Please answer all sub-questions 1-8
In this problem, we are going to investigate how to construct a basis called B-splines for the splines of order l1 Bil(x) Bie-1(x) + -Bi+1,2-1(x) Vie [1; k + 2m - -1]. (2) Xite- Xi+2+1 - Xi+1 This definition is very mathematical, and you should get in the habit of specializing the definition to get some intuition. In this problem, you are guided through that process, but this should become a natural reflex. [Q1] Sketch the order 0 splines in a situation with k = 2. This requires setting at least m= 0. (Sketching mean graphing a non exact but representative plot of what's going on) [Q2] Sketch the order 1 splines in the same situation with k = 2. This requires setting at least m = 1. (Sketching mean graphing a non exact but representative plot of what's going on) [Q3] Show by induction that Bil(x) = 0 for x [i, i+e+1). This shows that there is a bound on the support of a spline. [Q4] Show by induction that Bie(x) > 0 for x (Xi, ite+1). This shows that B splines are positive in their support when excluding the boundary points (that's called the interior of the support). [Q5] Show by induction that Dk+2m-b-1 Bi.e(x) = 1 for all x [x1;XK) [Q6] Show that Bim is a piece wise polynomial of degree mon (Xo; Xk+1] with "breaks" only at the knots {x}}}=1 [Q7] Show that an order m B-spline basis functions is obtained as the convolution of m +1 order 0 B-spline basis functions. [Q8] Going back to our definition of polynomial splines, why are the B splines a basis? In this problem, we are going to investigate how to construct a basis called B-splines for the splines of order l1 Bil(x) Bie-1(x) + -Bi+1,2-1(x) Vie [1; k + 2m - -1]. (2) Xite- Xi+2+1 - Xi+1 This definition is very mathematical, and you should get in the habit of specializing the definition to get some intuition. In this problem, you are guided through that process, but this should become a natural reflex. [Q1] Sketch the order 0 splines in a situation with k = 2. This requires setting at least m= 0. (Sketching mean graphing a non exact but representative plot of what's going on) [Q2] Sketch the order 1 splines in the same situation with k = 2. This requires setting at least m = 1. (Sketching mean graphing a non exact but representative plot of what's going on) [Q3] Show by induction that Bil(x) = 0 for x [i, i+e+1). This shows that there is a bound on the support of a spline. [Q4] Show by induction that Bie(x) > 0 for x (Xi, ite+1). This shows that B splines are positive in their support when excluding the boundary points (that's called the interior of the support). [Q5] Show by induction that Dk+2m-b-1 Bi.e(x) = 1 for all x [x1;XK) [Q6] Show that Bim is a piece wise polynomial of degree mon (Xo; Xk+1] with "breaks" only at the knots {x}}}=1 [Q7] Show that an order m B-spline basis functions is obtained as the convolution of m +1 order 0 B-spline basis functions. [Q8] Going back to our definition of polynomial splines, why are the B splines a basis
