Interpolation with restricted arc length

Abstract

For given data (t_i, y_i), i=0,1,...,n,0=t₀<t₁<...<t_n=1 we study constrained interpolation problem of Favard type

$$\inf \left\{ {\left\| {f''} \right\|\left. {_\infty } \right|f \in W_{_\infty }^2 \left[ {0,1} \right],f\left( {t_i } \right) = y_i ,i = 0, \cdots n,l\left( {f;\left[ {0,1} \right]} \right) \leqslant l_0 } \right\}$$

where$l\left( {f;\left[ {0,1} \right]} \right) = \int_0^1 {\sqrt {1 + f'^2 \left( x \right)} d} x$ is the arc length of f in [0,1]. We prove the existence of a solution f, of the above problem, that is a quadratic spline with a second derivative f., which coincides with one of the constants$\left\| {f''.} \right\|_\infty ,0,\left\| {f''.} \right\|_\infty$ between every two consecutive knots. Thus, we extend a result of Karlin concerning Favard problem, to the case of restricted length interpolation.