Question

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Python
A quadratic B-spline is a smooth quadratic function, which can be defined interms of truncated polynomials.

B delta (x)= 1/(2delta ^ 2) ((x-0) + ^ 2 -3(x- delta) + ^ 2 +3(x-2 delta) + ^ 2 -(x-3 delta) + ^ 2 )

Write a function 'bspline' which takes arguments x and & and returns B_{delta}(x) Thedefault value of & should be 1. Compute the value of bspline (19, 10) andbspline (1:20, 5)

Answer 1

A quadratic B-spline function B_{δ}(x) is defined using truncated polynomials. To compute values for bspline(19, 10) and bspline(1:20, 5), one would use the bspline function, substituting the appropriate values for x and δ.

Step-by-step explanation:

The quadratic B-spline function, Bδ(x), can be defined using truncated polynomials as follows:

Bδ(x) = \frac{1}{2δ^2} ((x-0)_{+}^2 - 3(x-δ)_{+}^2 + 3(x-2δ)_{+}^2 - (x-3δ)_{+}^2)

Where (x-a)_{+} means that the expression is taken as x-a if x is greater than a, and 0 otherwise. This ensures the function is piecewise and its value is 'truncated' or set to zero whenever x is less than a.

The B-spline function is smooth and defined piecewise. To write the function 'bspline', we could use the following pseudo-code:

function bspline(x, δ=1){ return 1/(2*δ^2) * (max(0, x-0)^2 - 3*max(0, x-δ)^2 + 3*max(0,x-2*δ)^2 - max(0,x-3*δ)^2); }

To calculate bspline(19, 10) and bspline(1:20, 5), you would substitute x with 19 and delta with 10 for the first case, and x with 1 through 20 and delta with 5 for the second case, respectively, in the bspline function to obtain the values.