Let
f
f
be a two variable continuously differentiable real-valued function of certain order on
[
0
,
1
]
2
{[0,1]^2}
and let
L
L
be a linear differential operator involving mixed partial derivatives and suppose that
L
(
f
)
≥
0
L(f) \geq 0
. Then there exists a sequence of two-dimensional polynomials
Q
m
,
n
(
x
,
y
)
{Q_{m,n}}(x,y)
with
L
(
Q
m
,
n
)
≥
0
L({Q_{m,n}}) \geq 0
, so that
f
f
is approximated simultaneously and uniformly by
Q
m
,
n
{Q_{m,n}}
. This approximation is accomplished quantitatively by the use of a suitable two-dimensional first modulus of continuity.