(solution) ( This problem results in an orthonormal set that is a subset of the well-known Legen- dre

(solution) ( This problem results in an orthonormal set that is a subset of the well-known Legen- dre

( This problem results in an orthonormal set that is a subset of the well-known Legen- dre polynomials. As such they are not particularly important in communications. The problem is included to strengthen your understanding of the Gram–Schmidt procedure. ) Using the Gram–Schmidt procedure, construct an orthonormal basis for the space of quadratic polynomials { a 2 t 2 + a 1 t + a 0 ; a 0 , a 1 , a 2 ∈ R} over the interval −1 ≤ t ≤ 1. Hint The equivalent problem is to find an orthonormal basis for three signals {1, t , t 2 } over the interval −1 ≤ t ≤ 1.