In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, in the field of
harmonic analysis
Harmonic analysis is a branch of mathematics concerned with investigating the connections between a function and its representation in frequency. The frequency representation is found by using the Fourier transform for functions on unbounded do ...
,
the van der Corput lemma is an estimate for
oscillatory integral
In mathematical analysis an oscillatory integral is a type of distribution. Oscillatory integrals make rigorous many arguments that, on a naive level, appear to use divergent integrals. It is possible to represent approximate solution operators for ...
s
named after the
Dutch
Dutch or Nederlands commonly refers to:
* Something of, from, or related to the Netherlands
** Dutch people as an ethnic group ()
** Dutch nationality law, history and regulations of Dutch citizenship ()
** Dutch language ()
* In specific terms, i ...
mathematician
J. G. van der Corput.
The following result is stated by
E. Stein:
Suppose that a real-valued function
is smooth in an open interval
,
and that
for all
.
Assume that either
, or that
and
is monotone for
.
Then there is a constant
, which does not depend on
,
such that
:
for any
.
Sublevel set estimates
The van der Corput lemma is closely related to the
sublevel set estimates,
[M. Christ, ''Hilbert transforms along curves'', Ann. of Math. 122 (1985), 575–596]
which give the upper bound on the
measure of the set
where a function takes values not larger than
.
Suppose that a real-valued function
is smooth
on a finite or infinite interval
,
and that
for all
.
There is a constant
, which does not depend on
,
such that
for any
the measure of the sublevel set
is bounded by
.
References
Inequalities (mathematics)
Harmonic analysis
Fourier analysis