Mu
.
c lu
.
c
Mo
.
’
dˆa
`
u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
C´ac k´y hiˆe
.
u d`ung trong luˆa
.
n ´an . . . . . . . . . . . . . . . . . . 11
L`o
.
i ca
’
m o
.
n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Chu
.
o
.
ng 1 . T´ıch chˆa
.
p c´o h`am tro
.

.
i ph´ep biˆe
´
n d ˆo
’
i t´ıch phˆan Fourier sine 32
Chu
.
o
.
ng 2 . T´ıch chˆa
.
p suy rˆo
.
ng dˆo
´
i v´o
.
i hai ph´ep biˆe
´
n
dˆo
’
i t´ıch phˆan 41
2.1 T´ıch chˆa
.
p suy rˆo
.
ng c´o h`am tro
.

ng dˆo
´
i v´o
.
i c´ac ph´ep biˆe
´
n dˆo
’
i t´ıch phˆan Fourier v`a
Fourier sine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
2.4 T´ıch chˆa
.
p suy rˆo
.
ng c´o h`am tro
.
ng dˆo
´
i v´o
.
i c´ac ph´ep biˆe
´
n dˆo
’
i t´ıch phˆan
Fourier v`a Fourier cosine . . . . . . . . . . . . . . . . . . . . . . . 80
Chu
.
o
.

3.2 T´ıch chˆa
.
p suy rˆo
.
ng c´o h`am tro
.
ng dˆo
´
i v´o
.
i c´ac ph´ep biˆe
´
n dˆo
’
i t´ıch phˆan
Fourier cosine, Fourier v`a Fourier sine . . . . . . . . . . . . . . . . 108
3.3 T´ıch chˆa
.
p suy rˆo
.
ng Fourier sine, Fourier v`a Fourier cosine . . . . . 126
Kˆe
´
t luˆa
.
n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
T`ai liˆe
.
u d˜a cˆong bˆo
´

’
i t´ıch
to´an ho
.
c v`a du
.
o
.
.
c ph´at triˆe
’
n liˆen tu
.
c trong suˆo
´
t gˆa
`
n mˆa
´
y tr˘am n˘am qua. Ph´ep
biˆe
´
n dˆo
’
i t´ıch phˆan d´ong vai tr`o quan tro
.
ng trong to´an ho
.
c c˜ung nhu
.

n biˆen cu
’
a phu
.
o
.
ng tr`ınh vi phˆan, phu
.
o
.
ng tr`ınh
da
.
o h`am riˆeng, phu
.
o
.
ng tr`ınh t´ıch phˆan v`a c´ac b`ai to´an cu
’
a vˆa
.
t l´y to´an. C´ac
ph´ep biˆe
´
n dˆo
’
i t´ıch phˆan l`a nh˜u
.
ng cˆong cu
.

’
n´oi trong l´o
.
p nh˜u
.
ng ph´ep biˆe
´
n dˆo
’
i t´ıch phˆan phˆo
’
biˆe
´
n nhˆa
´
t, c´o
´u
.
ng du
.
ng rˆo
.
ng r˜ai nhˆa
´
t th`ı c´ac ph´ep biˆe
´
n dˆo
’
i Fourier, Fourier cosine v`a Fourier
sine ra d`o

bˆa
´
y gi`o
.
nhu
.
Lagrange, Poisson, Laplace, nˆen pha
’
i dˆe
´
n n˘am 1815 cˆong tr`ınh
khai s´ang cho l´y thuyˆe
´
t c´ac ph´ep biˆe
´
n dˆo
’
i t´ıch phˆan cu
’
a Fourier m´o
.
i du
.
o
.
.
c cˆong
bˆo
´
. L´y thuyˆe

nhu
.
mˆo
.
t minh ch´u
.
ng thuyˆe
´
t phu
.
c cho su
.
.
g˘a
´
n kˆe
´
t gi˜u
.
a c´ac ph´ep biˆe
´
n dˆo
’
i t´ıch
phˆan cu
’
a to´an ho
.
c v´o
.

t hu
.
´o
.
ng
ph´at triˆe
’
n m´o
.
i cu
’
a l´y thuyˆe
´
t c´ac ph´ep biˆe
´
n dˆo
’
i t´ıch phˆan l`a t´ıch chˆa
.
p cu
’
a c´ac
ph´ep biˆe
´
n dˆo
’
i t´ıch phˆan xuˆa
´
t hiˆe
.

.
p cu
’
a ph´ep biˆe
´
n dˆo
’
i Laplace [40, 44], t´ıch chˆa
.
p cu
’
a ph´ep
biˆe
´
n dˆo
’
i Mellin [40] v`a sau d´o l`a c´ac t´ıch chˆa
.
p cu
’
a c´ac ph´ep biˆe
´
n dˆo
’
i Hilbert
[16, 44], ph´ep biˆe
´
n dˆo
’
i Hankel [20, 51], ph´ep biˆe

