Interpolation Degree

The value of ao is always 1. The interpolating degree is determined by the subscript, i, of the first non-zero a. ... 

If a given polynomial is to be reproduced, it must be within the span of the scheme, and it must also be interpolated. We can therefore see without any sophisticated argument that the degree of reproduction is just the lower of the spanning degree and the interpolation degree. 

These are the support, the generation degree and the interpolating degree. 

The interpolation degree is given by one less than the first (non-zero) power of 6 with a non-zero coefficient in this expansion of the symbol, and so for linear interpolation we have effectively no constraint. 

This linear subspace can usefully be made more explicit by taking as the corners of our barycentric combination not the B-splines, but schemes satisfying the interpolation degree constraints. 

The interpolation degree can also be expressed in terms of the presentation of the unit row eigenvector as a polynomial in S2. Since the unit row eigenvector is often shorter than the mask, this might be slightly advantageous. 

Because the unit row eigenvector is not a linear function of the mask, the quasi-interpolation degree is not necessarily a linear constraint in design space. 

However, for primal schemes of any arity, any scheme with interpolation degree c will leave original vertices unchanged if they lie on a polynomial of degree less than or equal to c . A weighted mean of equal points is always the same point, and so this condition remains a linear one. 

The simple result here is that each of the polynomial degrees we considered is no lower than the lowest of the individual steps. If a given generation or interpolation degree is required, then the trajectory of the scheme should lie entirely within the set of schemes that has that degree. 

It is also possible to increase the interpolation degree without going all the way, by using narrow-banded approximations to the inverse of E. 

End-conditions for Schemes with Higher Quasi-interpolation Degree... 

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