Eigenfunctions of the Cosine and Sine Transforms
A description of the eigensubspaces of the cosine and sine operators is given. The spectrum of each of these two operators consists of two eigen- values 1, -1 and their eigensubspaces are infinite{dimensional. There are many possible bases for these subspaces, but most popular are the ones construct...
Gespeichert in:
| Veröffentlicht in: | Журнал математической физики, анализа, геометрии |
|---|---|
| Datum: | 2013 |
| 1. Verfasser: | |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2013
|
| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/106768 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Eigenfunctions of the Cosine and Sine Transforms / V. Katsnelson // Журнал математической физики, анализа, геометрии. — 2013. — Т. 9, № 4. — С. 476-495. — Бібліогр.: 7 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
nasplib_isofts_kiev_ua-123456789-106768 |
|---|---|
| record_format |
dspace |
| spelling |
Katsnelson, V. 2016-10-04T20:01:55Z 2016-10-04T20:01:55Z 2013 Eigenfunctions of the Cosine and Sine Transforms / V. Katsnelson // Журнал математической физики, анализа, геометрии. — 2013. — Т. 9, № 4. — С. 476-495. — Бібліогр.: 7 назв. — англ. 1812-9471 https://nasplib.isofts.kiev.ua/handle/123456789/106768 A description of the eigensubspaces of the cosine and sine operators is given. The spectrum of each of these two operators consists of two eigen- values 1, -1 and their eigensubspaces are infinite{dimensional. There are many possible bases for these subspaces, but most popular are the ones constructed from the Hermite functions. We present other "bases" which are not discrete orthogonal sequences of vectors, but continuous orthogo- nal chains of vectors. Our work can be considered to be a continuation and further development of the results obtained by Hardy and Titchmarsh: "Self-reciprocal functions"(Quart. J. Math., Oxford, Ser. 1 (1930)). Представлено описание собственных подпространств косинус- и синус-опе- раторов. Спектр каждого из этих двух операторов состоит из двух собственных значений 1, -1 , а их собственные подпространства бесконечномерные. Есть много возможных базисов этих подпространств, но наиболее популярными являются те, которые построены из функций Эрмита. Мы представляем другие "базисы", которые являются не дискретными ортогональными последовательностями векторов, а непрерывными ортогональными цепочками векторов. Работу можно считать продолжением и дальнейшим развитием результатов, приведенных Харди и Титчмаршем в статье "Self-reciprocal functions" (Quart. J. Math., Oxford Ser. 1 (1930)). I thank Armin Rahn for his careful reading of the manuscript and his help in improving English in this paper. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Журнал математической физики, анализа, геометрии Eigenfunctions of the Cosine and Sine Transforms Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Eigenfunctions of the Cosine and Sine Transforms |
| spellingShingle |
Eigenfunctions of the Cosine and Sine Transforms Katsnelson, V. |
| title_short |
Eigenfunctions of the Cosine and Sine Transforms |
| title_full |
Eigenfunctions of the Cosine and Sine Transforms |
| title_fullStr |
Eigenfunctions of the Cosine and Sine Transforms |
| title_full_unstemmed |
Eigenfunctions of the Cosine and Sine Transforms |
| title_sort |
eigenfunctions of the cosine and sine transforms |
| author |
Katsnelson, V. |
| author_facet |
Katsnelson, V. |
| publishDate |
2013 |
| language |
English |
| container_title |
Журнал математической физики, анализа, геометрии |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| format |
Article |
| description |
A description of the eigensubspaces of the cosine and sine operators is given. The spectrum of each of these two operators consists of two eigen- values 1, -1 and their eigensubspaces are infinite{dimensional. There are many possible bases for these subspaces, but most popular are the ones constructed from the Hermite functions. We present other "bases" which are not discrete orthogonal sequences of vectors, but continuous orthogo- nal chains of vectors. Our work can be considered to be a continuation and further development of the results obtained by Hardy and Titchmarsh: "Self-reciprocal functions"(Quart. J. Math., Oxford, Ser. 1 (1930)).