.
ng du
.
ng l´y th´u
trong t´ınh to´an t´ıch phˆan, t´ınh tˆo
’
ng cu
’
a chuˆo
˜
i, gia
’
i c´ac b`ai to´an Vˆa
.
t l ´y to´an,
phu
.
o
.
ng tr`ınh vi phˆan da
.
o h`am riˆeng, phu
.
o
.
ng tr`ınh t´ıch phˆan, hˆe
.
phu
.
o

iyx
dy, y ∈ R
ta c´o ph´ep biˆe
´
n dˆo
’
i ngu
.
o
.
.
c
f(y) = (F
−1
f)(y) =
1
√
2π

+∞
−∞
˜
f(x)e
−iyx
dy.
T´ıch chˆa
.
p dˆo
´
i v´o

Ph´ep to´an t´ıch chˆa
.
p n`ay thoa
’
m˜an d˘a
’
ng th´u
.
c nhˆan tu
.
’
h´oa
F (f ∗
F
g)(y) = (F f)(y).(F g)(y), ∀y ∈ R. (0.2)
V´o
.
i ph´ep biˆe
´
n dˆo
’
i Fourier cosine F
c
[12, 35, 40]
˜
f(y) = (F
c
f)(y) =

2

n dˆo
’
i t´ıch phˆan Fourier cosine F
c
cu
’
a hai h`am f v`a g
du
.
o
.
.
c di
.
nh nghi˜a b˘a
`
ng biˆe
’
u th´u
.
c [12, 40]
(f ∗
F
c
g)(x) =
1
√
2π

+∞

.
i ph´ep biˆe
´
n dˆo
’
i t´ınh phˆan Laplace L c´o
da
.
ng [40, 52]
(f ∗
L
g)(x) =

x
0
f(x − t)g(t)dt, x > 0 (0.5)
v`a tho
’
a m˜an d˘a
’
ng th´u
.
c nhˆan tu
.
’
h´oa
L(f ∗
L
g)(y) = (Lf)(y)(Lg)(y), ∀y > 0. (0.6)
O

ng n˘am 50 cu
’
a thˆe
´
ky
’
tru
.
´o
.
c c´ac t´ıch chˆa
.
p d˜a du
.
o
.
.
c biˆe
´
t
l`a c´ac t´ıch chˆa
.
p khˆong c´o h`am tro
.
ng v`a nhiˆe
`
u ph´ep biˆe
´
n dˆo
’

phˆan rˆa
´
t ha
.
n chˆe
´
v`a gˆa
`
n nhu
.
khˆong ph´at triˆe
’
n du
.
o
.
.
c. Nhu
.
ng nh˜u
.
ng bˆe
´
t˘a
´
c
n`ay d˜a du
.
o
.

i. D´o l`a t´ıch
chˆa
.
p v´o
.
i h`am tro
.
ng γ
0
(x) =
π
xsh(πx)

Γ

P + ix +
1
2

−2
dˆo
´
i v´o
.
i ph´ep biˆe
´
n dˆo
’
i
t´ıch phˆan Mehler - Fox [46] du

’
di
.
nh ngh˜ıa t´ıch chˆa
.
p cu
’
a ph´ep biˆe
´
n dˆo
’
i t´ıch phˆan
K v´o
.
i h`am tro
.
ng γ(x) du
.
.
a trˆen d˘a
’
ng th´u
.
c nhˆan tu
.
’
h´oa
K(f
γ
∗ g)(x) = γ(x)(Kf)(x).(Kg)(x).

´
i v´o
.
i c´ac ph´ep biˆe
´
n dˆo
’
i t´ıch phˆan kh´ac. C´ac t´ıch
chˆa
.
p c´o h`am tro
.
ng du
.
o
.
.
c t`ım ra, ch˘a
’
ng ha
.
n nhu
.
t´ıch chˆa
.
p dˆo
´
i v´o
.
i ph´ep biˆe

i t´ıch phˆan Fourier sine F
s
[20, 31]
(f
γ
1
∗
F
s
g)(x) =
1
2
√
2π

+∞
0
f(x)[g(x + 1 + t) + g(|x + 1 − t|) sign ( x + 1 − t)
4
+ g(|x − 1 + t|) sign (x − 1 + t)+g(|x − 1 − t|) sign (x − 1 − t)]dt.
(0.7)
V´o
.
i t´ıch chˆa
.
p n`ay ta c´o d˘a
’
ng th´u
.
c nhˆan tu

2
π

+∞
−∞
f(x) sin(yx)dx.
T´ıch chˆa
.
p v´o
.
i h`am tro
.
ng γ
2
(y) = y
−ν
cu
’
a 2 h`am f, g dˆo
´
i v´o
.
i ph´ep biˆe
´
n dˆo
’
i t´ıch
phˆan Hankel H du
.
o

+∞
0
u
ν+1
f(u).g(
√
x
2
+ u
2
− 2xu cos t)
(x
2
+ u
2
− 2xu cos t)
ν
2
du.
D˘a
’
ng th´u
.
c nhˆan tu
.
’
h´oa cu
’
a t´ıch chˆa
.

