Dual binary schemes

Here the mask has an even number of entries. The calculations above follow through in exactly the same way, except that the range of influence of an original control point reaches to a point half-way between images of control points. This means that we expect there to be discontinuities of some derivative at these half-way points, rather than at the images of control points. 

For example, the binary scheme whose mask is [1,3,3, l]/4 (the binary quadratic B-spline) has a support width of (4 — l)/2 = 3/2 spans on each side. 

We see the same effect in the growth of the extent of influence of one control point after 0,1,oo refinements in the quadratic B-spline scheme. Again, the refined polygons converge towards the basis function. 

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Note that for dual binary schemes the distribution of control points gives an extra half-integer-worth of curve. 

What the choice of a diagonal does is to imply a labelling, giving a correspondence between a sequence of points of the old polygon and a sequence of the refined one. In particular it implies a mark point which is an abscissa value which maps into itself under the map from old abscissa values to new ones. In the case of a primal binary scheme, the mark point is at a point of both new and old polygons. In the case of a dual scheme the mark point is at a mid-edge in both old and new. 

We consider first binary schemes primal and dual need not be distinguished. If the symbol has d+ 1 factors of (1 + z)j2 then... 

Note that t here might not be an integer for binary dual schemes each t will be an odd half-integer, s will be an integral multiple of the arity. These complicated conventions are designed exactly to make these last two equations simple. 

The space of binary uniform stationary schemes is therefore understandable in terms of two components one contains all primal schemes, with an odd number of entries in the mask, which is a linear combination of odd degree B-splines the other all dual schemes, with an even number of entries in the mask, which is a linear combination of even degree B-splines. In principle each is only a countably infinite-dimensional space, and in practice each is only a finite dimensional space because we shall not wish to include B-splines above some maximum degree in order to keep the support limited. 

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