Представлено описание собственных подпространств косинус- и синус-опе- раторов. Спектр каждого из этих двух операторов состоит из двух собственных значений 1, -1 , а их собственные подпространства бесконечномерные. Есть много возможных базисов этих подпространств, но наиболее популярными являются те, которые построены из функций Эрмита. Мы представляем другие "базисы", которые являются не дискретными ортогональными последовательностями векторов, а непрерывными ортогональными цепочками векторов. Работу можно считать продолжением и дальнейшим развитием результатов, приведенных Харди и Титчмаршем в статье "Self-reciprocal functions" (Quart. J. Math., Oxford Ser. 1 (1930)).
|
| issn |
1812-9471 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/106768 |
| citation_txt |
Eigenfunctions of the Cosine and Sine Transforms / V. Katsnelson // Журнал математической физики, анализа, геометрии. — 2013. — Т. 9, № 4. — С. 476-495. — Бібліогр.: 7 назв. — англ. |
| work_keys_str_mv |
AT katsnelsonv eigenfunctionsofthecosineandsinetransforms |
| first_indexed |
2025-11-27T01:26:23Z |
| last_indexed |
2025-11-27T01:26:23Z |
| _version_ |
1850790849315602432 |
| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2013, vol. 9, No. 4, pp. 476–495
Eigenfunctions of the Cosine and Sine Transforms
V. Katsnelson
The Weizmann Institute of Science
Rehovot 76100, Israel
E-mail: victor.katsnelson@weizmann.ac.il; victorkatsnelson@gmail.com
Received October 29, 2012, revised December 17, 2012
A description of the eigensubspaces of the cosine and sine operators is
given. The spectrum of each of these two operators consists of two eigen-
values 1, −1 and their eigensubspaces are infinite–dimensional. There are
many possible bases for these subspaces, but most popular are the ones
constructed from the Hermite functions. We present other ”bases” which
are not discrete orthogonal sequences of vectors, but continuous orthogo-
nal chains of vectors. Our work can be considered to be a continuation
and further development of the results obtained by Hardy and Titchmarsh:
”Self-reciprocal functions”(Quart. J. Math., Oxford, Ser. 1 (1930)).
Key words: Fourier transform, cosine-sine transforms, eigenfunctions,
Melline transform.
Mathematics Subject Classification 2010: 47A38 (primery); 47B35, 47B06,
47A10 (secondary).
1. The cosine transform C and the sine transform S are defined by the
formulas
(Cx)(t) =
√
2
π
∫
R+
cos(tξ)x(ξ) dξ, t ∈ R+ , (1a)
(Sx)(t) =
√
2
π
∫
R+
sin(tξ) x(ξ) dξ, t ∈ R+ , (1b)
where R+ is the positive half-axis, R+ = {t ∈ R : t > 0}.
For x ∈ L1(R+), the integrals in (1) are well defined as Lebesgue integrals. If
x(t) ∈ L2(R+) ∩ L1(R+), then the Parseval equalities hold:
∫
R+
|(Cx)(t)|2dt =
∫
R+
|x(t)|2dt, (2a)
∫
R+
|(Sx)(t)|2dt =
∫
R+
|x(t)|2dt. (2b)
c© V. Katsnelson, 2013
Eigenfunctions of the Cosine and Sine Transforms
Thus, the transforms C and S can both be considered as the linear operators de-
fined on the linear manifold L1(R+)∩L2(R+) of the Hilbert space L2(R+), map-
ping this linear manifold into L2(R+) isometrically. Since the set L1(R+)∩L2(R+)
is dense in L2(R+), each of these operators can be extended to an operator defined
on the whole space L2(R+), which maps L2(R+) into L2(R+) isometrically. We
retain the notation C and S for the extended operators. In an even broader con-
text, the transformation (1) can be considered for those x, for which the integrals
on the right–hand sides are meaningful.
