On High-order Interpolation and Non-linear Interpolation

Defining a mechanism to represent polynomials in OtVar’s mechanism is possible, if one axis value could be assigned to multiple axes. From the equation (2) in the previous article, if in v we have v{0}=v{1}=t, then we could define a region m that satisfies W(m,v)=t{2} by making the axis region m{0} and m{1} as axis regions that peaked at 1. Terms with higher degree could also be constructed in the similar manner.

Actually, with duplicate axis, we could define a ring for variable quantities, since the multiplication:

{(x̂⋅ŷ)(v)}	{=x̄⋅ȳ}	{(1)}
{}	{+x̄⋅{{∑}m∈𝔐{y}}D{y}[m]W(m,v)}	{(2)}
{}	{+ȳ⋅{{∑}m∈𝔐{x}}D{x}[m]W(m,v)}	{(3)}
{}	{+{{∑}m{x}∈𝔐{x}} {{∑}m{y}∈𝔐{y}}D{x}[m{x}]D{y}[m{y}]W(m{x},v)W(m{y},v)}	{(4)}

Part (1), (2), (3) are already representable in OtVar, and in the part (4), we could use duplicate axes to ensure that for any region pair m{x} and m{y}, their axis index set does not overlap, which made the product W(m{x},v)W(m{y},v) representable in the OtVar form.

If OtVar could further extended to support non-linear variation, the simplest change is to add a degree parameter to each region, which changes the region-weighting function into:

{W′(m,v)={{∏}a∈𝔄}Λ(m[a],v[a]){d[m[a]]}}

{(5)}

where d[m[a]] denotes the degree parameter of axis region m[a].