Higher arities

As far as can be seen, all the interesting effects are covered above, and higher arities introduce no new features. 

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As we look at different analyses in the following chapters we shall take the binary primal subset first, to establish ideas, and then generalise to duals and to higher arities. 

It then turns out that the discontinuities found at such points may be totally different from those at the dyadic points. We can always make schemes of higher arity by considering two or more refinements as a single step. We call this squaring or taking a higher power of the scheme. 

For higher arity schemes we have the two complications noted in the previous chapter, that the equivalent of the (l + z)/2 found in the binary case becomes (1 + z +. .. + za )/a. 

The same result holds for higher arities. The smoothing factor is and... 

With higher arity still the kernel width can be higher. The fraction of each span with the lower number of support points is (k — 1 )/(a — 1), and this fraction is filled at the first step of refinement by a polynomial piece. The remainder is a fraction r = (a — k)/(a — 1) which is less than 1. It has that same fraction as the original filled at the next step by one or more polynomial pieces, and thus the amount remaining unfilled by polynomial pieces after m steps is rm which converges to zero. Thus the entire limit curve consists of polynomial pieces (of degree n — 1), but an unbounded number of them. 

Quaternary and higher arities are significantly more complex. Quaternary schemes can, of course be created by squaring a binary scheme, but there are others which are not so created. Note that the square of a weighted mean of two binary schemes is not the same as the weighted mean of the squares of those schemes. 

The situation gets considerably more complex for higher arities. 

The artifact analysis above is limited, essentially to binary and ternary schemes. Higher arities can bring in artifacts at higher frequencies which might spoil or improve the shapes of the limit curves, and it is not obvious how these can most sensibly be handled. 

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