Question 1Consider i) interpolation and ii) regression of a two-dimensional functionf(x) wheref:R2!R. Assume that the approximating functionp(x) in both cases is a linear combination ofNdistinct basis functionsφi(x), and thatfis sampled atMdistinct locations. The interpolationconditions for i) lead to a system of linear equations, with system matrixA.Consider the existence and uniqueness of solutions to these two problems.What condition isrequired on i) and ii) respectively, to guarantee unique solutions for each problem?

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Question 2Steady laminar flow through a channel with parallel walls has aparabolicvelocityprofile (“Poiseuille” flow).In a channel of unit height on the intervaly2[0,1] we have twoapproximatemeasurements of the velocity, aty0=13andy1=23with valuesv0= 7.3 andv1= 8.7.In addition we know that the velocity at both channel walls isexactlyzero (by the no-slip boundarycondition), i.e.v(0) =v(1) = 0. Solve a mixed interpolation/regression problem to approximatev(y), using a quadratic polynomial. What is the maximum velocity in the approximation ofv?A: UnansweredB: 7.5C: 8D: 8.5E: 9F: 9.5G: 10Question 3Consider a generalized spline interpolants(x) on the interval [0,10]. The intervalis divided into sub-intervals withN+ 1 nodes:0 =x0< x1<· · ·< xN-1< xN= 10at which sample data are given. Assume that the spline consists of polynomials of degreedoneach interval, and that the spline is required to bed-1-times continuously di↵erentiable on theentire interval.We impose 6 constraints at the boundaries:f0(0) = 1,f00(0) = 0,f000(0) = 0,f0(10) = 0,f00(10) = 0 andf000(10) = 0. What is the degreedof the polynomials needed to construct such aspline?

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