.
.
c x´ac di
.
nh nhu
.
sau
(H
ν
φ)(t) =

+∞
0
τJ
ν
(tτ)φ(τ )dτ.
v´o
.
i J
ν
l`a h`am Bessel loa
.
i mˆo
.
t [5]. Nhu
.
d˜a biˆe
´
t t´ıch chˆa
.

nh`a to´an ho
.
c d˜a nghiˆen c´u
.
u ph´ep biˆe
´
n dˆo
’
i t´ıch phˆan kiˆe
’
u t´ıch chˆa
.
p kiˆe
’
u
Mellin, kiˆe
’
u Fourier, kiˆe
’
u Kontovich - Lebedev [7, 26, 49, 50]. Trong l´y thuyˆe
´
t
v`anh di
.
nh chuˆa
’
n ph´ep to´an t´ıch chˆa
.
p du
.

ng ´u
.
ng
du
.
ng l´y th´u v`a d˜a c´o nhiˆe
`
u cˆong tr`ınh khoa ho
.
c g˘a
´
n v´o
.
i nh˜u
.
ng ´u
.
ng du
.
ng n`ay
[13, 14, 28, 39, 42, 45, 47]. Nh`o
.
c´ac t´ıch chˆa
.
p v´o
.
i h`am tro
.
ng ra d`o
.

sung cu
’
a l´o
.
p t´ıch chˆa
.
p suy rˆo
.
ng, nhiˆe
`
u diˆe
`
u l´y th´u trong
l˜ınh vu
.
.
c n`ay m´o
.
i du
.
o
.
.
c ph´at hiˆe
.
n.
5
T´ıch chˆa
.
p suy rˆo

.
ng
cu
’
a hai h`am f v`a g dˆo
´
i v´o
.
i c´ac ph´ep biˆe
´
n dˆo
’
i t´ıch phˆan Fourier sine v`a Fourier
cosine [40].
(f ∗
1
g)(x) =
1
√
2π

+∞
0
f(y)[g(|x −y|) −g(x + y)]dy, x > 0 (0.9)
v´o
.
i d˘a
’
ng th´u
.

.
`o
.
ng ho
.
.
p cu
’
a
t´ıch chˆa
.
p suy rˆo
.
ng dˆo
´
i v´o
.
i c´ac ph´ep biˆe
´
n dˆo
’
i t´ıch phˆan m´o
.
i du
.
o
.
.
c tiˆe
´

´
i v´o
.
i c´ac ph´ep biˆe
´
n dˆo
’
i t´ıch phˆan kiˆe
’
u Melin [54]
(f ∗
2
g) =
1
(2πi)
2

σ
s

σ
t
k
∗
1
(s)k
∗
2
(t)
k

.
’
dˆay
(K
i
f) =

+∞
0
k
i

y
t

f(t)
dt
t
, i = 0, 1, 2, σ
s
= {z : Re z = γ}.
N˘am 1991 Yakubovich S. B. tiˆe
´
p tu
.
c cˆong bˆo
´
t´ıch chˆa
.
p suy rˆo

(s + t)
f
∗
(s)g
∗
(t)x
−s−t
dsdt, x > 0
v´o
.
i d˘a
’
ng th´u
.
c nhˆan tu
.
’
h´oa
G
1
(f ∗
3
g)(y) = (G
2
f)(y)(G
3
g)(y), ∀y > 0
6
o
.

+ s)
n
j

k=1
Γ(1 − α
j
k
− s)
p
j

k=n
j
+1
Γ(α
j
k
+ s)
q
j

k=m
j
+1
Γ(1 − β
γ
k
− s)
.

θ(x, y, z) = −
1
2

+∞
0

+∞
0

+∞
0
exp

1
2

uω
v
+
uω
u
+
uω
ω

×
k
3
(xu)k

I
k
2
iτ
g)
o
.
’
dˆay
I
k
γ
iτ
=

+∞
0
I
k
j
(τ, u)f(u)du
I
k
j
(τ, u) =

+∞
0
K
iτ

x´ac di
.
nh t´ıch chˆa
.
p suy rˆo
.
ng dˆo
´
i v´o
.
i 3 ph´ep biˆe
´
n dˆo
’
i t´ıch phˆan bˆa
´
t
k`y v´o
.
i h`am tro
.
ng γ(y) m`a dˆo
´
i v´o
.
i ch´ung luˆon c´o d˘a
’
ng th´u
.
c nhˆan tu

.
ng ph´ap n`ay mo
.
’
du
.
`o
.
ng cho mˆo
.
t sˆo
´
t´ıch chˆa
.
p
suy rˆo
.
ng m´o
.
i tiˆe
´
p tu
.
c xuˆa
´
t hiˆe
.
n. O
.
’

´
n dˆo
’
i
t´ıch phˆan. C`on nˆe
´
u K
1
= K
2
= K
3
ta c´o t´ıch chˆa
.
p suy rˆo
.
ng m´o
.
i du
.
o
.
.
c cˆong bˆo
´
gˆa
`
n dˆay nhu
.
: t´ıch chˆa

.
p suy rˆo
.
ng kh´ac. Ch˘a
’
ng ha
.
n nhu
.
t´ıch chˆa
.
p suy rˆo
.
ng dˆo
´
i
v´o
.
i ph´ep biˆe
´
n dˆo
’
i t´ıch phˆan Fourier cosine v`a ph´ep biˆe
´
n dˆo
’
i t´ıch phˆan Fourier
sine [30] du
.
o

g)(y) = (F
s
f)(y).(F
s
g)(y), ∀y > 0. (0.12)
T´ıch chˆa
.
p suy rˆo
.
ng v´o
.
i h`am tro
.
ng γ
3
(y) = sh
−1
(πy) dˆo
´
i v´o
.
i c´ac ph´ep biˆe
´
n dˆo
’
i
t´ıch phˆan Fourier cosine, Kontorovich - Lebedev, Fourier sine [33]
(f
γ
3

γ
3
∗
6
g)(y) = γ
3
(y).(K
−1
f)(y).(F
s
g)(y), ∀y > 0 (0.14)
o
.
’
dˆay K
−1
l`a to´an tu
.
’
Kontorovich - Lebedev nghi
.
ch da
’
o [5].
T´ıch chˆa
.
p suy rˆo
.
ng v´o
.

