 2 conditions deg(pn) = n; 8n ≥ 0; (15) fw11g (pn; pm) = 0; n 6= m; (pn; pn) 6= 0; 8n ≥ n: (16) fw12g When scalar product is defined by a Stieltjes integral When the scalar product ( ; ) is defined by a Stieltjes integral Z b (p; q) = p(x) q(x) dα(x); (17) fw13g a where α is an appropriate real-valued function, then (16)
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