Shepard operators of Birkhoff-type

An Iterative Modification of Shepard–Bernoulli Operator

“…We study the numerical behavior in an ideal setting when all the required values of f and its derivatives are known, as in others papers (see, e.g., [1,4,5,7]). …”

“…This modified Shepard method is one of the most powerful software tools for the multivariate interpolation of large scattered data sets. Also, for improving the qualities of Shepard method, several authors (see, e.g., [1][2][3][4][5]7,15]), starting with Shepard himself, have suggested to apply (Sf )(x, y) not directly to f (x i , y i ), but to some interpolation operator P [f, (x i , y i )](x, y) at (x i , y i ) by considering the so-called combined operator: Using this method there have been constructed several types of combined Shepard operators. In [3] we have introduced the combined Shepard operators of Bernoulli type, using both the classical and the quadratic Shepard methods.…”

Fast computation of triangular Shepard interpolants

“…However, a drawback of the Shepard method is its low polynomial precision (only constant) that badly affects the reconstructed surface. To overcome this drawback several variants of the Shepard method have been introduced [11,12,13,14,15,8,16] by combining Shepard basis functions with local polynomial interpolants with higher polynomial reproduction properties, even by using supplementary derivative data, if given. The triangular Shepard method [17] combines triangle-based weight functions with local linear interpolants and reaches linear precision without using derivative data.…”