The one-dimensional case can be extended to multiple dimensions through
the use of the tensor product spline constructs. A spline subspace
is defined for each dimension as discussed in the preceding section.
Thereafter a tensor product function in the 
th-dimensional space can
be expressed as:
Tensor product splines provide a straightforward and simple generalization of the one-dimensional spline functions. Multivariate approximation of functions by tensor products has a strong directional bias along lines parallel to the axis direction. Difficult functional features running along different directions require a fine mesh in all the functional space in order to achieve accurate approximation. This increases the computational demands of the problem. Recently, advances in mathematical research on the theory and computational methods of multivariate splines [20] have shown great potential in various applications without the constraints of tensor product constructs.
In the two-dimensional space, the equivalent TPS formulation of (2.73) can be written as:
Where 
is the ith B-spline of order 
for the knot
sequence 
, 
and 
the number of knots in the X and
Y direction respectively.
As in the one-dimensional case, the coefficients 
can be
selected to enforce interpolating or more general conditions.