2π
2

+∞
0
[e
−uch(x−1+v)
+ e
−uch(x−1−v)
− e
−uch(x+1+v)
− e
−uch(x+1−v)
] × f(u)g(v)dudv, x > 0
8
c´o d˘a
’
ng th´u
.
c nhˆan tu
.
’
ho´a
F
s
(f
γ
4
∗
7

.
p
suy rˆo
.
ng. Xˆay du
.
.
ng v`a nghiˆen c´u
.
u c´ac t´ıch chˆa
.
p suy rˆo
.
ng thu
.
.
c su
.
.
c´o ´y ngh˜ıa
khoa ho
.
c trong l˜ınh vu
.
.
c l´y thuyˆe
´
t t´ıch chˆa
.
p v`a phu

i. Ch´ung tˆoi d˜a cho
.
n hu
.
´o
.
ng nghiˆen c´u
.
u cu
’
a luˆa
.
n ´an l`a xˆay du
.
.
ng v`a nghiˆen
c´u
.
u t´ıch chˆa
.
p, t´ıch chˆa
.
p suy rˆo
.
ng dˆo
´
i v´o
.
i c´ac ph´ep biˆe
´

p m´o
.
i dˆe
’
gia
’
i phu
.
o
.
ng tr`ınh t´ıch phˆan v`a hˆe
.
phu
.
o
.
ng tr`ınh t´ıch phˆan.
Nˆo
.
i dung cu
’
a luˆa
.
n ´an, ngo`ai phˆa
`
n mo
.
’
dˆa
`

i v´o
.
i 1 trong 2 ph´ep
biˆe
´
n biˆe
´
n dˆo
’
i t´ıch phˆan Fourier cosine, Fourier sine, du
.
a ra ´u
.
ng du
.
ng gia
’
i
phu
.
o
.
ng tr`ınh t´ıch phˆan kiˆe
’
u t´ıch chˆa
.
p v`a biˆe
’
u diˆe
˜

i d˜a du
.
o
.
.
c xˆay du
.
.
ng v`a nghiˆen c´u
.
u o
.
’
dˆay l`a: T´ıch chˆa
.
p c´o
h`am tro
.
ng dˆo
´
i v´o
.
i ph´ep biˆe
´
n dˆo
’
i t´ıch phˆan Fourier cosine; T´ıch chˆa
.
p c´o h`am
tro

i dˆo
´
i v´o
.
i 2 trong 3 ph´ep biˆe
´
n dˆo
’
i t´ıch phˆan Fourier, Fourier cosine,
Fourier sine. Nghiˆen c´u
.
u cˆa
´
u tr´uc ph´ep to´an nhˆan chˆa
.
p, nˆeu mˆo
´
i liˆen hˆe
.
v´o
.
i
c´ac t´ıch chˆa
.
p d˜a biˆe
´
t. Du
.
a ra c´ac ´u
.

`
n
nhiˆe
.
t theo cˆong th´u
.
c t´ıch chˆa
.
p tu
.
o
.
ng ´u
.
ng.
C´ac t´ıch chˆa
.
p suy rˆo
.
ng m´o
.
i d˜a du
.
o
.
.
c xˆay du
.
.
ng v`a nghiˆen c´u

i c´ac ph´ep biˆe
´
n
dˆo
’
i t´ıch phˆan Fourier cosine v`a Fourier sine. T´ıch chˆa
.
p suy rˆo
.
ng dˆo
´
i v´o
.
i c´ac
9
ph´ep biˆe
´
n dˆo
’
i t´ıch phˆan Fourier v`a Fourier sine. T´ıch chˆa
.
p suy rˆo
.
ng c´o h`am
tro
.
ng dˆo
´
i v´o
.

i v´o
.
i 3 ph´ep biˆe
´
n dˆo
’
i t´ıch phˆan riˆeng biˆe
.
t l`a Fourier, Fourier cosine,
Fourier sine. Nghiˆen c´u
.
u cˆa
´
u tr´uc cu
’
a ph´ep to´an nhˆan chˆa
.
p, nˆeu mˆo
´
i liˆen hˆe
.
v´o
.
i c´ac t´ıch chˆa
.
p d˜a biˆe
´
t. Du
.
a ra c´ac ´u

c biˆe
’
u diˆe
˜
n nghiˆe
.
m cu
’
a
phu
.
o
.
ng tr`ınh truyˆe
`
n nhiˆe
.
t theo cˆong th´u
.
c cu
’
a t´ıch chˆa
.
p tu
.
o
.
ng ´u
.
ng. C´ac t´ıch

i v´o
.
i c´ac ph´ep biˆe
´
n dˆo
’
i t´ıch phˆan Fourier, Fourier
cosine v`a Fourier sine; T´ıch chˆa
.
p suy rˆo
.
ng v´o
.
i h`am tro
.
ng dˆo
´
i v´o
.
i c´ac ph´ep biˆe
´
n
dˆo
’
i t´ıch phˆan Fourier cosine, Fourier v`a Fourier sine; T´ıch chˆa
.
p suy rˆo
.
ng c´o
h`am tro

