Analysis in Theory and Applications

, Volume 19, Issue 2, pp 153–159 | Cite as

Interpolation with restricted arc length



For given data (ti, yi), i=0,1,...,n,0=t0<t1<...<tn=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.

Key Words

interpolation knot quadratic spline 

AMS (2000) subject classification



Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Bojanov B., Hakopian, H. and Sahakian, A., Spline Functions and Multivariate Interpolations, Mathematics and Its Applications, Vol. 248, Kluwer Academic Publishers, Dordrecht-Boston-London, 1993.MATHGoogle Scholar
  2. [2]
    De Boor C. A Remark Concerning Perfect Splines, Bull. Amer. Math. Soc., 80(1974), 724–727.MATHMathSciNetCrossRefGoogle Scholar
  3. [3]
    Favard, J. Sur I’interpolation. J. Math. Pures Appl., 19(1940), 281–306.MATHMathSciNetGoogle Scholar
  4. [4]
    Glaeser, G., Prolongement Extrémal de Functions Différentiables D’une Variable, J. Approx. Theory, 8(1973), 249–261.MATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    Karlin, S., Some Variational Problems on Certain Sobolev Spaces and Perfect Splines, Bull. Amer. Math. Soc., 79(1973), 124–128.MATHMathSciNetGoogle Scholar

Copyright information

© Springer 2003

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of SofiaSofiaBulgaria

Personalised recommendations