Answer each question with a minimum of 50 words per question: Discuss the discrete and orthogonal polynomials to least square approximations. Provide at least one example on each case to support your...

1) Sometimes we may be confronted with finding a function which may represent a
set of data points which are given for both arguments x and y. Often it may be
difficult to find such a function y = y(x) except by certain techniques. One of the
known methods of fitting a polynomial function to this set of data is the Least
squares approach. The least squares approach is a technique which is
developed to reduce the sum of squares of errors in fitting the unknown function.
The Least Squares Approximation methods can be classified into two, namely
the discrete least square approximation and the continuous least squares
approximation. The first involves fitting a polynomial function to a set of data
points using the least squares approach, while the latter requires the use of
orthogonal polynomials to determine an appropriate polynomial function that fits
a given function.
a) Discrete Least squares approximation:
The basic idea of least square approximation is to fit a polynomial function
P(x) to a set of data points (xi, yi) having a theoretical solution
y = f(x)……………………..(1)
The aim is to minimize the squares of the errors. In order to do this, suppose
the set of data satisfying the theoretical solution (1) are (x1, y1), (x2, y2), . . . ,
(xn, yn). Attempt will be made to fit a polynomial using these set of data
points to approximate the theoretical solution f(x).
The polynomial to be fitted to these set of points will be denoted by P(x) or
sometimes Pn(x) to denote a polynomial of degree n. The curve or line P(x)
fitted to the observation y1, y2, . . . , yn will be regarded as the best fit to f(x),
if the difference between P(xi) and f(xi) , i = 1, 2, . . . , n is least. That is, the
sum of the differences ei = f(xi) – P(xi), i = 1, 2, . . . , n should be the
minimum. The differences obtained from ei could be negative or positive and
when all these ei are summed up, the sum may add up to zero. This will not
give the true error of the approximating polynomial. Thus to estimate the
exact error sum, the square of these differences...