Geodesic
Averaging Interpolation of Hermite-Fejér Type
Canadian mathematical bulletin, Tome 19 (1976) no. 3, pp. 315-321

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Averaging interpolation generalizes polynomials of Lagrange interpolation and also the next-to-interpolatory polynomials. This notion has recently been introduced by Motzkin, Sharma and Straus [3]. Later in [5], Saxena and Sharma have considered the convergence problem of the averaging interpolators on Tchebycheff abscissas. It appears that averaging interpolators have convergence properties similar to those of Lagrange interpolation. It is therefore reasonable to look for an extension of these operators along the lines of Hermite-Fejér interpolation. This can be done in three ways: (i) by taking assigned averages of function-values and by taking the derivatives to be zero, (ii) by taking assigned function-values and by taking the averages of derivatives to be zero, or (iii) by taking averages of function-values and by taking the averages of derivatives to be zero. The object of this note is to take the first approach. The second approach has been the subject of study by M. Botto and A. Sharma [2]. 
Saxena, R. B. Averaging Interpolation of Hermite-Fejér Type. Canadian mathematical bulletin, Tome 19 (1976) no. 3, pp. 315-321. doi: 10.4153/CMB-1976-048-8
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     title = {Averaging {Interpolation} of {Hermite-Fej\'er} {Type}},
     journal = {Canadian mathematical bulletin},
     pages = {315--321},
     year = {1976},
     volume = {19},
     number = {3},
     doi = {10.4153/CMB-1976-048-8},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1976-048-8/}
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