B Splines

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Learning Project Summary[edit | edit source]

Content summary[edit | edit source]

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.

Goals[edit | edit source]

  • Define B Spline
  • Differentiate types of B Splines
  • Approximation using B Splines

Lessons[edit | edit source]

Lesson 0: Prerequisite[edit | edit source]

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: Definition[edit | edit source]

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 | edit source]

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 | edit source]

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 | edit source]

The higher degree B splines, and actually including , are defined as

Lesson 2: Approximation[edit | edit source]

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.

Schoenberg's Approximation[edit | edit source]

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[edit | edit source]

Active participants in this Learning Group

Mathematics since 00:11, 11 November 2013‎ --Marshallsumter (discusscontribs) 23:48, 11 May 2019 (UTC)

Inactive participants[edit | edit source]

  • Jlietz 18:04, 12 December 2006 (UTC)