+
= {(x, y) ∈ R
2
: x, y > 0}.
• L(R, e
x
) l`a tˆa
.
p ho
.
.
p tˆa
´
t ca
’
c´ac h`am f x´ac di
.
nh trˆen R :
+∞

−∞
e
x
|f(x)|dx <
+∞.
• L(R
+
) l`a tˆa
.
p ho


0
e
x
|f(x)|dx < +∞.
• L(R,
√
1 + x
2
) l`a tˆa
.
p ho
.
.
p tˆa
´
t ca
’
c´ac h`am f x´ac di
.
nh trˆen R sao cho:
+∞

−∞
√
1 + x
2
|f(x)|dx < +∞.
11
C´ac h`am tro

(y) = cos y.
• H`am γ
6
(y) = sign y.
• H`am γ
7
(y) = e
−y
sin y.
• H`am γ
8
(y) =
y
1 + y
2
.
12
L`o
.
i ca
’
m o
.
n
Dˆo
´
i v´o
.
i mˆo
.

n trung trong kh´oi lu
.
’
a cu
’
a cuˆo
.
c kh´ang chiˆe
´
n chˆo
´
ng M˜y c´u
.
u nu
.
´o
.
c oanh liˆe
.
t
cu
’
a dˆan tˆo
.
c, du
.
o
.
.
c bu

m vinh ha
.
nh l´o
.
n. Dˆa
´
t nu
.
´o
.
c v`u
.
a tra
’
i qua kh´oi lu
.
’
a cu
’
a
chiˆe
´
n tranh t`an khˆo
´
c, cuˆo
.
c sˆo
´
ng pha
’

’
khoa
to´an da
.
y dˆo
˜
, truyˆe
`
n thu
.
cho kiˆe
´
n th´u
.
c v`a yˆeu qu´y hˆe
´
t m`ınh. Tˆo
´
t nghiˆe
.
p da
.
i
ho
.
c kh´a tˆo
´
t trong ho`an ca
’
nh nhu

nhiˆe
.
t huyˆe
´
t nghiˆen c´u
.
u khoa ho
.
c m`a c´ac thˆa
`
y cˆo d˜a th˘a
´
p s´ang. Mu
.
`o
.
i n˘am
sau, n˘am 1995 tˆoi c´o du
.
o
.
.
c vinh du
.
.
tro
.
’
la
.

˜
n tˆa
.
n t`ınh, chu d´ao cu
’
a c´ac thˆa
`
y cˆo,
c´ac gi´ao su
.
h`ang dˆa
`
u cu
’
a dˆa
´
t nu
.
´o
.
c, tˆoi d˜a ho`an th`anh xuˆa
´
t s˘a
´
c luˆa
.
n v˘an Tha
.
c
s˜y v´o

.
c s˜y, tˆoi la
.
i du
.
o
.
.
c tro
.
’
la
.
i di tiˆe
´
p con du
.
`o
.
ng d˜a
cho
.
n cu
’
a m`ınh, ho`an th`anh luˆa
.
n ´an Tiˆe
´
n s˜y To´an ho
.

khoa to´an, o
.
’
ph`ong sau da
.
i ho
.
c v`a nh`a tru
.
`o
.
ng
l`ong biˆe
´
t o
.
n chˆan th`anh v`a sˆau s˘a
´
c.
D
˘a
.
c biˆe
.
t t`u
.
trong tˆam kha
’
m, tˆoi xin b`ay to
’

˜
n nhiˆe
.
t t`ınh, tˆa
.
n tˆam v`a truyˆe
`
n da
.
t cho tˆoi nhiˆe
`
u
kinh nghiˆe
.
m qu´y b´au trong nghiˆen c´u
.
u khoa ho
.
c.
Tˆoi xin chˆan th`anh ca
’
m o
.
n GS. TSKH Pha
.
m K`y Anh, GS. TS. Nguyˆe
˜
n
H˜u
.

˜
n Ca
’
nh Lu
.
o
.
ng, TS. Trˆa
`
n D´u
.
c
Long, TS. Nguyˆe
˜
n V˘an Ngo
.
c, d˜a gi´up d˜o
.
v`a chı
’
gi´ao cho tˆoi nhiˆe
`
u ´y kiˆe
´
n qu´y
13
b´au. C´ac kˆe
´
t qua
’

c gia H`a Nˆo
.
i, Xˆemina phu
.
o
.
ng tr`ınh da
.
o
h`am riˆeng DHKHTN - DHQG H`a Nˆo
.
i; Xˆemina cu
’
a khoa to´an tin DHBK H`a
Nˆo
.
i, Xˆemina gia
’
i t´ıch Da
.
i ho
.
c Thuy
’
lo
.
.
i; Hˆo
.
i tha

