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I have a infinitely differentiable, bijective function $f:[0,1]\to[0,1]$, and I would like to approximate this function by a series of other functions $T_i$ (think Taylor) – with the conditions that

  • all the approximants are also bijective and
  • its inverse must be representable by elementary functions (addition, multiplication, exponentation, $\exp$, $\log$, $\sin$ etc.).

Taylor polynomials do not fit this bill since, as polynomials, root finding is not elementary.

Any hints here? Links to articles could suffice.

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  • $\begingroup$ I do not understand why Taylor polynomials do not fit. The inverse of a polynomial is an elementary function. $\endgroup$ – Levent Jan 26 '18 at 0:02
  • $\begingroup$ Only up to degree 4. By the Abel–Ruffini theorem, there is no algebraic solution to general polynomials equations of degree 5 or higher. $\endgroup$ – Nico Schlömer Jan 26 '18 at 0:05
  • $\begingroup$ As crude as it is, do you consider piecewise linear functions elementary? $\endgroup$ – orlp Jan 26 '18 at 0:09
  • $\begingroup$ @orlp A good suggestion. For the actual application I'm working towards (the same thing in higher dimensions) piecewise linear functions probably aren't that useful though. Given a point in a discretized $n$D space, it'll be computationally expensive to find which segment (/simplex) the point is in. Nevertheless: A good suggestion. $\endgroup$ – Nico Schlömer Jan 26 '18 at 0:20
  • $\begingroup$ @NicoSchlömer Well, you can setup piecewise linear functions such that the output domain is divided into equidistant sections, which makes the inverse trivial to compute (at the cost of forward computation time). It depends on whether you mostly do inverses or forward invocations. $\endgroup$ – orlp Jan 26 '18 at 0:24

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