<s>
In	O
mathematics	O
,	O
the	O
Whitney	B-Algorithm
inequality	I-Algorithm
gives	O
an	O
upper	O
bound	O
for	O
the	O
error	O
of	O
best	O
approximation	O
of	O
a	O
function	O
by	O
polynomials	O
in	O
terms	O
of	O
the	O
moduli	B-Algorithm
of	I-Algorithm
smoothness	I-Algorithm
.	O
</s>
<s>
It	O
was	O
first	O
proved	O
by	O
Hassler	O
Whitney	O
in	O
1957	O
,	O
and	O
is	O
an	O
important	O
tool	O
in	O
the	O
field	O
of	O
approximation	B-Algorithm
theory	I-Algorithm
for	O
obtaining	O
upper	O
estimates	O
on	O
the	O
errors	O
of	O
best	O
approximation	O
.	O
</s>
<s>
The	O
moduli	B-Algorithm
of	I-Algorithm
smoothness	I-Algorithm
of	O
order	O
of	O
a	O
function	O
are	O
defined	O
as	O
:	O
</s>
<s>
where	O
is	O
the	O
finite	B-Algorithm
difference	I-Algorithm
of	O
order	O
.	O
</s>
<s>
The	O
theorem	O
is	O
particularly	O
useful	O
when	O
applied	O
on	O
intervals	O
of	O
small	O
length	O
,	O
leading	O
to	O
good	O
estimates	O
on	O
the	O
error	O
of	O
spline	B-Algorithm
approximation	O
.	O
</s>
<s>
The	O
original	O
proof	O
given	O
by	O
Whitney	O
follows	O
an	O
analytic	O
argument	O
which	O
utilizes	O
the	O
properties	O
of	O
moduli	B-Algorithm
of	I-Algorithm
smoothness	I-Algorithm
.	O
</s>
