}

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Introduction to Linear Algebra with Mathematica

Glossary

Preface

There are known many orthogonal polynomials, for example, Zernike polynomials, Walsh polynomials, Haar wavelets, Jacobi polynomials, Rational orthogonal functions

Orthogonal polynomials

   
Example 1:    ■
End of Example 1

 

Recurrence relation

Let { pₙ }n≥0 be a linearly independent sequence of polynomials with deg pₙ = n, and let ⟨ · , · ⟩ (also denoted by ⟨ · ∣ · ⟩) be an inner product on polynomials (e.g., span class="math">\( \displaystyle \quad \langle f,g\rangle =\int _a^bf(x)g(x)w(x)\, {\text d}x \) ).

Apply Gram–Schmidt to { pₙ }n≥0 to obtain an orthogonal sequence { ψₙ }. The two‑term recurrence becomes

\[ \begin{split} \psi_0(x) &= p_0(x), \\ \psi_n (x) &= p_n(x)-\frac{\langle p_n,\psi _{n-1}\rangle }{\langle \psi _{n-1},\psi _{n-1}\rangle }\, \psi _{n-1}(x),\quad n\geq 1. \end{split} \]
If we choose pₙ(x) = xⁿ, this is the Gram–Schmidt construction of an orthogonal polynomial system with respect to ⟨ · , · ⟩.

Assume now that { ψₙ(x) } is an orthogonal polynomial system with degree ψₙ(x) = n, and that we have chosen a normalization (e.g., monic or orthonormal). Then multiplication by x maps ψₙ into the span of { ψ₀, ψ₁, … , ψn+1 }. Orthogonality plus degree considerations imply

\[ x\psi_n (x) = \alpha _{n+1}\psi _{n+1}(x)+\beta _n\psi _n(x)+\gamma _n\psi _{n-1}(x),\quad n\geq 1, \]
with suitable αn+1, βₙ, γₙ determined by inner products:
\[ \beta_n =\frac{\langle x\psi_n,\psi _n\rangle }{\langle \psi _n,\psi _n\rangle },\quad \gamma_n =\frac{\langle x\psi _n,\psi _{n-1}\rangle }{\langle \psi _{n-1}, \psi _{n-1}\rangle },\quad \alpha_{n+1}=\frac{\langle x\psi _n,\psi _{n+1}\rangle}{\langle \psi_{n+1},\psi_{n+1}\rangle }. \]
Solving for ψn+1 gives the classical three‑term recurrence
\[ ???? \]
Conceptually: the two‑term recurrence describes how to orthogonalize each new input polynomial pₙ against ψn-1; the three‑term recurrence describes how the multiplication operator x acts on the orthogonal basis { ψₙ }.

TwoTermGS[v_List, ip_, opts : OptionsPattern[]] := Module[{psi}, If[Length[v] == 0, Return[{}]]; psi[0] = v[[1]]; Do[ psi[n] = v[[n + 1]] - (ip[v[[n + 1]], psi[n - 1]]/ip[psi[n - 1], psi[n - 1]]) psi[n - 1], {n, 1, Length[v] - 1} ]; Table[psi[n], {n, 0, Length[v] - 1}] ]; (* Polynomial-specialized front-end *) Clear[TwoTermGSPolynomials]; Options[TwoTermGSPolynomials] = {Assumptions -> True}; TwoTermGSPolynomials[n_Integer?NonNegative, x_Symbol, {a_, b_}, w_, opts : OptionsPattern[]] := Module[ {asm, ip, vList}, asm = OptionValue[Assumptions]; ip[f_, g_] := Assuming[asm, Integrate[f g w, {x, a, b}]]; vList = Table[x^k, {k, 0, n}]; TwoTermGS[vList, ip] ]; End[]; EndPackage[];
(* Example: Legendre-like system on [-1,1] with weight 1 *) psiList = TwoTermGSPolynomials[4, x, {-1, 1}, 1, Assumptions -> True]; (* Check orthogonality *) Table[ Integrate[psiList[[i]] psiList[[j]], {x, -1, 1}], {i, 1, Length[psiList]}, {j, 1, Length[psiList]} ] // Simplify
   
Example 1:
   ■
End of Example 1

 

  1. Agarwal, R.P. and O'Regan, D., Ordinary and Partial Differential Equations with Special Functions, Fourier Series, and Boundary Value Problems, Springer, 2009, NY.
  2. Bayatbabolghani, F. and Parand, K., A Comparison between Laguerre, Hermite, and Sinc orthogonal functions, 2017, 31 pages.
  3. Ismail, M.E.H., Classical and Quantum Orthogonal Polynomials in One Variable, Cambridge University Press, 2003, https://doi.org/10.1017/CBO9781107325982

 

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