B Splines

Please see Directions for use for more information.

Learning Project SummaryEdit

Content 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.

GoalsEdit

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


LessonsEdit

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.

Schoenberg's ApproximationEdit

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 participantsEdit

Active participants in this Learning Group

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

Inactive participantsEdit

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