Considered as operators in the Hilbert space L2(R+), the operators C and S
are self-adjoint operators which satisfy the equalities
C2 = I, S2 = I, (3)
where I is the identity operator in L2(R+). Each of the spectra σ(C) and σ(S)
of these operators consists of two points: +1 and −1. By Cλ and Sλ we denote
the spectral subspaces of the operators C and S, respectively, corresponding to
the points λ = 1 and λ = −1 of their spectra. These spectral subspaces are
eigensubspaces:
C1 = {x ∈ L2(R+) : Cx = x}, C−1 = {x ∈ L2(R+) : Cx = −x}; (4a)
S1 = {x ∈ L2(R+) : Sx = x}, S−1 = {x ∈ L2(R+) : Sx = −x}. (4b)
Moreover, two orthogonal decompositions hold
L2(R+) = C1 ⊕ C−1, L2(R+) = S1 ⊕ S−1 . (5)
The spectra of the operators C and S are highly degenerated: the eigensubspaces
Cλ and Sλ are infinite-dimensional. Many bases are possible in these subspaces.
The best known are the bases formed by the Hermite functions hk(t) restricted
onto R+.
The Hermite functions hk(t) are defined as
hk(t) = e
t2
2
dk (e−t2)
dtk
, t ∈ R, k = 0, 1, 2, . . . . (6)
It is known that the system {hk}k=0,1,2, ... forms an orthogonal basis in the Hilbert
space L2(R). The properties of the Hermite functions hk as eigenfunctions of the
Fourier transform were established by N. Wiener, [1, Ch. 1]. In [1], N. Wiener
developed the L2-theory of the Fourier transform which was based on these prop-
erties of the Hermite functions.
The Hermite functions hk are originally defined on the whole real axis R.
The restrictions hk |R+
of the Hermite functions hk onto R+ are considered as the
vectors of the Hilbert space L2(R+). Each of two systems {h2k |R+
}k=0,1,2, ... and
Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 4 477
V. Katsnelson
{h2k+1|R+
}k=0,1,2, ... is an orthogonal basis in L2(R+). The systems
{
h4l|R+
}
l=0,1,2, ...
,{
h4l+2|R+
}
l=0,1,2, ...
,
{
h4l+1|R+
}
l=0,1,2, ...
, and
{
h4l+3|R+
}
l=0,1,2, ...
are orthogonal
bases of the eigensubspaces C1, C−1, S1 and S−1, respectively. We present other
”bases” which are not discrete orthogonal sequences of vectors, but continuous
orthogonal chains of (generalized) vectors. This is the main goal of this paper.
Our work may be considered as a further development of the results given in [2]
by Hardy and Titchmarsh. (The contents of [2] and [3] were reproduced in the
book [4].)
2. First we discuss the eigenfunctions of the transforms C and S in the broad
sense. These transforms are of the form x → Kx, where
(Kx)(t) =
∫
R+
k(tξ)x(ξ)dξ, (7)
and k is a function of one variable defined on R+. (It should be mentioned that
some operational calculus related to the operators of the form (7) was developed
in [5].)
