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Learning Project SummaryEdit
This learning project aims to provide as an introduction to the specialized spline functions known as B (or Basis) splines. B splines have varying applications, including numerical analysis.
- Define B Spline
- Differentiate types of B Splines
- Approximation using B Splines
Lesson 0: PrerequisiteEdit
B splines derive from Splines, therefore an understanding of Splines in general is beneficial to completing this lesson. In short, a Spline function approximates another function by defining a set of polynomials
where each of these polynomials defines a specific piece of the resulting Spline. might exist on the interval , might exist on the interval , and so on. The result will be a piecewise approximation to some other exact function.
Lesson 1: DefinitionEdit
A definition of B splines assumes:
- an infinite set of knots are defined at points along the x-axis (can be spaced uniformally or not), that is,
Degree 0 (or constant)Edit
With that in mind, we can now move on to the simplest of B splines, those of degree 0, which are defined as
In other words, a degree 0 B spline is equal to 0 at all points except on the interval .
It should now be easy to see that a degree 0 Spline can be formed as a weighted linear combination of degree 0 B splines so that,
Degree 1 (or linear)Edit
Logically, the next B spline are those of degree 1, defined as
This might seem difficult to visualize at first glance, but its actually quite easy. Just like , it is 0 at quite nearly all points. However, we now have the two intervals, and , at which .
On the first interval it is easy to see that, substituting and give 0 and 1, respectively. Thus, this function yields an upward sloping line, with a maximum height of 1. Similarly, the second interval yields a downward sloping line, starting from the point that the first interval terminates.
Again, similarly to , it should now be easy to see that a degree 1 Spline can be formed as a weighted linear combination of degree 1 B splines so that,
Degree k (or quadratic and above)Edit
The higher degree B splines, and actually including , are defined as
Lesson 2: ApproximationEdit
We have seen how B splines can be used to construct general Spline. Now we will discuss a process for approximating a generic function by using B splines.
This specific approximation utilizes (or quadratic B splines) to approximate a function with (or a quadratic Spline). The approximation is defined as
- , or the average of the next two knots
In real life, we would only approximate the function over a specific interval .
Active participants in this Learning Group
- Jlietz 18:04, 12 December 2006 (UTC)