.
ng (27-29 th´ang 9 n˘am 2004) Ha
.
Long, Hˆo
.
i
nghi
.
khoa ho
.
c - DHKHTN n˘am 2004; Hˆo
.
i nghi
.
quˆo
´
c tˆe
´
lˆa
`
n th´u
.
II vˆe
`
gia
’
i t´ıch
tr`u
.
u tu

i nghi
.
to´an ho
.
c tru
.
`o
.
ng Da
.
i ho
.
c
su
.
pha
.
m H`a Nˆo
.
i th´ang 9 n˘am 2005; Hˆo
.
i nghi
.
Khoa ho
.
c DHTL (th´ang 11 n˘am
2004); Hˆo
.
i tha
’

i th´ang 11 - 2005. Hˆo
.
i nghi
.
quˆo
´
c tˆe
´
to´an trong mˆoi tru
.
`o
.
ng t`u
.
n˘am 2006
- Ba V`ı; Hˆo
.
i nghi
.
khoa ho
.
c DHKHTN - DHQG H`a Nˆo
.
i n˘am 2006; Hˆo
.
i nghi
.
khoa ho
.
c lˆa

.
n chiˆe
`
u (Huˆe
´
2006).
Tˆoi xin du
.
o
.
.
c b`ay to
’
l`ong biˆe
´
t o
.
n chˆan th`anh dˆe
´
n Ban gi´am dˆo
´
c Da
.
i ho
.
c
Quˆo
´
c gia, Ban gi´am hiˆe
.

i ho
.
c Da
.
i ho
.
c KHTN H`a
Nˆo
.
i, Ban chu
’
nhiˆe
.
m khoa To´an Co
.
- Tin ho
.
c Da
.
i ho
.
c Khoa ho
.
c tu
.
.
nhiˆen H`a
Nˆo
.
i, Bˆo

ng ca
’
m o
.
n Ban gi´am hiˆe
.
u tru
.
`o
.
ng Da
.
i ho
.
c giao thˆong vˆa
.
n
ta
’
i H`a Nˆo
.
i, Ph`ong tˆo
’
ch´u
.
c c´an bˆo
.
, Ban chu
’
nhiˆe

o diˆe
`
u kiˆe
.
n thuˆa
.
n lo
.
.
i cho tˆoi ho
.
c tˆa
.
p v`a nghiˆen c´u
.
u.
H`a Nˆo
.
i, ng`ay 20 th´ang 06 n˘am 2007.
Nghiˆen c´u
.
u sinh
Nguyˆe
˜
n Minh Khoa
14
Chu
.
o
.

’
i Fourier, Fourier cosine du
.
o
.
.
c
nghiˆen c´u
.
u. C´ac t´ıch chˆa
.
p n`ay c´o h`ang loa
.
t t´ınh chˆa
´
t l´y th´u v`a dˆo
`
ng th`o
.
i du
.
o
.
.
c
su
.
’
du
.

i t´ıch chˆa
.
p
v´o
.
i h`am tro
.
ng dˆo
´
i v´o
.
i ph´ep biˆe
´
n dˆo
’
i t´ıch phˆan du
.
o
.
.
c V. A. Kakichev du
.
a ra v`a
nghiˆen c´u
.
u v`a nhˆa
.
n du
.
o

´
i v´o
.
i mˆo
.
t ph´ep biˆe
´
n dˆo
’
i t´ıch phˆan n´oi chung v`a mˆo
.
t trong ba ph´ep biˆe
´
n dˆo
’
i
t´ıch phˆan Fourier, Fourier sine, Fourier cosine n´oi riˆeng l`a mˆo
.
t cˆong viˆe
.
c dˆa
`
y
kh´o kh˘an. D´o c˜ung ch´ınh l`a l´y do m`a cho dˆe
´
n nay, d`u d˜a ho
.
n mˆo
.
t thˆe

m, xˆay du
.
.
ng v`a nghiˆen
c´u
.
u 2 t´ıch chˆa
.
p v´o
.
i h`am tro
.
ng m´o
.
i dˆo
´
i v´o
.
i mˆo
.
t trong hai ph´ep biˆe
´
n dˆo
’
i t´ıch
phˆan Fourier cosine, Fourier sine. C´ac t´ıch chˆa
.
p n`ay c´o d˘a
’
ng th´u

mˆo
´
i liˆen hˆe
.
v´o
.
i c´ac t´ıch chˆa
.
p d˜a biˆe
´
t du
.
o
.
.
c du
.
a ra. C´ac ´u
.
ng du
.
ng gia
’
i phu
.
o
.
ng
tr`ınh t´ıch phˆan kiˆe
’

t theo cˆong th´u
.
c t´ıch chˆa
.
p tu
.
o
.
ng ´u
.
ng.
15
1.1 T´ıch chˆa
.
p c´o h`am tro
.
ng dˆo
´
i v´o
.
i ph´ep biˆe
´
n dˆo
’
i t´ıch
phˆan Fourier cosine
1.1.1. Di
.
nh ngh˜ıa v`a c´ac t´ıch chˆa
´

