Here is an interesting fact about Legendre series. Sometimes we want to fit given curve as...

    

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Here is an interesting fact about Legendre series. Sometimes we want to fit given curve as closely as possible by a polynomial of a given degree, say a cubic. the criterion of "Least Squares" is often used to determine the best fit. This means hat if, say, we want to fit a given f(x) on (-1, 1) by a cubic, we find the coefficients a, b, c, d so that Ľ [₁ [ƒ(x) − (azª³ + bx³² + cx + d)]² da - (9.5) is as small as possible. Then (9.6) f(x) = ax³ + bx² + cx+d is called the best approximation (by a cubic) in the least squares sense. It can be proved that an expansion (as far as the desired degree of the polynomial ap proximation) in Legendre polynomials gives this best least squares approximation Find the 3rd-degree polynomial that minimizes the square of the differ- ences (least square) between the function sin x and the polynomial in the interval −1≤ x ≤ 1. Make a plot showing the function and your least- square fit and on a plot below show the differences. Here is an interesting fact about Legendre series. Sometimes we want to fit given curve as closely as possible by a polynomial of a given degree, say a cubic. the criterion of "Least Squares" is often used to determine the best fit. This means hat if, say, we want to fit a given f(x) on (-1, 1) by a cubic, we find the coefficients a, b, c, d so that Ľ [₁ [ƒ(x) − (azª³ + bx³² + cx + d)]² da - (9.5) is as small as possible. Then (9.6) f(x) = ax³ + bx² + cx+d is called the best approximation (by a cubic) in the least squares sense. It can be proved that an expansion (as far as the desired degree of the polynomial ap proximation) in Legendre polynomials gives this best least squares approximation Find the 3rd-degree polynomial that minimizes the square of the differ- ences (least square) between the function sin x and the polynomial in the interval −1≤ x ≤ 1. Make a plot showing the function and your least- square fit and on a plot below show the differences.