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The Convergence and Stability of Difference Methods 556
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assumes the form assumes the values best approximation Chebyshev polynomials coefficients column Consequently consider constant continuous function decimal places defined denote derivative differential digits divided differences elements En(f equal equation estimate exact value example expression f(xi finite differences forward interpolation formula Fourier series function f(x Gauss formulae given Hence Hermite interpolation Hilbert space Hn(P improper integrals inequality integrand interpolation function interval irremovable error Lagrange formula Lagrange interpolation polynomial linear combination linearly linearly independent Ln(x method multipliers nodes numerical integration formulae obtain operations order differences ortho-normal set orthogonal polynomials Pn(x points of interpolation points x0 polation polynomial of degree problem produce prove quantity relative errors required to find residual term result satisfies sequence substitute Suppose we put theorem Tn(x trigonometrical polynomial uniformly convergent vanish variable whilst xi+1