.
c cho bo
.
’
i
(f
γ
5
∗
15
g)(x) =
1
2
√
2π
+∞

0
f(t)

g(x + 1 + t)+ (1.1.1)
+ g(|x + 1 − t|) + g(|x − 1 + t|) + g(|x − 1 − t|)

dt
D
-
i
.
nh l´y 1.1.1. Gia
’

’
ng th´u
.
c nhˆan tu
.
’
h´oa
F
c
(f
γ
5
∗
15
g)(y) = cos y (F
c
f)(y) · (F
c
g)(y), ∀y ∈ R
+
. (1.1.2)
Ch´u
.
ng minh. Ta c´o
+∞

0


(f

≤
1
2
√
2π
+∞

0
|f(t)|

+∞

0
|g(x + 1 + t)|dx +
+∞

0


g(|x + 1 − t|)


dx+
+
+∞

0


g(|x − 1 + t|)


g(|x − 1 − t|)


dx =
+∞

t+1
|g(u)|du +
+∞

−1−t


g(|u|)


du
16
=
+∞

t+1


g(u)


du +
+∞


0


g(|x + 1 − t|)


dx +
+∞

0


g(|x − 1 + t|)


dx =
+∞

1−t


g(|u|)


du +
+∞

t−1


(1.1.4) v`a (1.1.5) ta nhˆa
.
n du
.
o
.
.
c
+∞

0



g(x+1+t)


+


g(|x+1–t|)


+


g(|x–1+t|)


+



dx ≤

2
π
+∞

0
|f(t)|dt
+∞

0
|g(t)|dt < +∞
v`ı vˆa
.
y (f
γ
5
∗
15
g)(x) ∈ L(R
+
).
Bˆay gi`o
.
ta ch´u
.
ng minh d˘a
’

ta nhˆa
.
n du
.
o
.
.
c
cos x(F
c
f)(x)(F
c
g)(x) =
1
2π
+∞

0
+∞

0

cos x(u + 1 + v) + cos x(u + 1 − v)
+ cos x(u − 1 + v) + cos x(u − 1 − v)

f(u)g(v)dudv. (1.1.7)
B˘a
`
ng ph´ep dˆo
’

+∞

0
+∞

0
cos xtf (y)g(|t − y −1|)dtdy
−
1
2π
+∞

0
y+1

0
cos xtf (y)g(y + 1 −t)dtdy. (1.1.8)
Tu
.
o
.
ng tu
.
.
, v´o
.
i ph´ep dˆo
’
i biˆe
´

18
+
1
2π
+∞

0
0

−1−y
cos xtf (y)g(t + y + 1)dtdy (1.1.9)
Ho
.
n n˜u
.
a
+∞

0
0

−1−y
cos xtf (y)g(t + y + 1)dtdy = −
+∞

0
0

1+y
cos xtf (y)g(y + 1 −t)dtdy

=
1
2π
+∞

0
+∞

0
cos xt

g(|t − y −1|) + g(t + y + 1)

f(y)dtdy. (1.1.11)
Tu
.
o
.
ng tu
.
.
1
2π
+∞

0
+∞

0
cos x(u − 1 + v)f(u)g(v)dudv

19
=
1
2π
+∞

0
+∞

0
cos xtf (y)g(|t − y + 1|)dtdy
+
1
2π
1

0
0

y−1
cos xtf (y)g(t − y + 1)dtdy
−
1
2π
+∞

1
y−1

0

1−y
cos xtf (y)g(t + y −1)dtdy
=
1
2π
1

0
+∞

1−y
cos xtf (y)g(t + y −1)dtdy
+
1
2π
+∞

1
+∞

1−y
cos xtf (y)g(t + y −1)dtdy
=
1
2π
+∞

0
+∞



y−1
cos xtf (y)g(t − y + 1)dtdy =
1

0
1−y

0
cos xtf (y)g(1 − y −t)dtdy,
(1.1.14)
+∞

1
0

1−y
cos xtf (y)g(t + y −1)dtdy =
+∞

1
y−1

0
cos xtf (y)g(y − 1 −t)dtdy.
(1.1.15)
T`u
.
(1.1.12), (1.1.13), (1.1.14), (1.1.15) ta di dˆe
´

cos x(F
c
f)(x)(F
c
g)(x) =
1
2π
+∞

0
cos xt

+∞

0
f(y)

g(t + 1 + y)
+ g(|t + 1 − y|) + g(|t − 1 + y|) + g(|t − 1 − y|)

dy

dt.
D˘a
’
ng th´u
.
c cuˆo
´
i c`ung v`a (1.1.1) dˆa

n x´et 1.1.1. Di
.
nh l´y 1.1.1 c´o vai tr`o then chˆo
´
t v´o
.
i t´ıch chˆa
.
p dang x´et.
Mˆo
.
t m˘a
.
t t`u
.
di
.
nh l´y n`ay ta nghiˆen c´u
.
u du
.
o
.
.
c c´ac t´ınh chˆa
´
t tiˆe
´
p theo cu
’

n du
.
o
.
.
c nghiˆe
.
m du
.
´o
.
i da
.
ng d´ong.
21
D
-
i
.
nh l´y 1.1.2. Trong khˆong gian c´ac h`am thuˆo
.
c L(R
+
), t´ıch chˆa
.
p c´o h`am
tro
.
ng dˆo
´

