The Fourier formula for discontinuous functions of several variables

THE FOURIER FORMULA FOR DISCONTINUOUS
FUNCTIONS OF SEVERAL VARIABLES
A.N. Podkorytov and Mai Van Minh
We would like to talk about following problem. Suppose Ω is convex compact
set in R m and χ Ω its indicator. Obviously its Fourier transformation ̂χ Ω is not
summable in R m , if Int(Ω) ≠ ∅. The problem is to interpret the integral in the
R.H.S. of the of inversion formula
∫
χ Ω (y 0 ) = ̂χ Ω (x)e −2πix·y0 dx,
R m
while keeping the equality true. For example if Ω = □ = [a 1 , b 1 ] × . . . × [a m , b m ] is
a rectangular parallelepiped it is not hard to verify that
∫
̂χ □ (x)e −2πix·y0 dx −→ χ □ (y 0 ) for y 0 /∈ ∂□.
[−R,R] m R→+∞
But as mentioned in [1], if Ω = ○ = {y ∈ R m | ‖y‖ ≤ 1} is a sphere then the
situation becomes more complicated. If y 0 ≠ 0 then
∫
̂χ ○ (x)e −2πix·y0 dx −→ χ ○ (y) for y 0 /∈ ∂○
R→+∞
‖x‖≤R
(in case ‖y 0 ‖ = 1 the limit equals 1 2 ). At the same time when m ≥ 3 at y 0 = 0 these
integrals do not have a limit. In [2] the sphere {‖x‖ ≤ R} have been replaced by a
cube and it has been ∫ shown that when m = 3 and y 0 = 0 the following equality is
correct 1 = lim ̂χ
R→+∞
[−R,R] 3 ○ (x) dx. In order to extend this result, we could prove
that for every convex compact set Ω ⊂ R m , m ≥ 2, the following equality is correct
∫
χ Ω (y 0 ) =
̂χ Ω (x)e −2πix·y0 dx for y 0 /∈ ∂Ω,
lim
R→+∞
x∈RW
if W is polyhedron in R m , 0 ∈ Int(W ) and the sides of W and their extensions do
not cross the origin. If m = 2 we can take as W any compact convex neighborhood
of the origin instead of polygon. In this case the inversion formula remains the
same for y 0 /∈ ∂Ω. And, what is more interesting, if W is simmetrical (relatively
the origin) at every point of y 0 ∈ ∂Ω there is finite limit of integrals on RW . It
equals 1 2 , if y 0 is not boundary angular point (vertex). For boundary angular point
this limits depends on choice of W .
[1] Pinsky M. A., Stanton N. K., Trapa P. E. Fourier series of radial functions
in several variables. J. Funct. Anal. 1993. V. 116. P. 111 – 122.
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2 A.N. PODKORYTOV AND MAI VAN MINH
[2] Harada C., Nakai E. The square partial sums of the Fourier transform of
radial functions in three dimensions. Sc. Math. Japon. 2002. V. 55, No. 3. P. 467
– 477.
[3] Podkorytov A.N., Mai Van Minh. The Fourier formula for discontinuous
functions of several variables. J. Math. Sci. 2004. Vol. 124. No. 3. P. 5018 –
5025.
[4] Mai Van Minh. The Fourier integral for discontinuous functions of two variables.
Vestnik S.-Peterburgskogo univ. 2006. Ser. 1. No. 1. P. 45 – 49.