Meshless methods using least-squares approximations and kernel approximatio
ns are based on non-shifted and shifted polynomial basis, respectively. We
show that, mathematically, the shifted and nonshifted polynomial basis give
rise to identical interpolation functions when the nodal volumes are set t
o unity in kernel approximations. This result indicates that mathematically
the least-squares and kernel approximations are equivalent. However, for l
arge point distributions or for higher-order polynomial basis the numerical
errors with a non-shifted approach grow quickly compared to a shifted appr
oach, resulting in violation of consistency conditions. Hence, a shifted po
lynomial basis is better suited from a numerical implementation point of vi
ew. Finally, we introduce an improved finite cloud method which uses a shif
ted polynomial basis and a fixed-kernel approximation for construction of i
nterpolation functions and a collocation technique for discretization of th
e governing equations. Numerical results indicate that the improved finite
cloud method exhibits superior convergence characteristics compared to our
original implementation [Aluru and Li (2001)] of the finite cloud method.