´
n dˆo
’
i
t´ıch phˆan Fourier cosine l`a kˆe
´
t ho
.
.
p t´u
.
c l`a
(f
γ
5
∗
15
g)
γ
5
∗
15
h = f
γ
5
∗
15
(g
γ
5

γ
5
∗
15
g)

(y) · (F
c
h)(y)
= cos y cos y(F
c
f)(y)(F
c
g)(y)(F
c
h)(y)
= cos y(F
c
f)(y)

cos y(F
c
g)(y)(F
c
h)(y)

= cos y(F
c
f)(y)


∗
15
g)
γ
5
∗
15
h = f
γ
5
∗
15
(g
γ
5
∗
15
h).
T´ınh chˆa
´
t giao ho´an, phˆan phˆo
´
i ch´u
.
ng minh tu
.
o
.
ng tu
.

.
nh dˆe
`
1.1.1. Nˆe
´
u f v`a g l`a c´ac h`am thuˆo
.
c L(R
+
), khi d´o bˆa
´
t d˘a
’
ng th´u
.
c
sau du
.
o
.
.
c tho
’
a m˜an
f
γ
5
∗
15
g ≤ f · g.

+∞

0
|f(x)|dx
+∞

0
|g(x)|dx.
22
Do d´o

2
π
+∞

0
|(f
γ
5
∗
15
g)(x)|dx ≤

2
π
+∞

0
|f(x)|dx ·


c L(R
+
), th`ı d˘a
’
ng th´u
.
c sau du
.
o
.
.
c
tho
’
a m˜an
(f
γ
5
∗
15
g)(x) =
1
2

(f ∗
F
c
g)(x + 1) + (f ∗
F
c

∗
15
g)(x) =
1
2

1
√
2π
+∞

0
f(t)

g(x + 1 + t) + g(|x + 1 − t|)

dt+
+
1
√
2π
+∞

0
f(t)

g(|x − 1 + t|) + g(|x − 1 − t|)

dt


i v´o
.
i 0 < x ≤ 1
g(|x−1|+t)+g(||x−1|−t|) = g(|1−x+t|)+g(|1−x−t|) = g(|x−1−t|)+g(|x−1+t|).
23
Do d´o
(f
γ
5
∗
15
g)(x) =
1
2

1
√
2π
+∞

0
f(t)

g(1 + x + t) + g(|1 + x − t|)

dt+
+
1
√
2π

.
c ch´u
.
ng minh.
D
-
i
.
nh l´y 1.1.3. Khˆong gian L(R
+
) du
.
o
.
.
c trang bi
.
bo
.
’
i ph´ep to´an t´ıch chˆa
.
p
(1.1.1) l`a mˆo
.
t v`anh di
.
nh chuˆa
’
n giao ho´an nhu



L(R
+
)
≤ f
L(R
+
)
g
L(R
+
)
.
C´ac t´ınh chˆa
´
t c`on la
.
i cu
’
a v`anh di
.
nh chuˆa
’
n l`a do
.
n gia
’
n. T´ınh giao ho´an cu
’

n vi
.
.
Gia
’
su
.
’
tˆo
`
n ta
.
i e l`a phˆa
`
n tu
.
’
do
.
n vi
.
cu
’
a ph´ep to´an t´ıch chˆa
.
p trong khˆong gian
L(R
+
), t´u
.

c
g)(y), ∀y > 0.
Do d´o
cos y(F
c
e)(y) · (F
c
g)(y) = (F
c
g)
(
y), ∀y > 0.
D˘a
’
ng th´u
.
c cuˆo
´
i tu
.
o
.
ng ´u
.
ng v´o
.
i d˘a
’
ng th´u
.

d˘a
’
ng th´u
.
c v`u
.
a dˆa
˜
n t´o
.
i tu
.
o
.
ng du
.
o
.
ng v´o
.
i
cos y(F
c
e)(y) − 1 = 0, ∀y > 0.
24
M˘a
.
t kh´ac do |cos y| ≤ 1, ∀y > 0 nˆen ´ap du
.
ng Di

’
ng th´u
.
c cuˆo
´
i o
.
’
trˆen dˆa
˜
n t´o
.
i vˆo l´y. Vˆa
.
y khˆong tˆo
`
n ta
.
i phˆa
`
n tu
.
’
do
.
n vi
.
cu
’
a ph´ep to´an t´ıch chˆa

.
nh l ´y 1.1.4. (Di
.
nh l´y kiˆe
’
u Titchmarch). Cho f, g ∈ L(R, e
x
). Nˆe
´
u (f
γ
5
∗
15
g)(x) ≡ 0, ∀x > 0, th`ı ho˘a
.
c f(x) ≡ 0 ho˘a
.
c g(x) ≡ 0.
Ch´u
.
ng minh. T`u
.
gia
’
thiˆe
´
t (f
γ
5


0
f(x) cos(yx)dx, y ∈ R
+
.
T`u
.



d
n
dy
n

cos(yx)f(x)




=



f(x) · x
n
· cos

yx + n
π

−x
x
n
n!
n! ≤ e
−x
e
x
n! = n!
ta c´o



d
n
dy
n
(F
c
f)(y)



≤

2
π
n!
+∞