R e m a r k 1. If the integral (7) does not exist as a Lebesgue integral, i.e.,
the function k(tξ)x(ξ) of the variable ξ is not summable, then a meaning can
be attached to the integral (7) by some regularization procedure. We use the
regularization procedure
∫
R+
k(tξ)x(ξ)dξ = lim
ε→+0
∫
R+
e−εξk(tξ)x(ξ)dξ, (8)
and the regularization procedure
∫
R+
k(tξ)x(ξ)dξ = lim
R→+∞
R∫
0
k(tξ)x(ξ)dξ . (9)
If for some a ∈ C both integrals
∫
R+
k(tξ)ξ−adξ and
∫
R+
k(tξ)ξa−1dξ (10)
have a meaning for every positive t, then, changing the variable tξ → ξ, we obtain
K t−a = κ(a) ta−1, K ta−1 = κ(1− a) t−a, (11)
478 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 4
Eigenfunctions of the Cosine and Sine Transforms
where
κ(a) =
∫
R+
k(ξ)ξ−adξ, κ(1− a) =
∫
R+
k(ξ)ξa−1dξ. (12)
Equalities (11) mean that the subspace (two-dimensional if a 6= 1/2) generated
by the functions t−a and ta−1 is invariant with respect to the transformation K
and that the matrix of this operator in the basis t−a, ta−1 is:
∥∥∥ 0 κ(1−a)
κ(a) 0
∥∥∥ . Thus,
assuming that κ(a) 6= 0, κ(1− a) 6= 0, we obtain that the functions
√
κ(1− a)t−a +
√
κ(a)ta−1 and
√
κ(1− a)t−a −
√
κ(a)ta−1 (13)
are the eigenfunctions of the transform K corresponding to the eigenvalues
λ+ =
√
κ(a)κ(1− a) and λ− = −
√
κ(a)κ(1− a), (14)
respectively.
To find the eigenfunctions of the form (13) for cosine and sine transforms C
and S, we have to calculate the constants (12) corresponding to the functions
kc(τ) =
√
2
π
cos τ and ks(τ) =
√
2
π
sin τ, (15)
which generate the kernels of these integral transforms. This is accomplished in
the following
Lemma 1. Let ζ belong to the strip 0 < Re ζ < 1.
Then
1.
∞∫
0
(cos s) sζ−1 ds =
(
cos
π
2
ζ
)
Γ(ζ) , (16a)
∞∫
0
(sin s) sζ−1 ds =
(
sin
π
2
ζ
)
Γ(ζ) , (16b)
where Γ is the Euler Gamma-function and the integrals in (16) are under-
stood in the sense
∞∫
0
{
cos s
sin s
}
s ζ−1 ds
= lim
R→+∞
R∫
0
{
cos s
sin s
}
s ζ−1 ds = lim
ε→+0
+∞∫
0
e−εs
{
cos s
sin s
}
s ζ−1 ds .
Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 4 479
V. Katsnelson
2. The above limits exist uniformly with respect to ζ from any fixed compact
subset of the strip 0 < Re ζ < 1.
3. Given δ > 0, then for any ζ from the strip δ < Re ζ < 1 − δ and for any
R ∈ (0, +∞), ε ∈ (0,+∞), the estimates
∣∣∣∣
R∫
0
{
cos s
sin s
}
sζ−1ds
∣∣∣∣ ≤ C(δ)e
π
2
|Im ζ|, (17a)
∣∣∣∣
+∞∫
0
e−εs
{
cos s
sin s
}
sζ−1ds
∣∣∣∣ ≤ C(δ)e
π
2
|Im ζ| (17b)
hold, where C(δ) < ∞ does not depend on ζ, R and ε.
We omit the proof of Lemma 1. This lemma can be proved by a standard
method using integration in the complex plane.
According to Lemma 1, the integrals
∞∫
0
{
kc(s)
ks(s)
}
s−a ds and
∞∫
0
{
kc(s)
ks(s)
}
sa−1 ds,
where kc and ks, (15), are the functions generating the kernels of the integral
transformations C and S, exist for every a such that 0 < Re a < 1, or, amounting
to the same, 0 < Re (1− a) < 1.
The constants κc(a) and κc(1 − a), corresponding to the function kc(τ) =√
2
π cos τ , are:
κc(a) =
√
2
π
sin
πa
2
Γ(1− a), κc(1− a) =
√
2
π
cos
πa
2
Γ(a). (18a)
The constants κs(a) and κs(1−a), corresponding to the function ks(τ) =
√
2
π sin τ ,
are:
κs(a) =
√
2
π
cos
πa
2
Γ(1− a), κs(1− a) =
√
2
π
sin
πa
2
Γ(a). (18b)
3. Later, we will have to transform the expression (18) for the constants κc
480 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 4
Eigenfunctions of the Cosine and Sine Transforms
and κs using the following identities for the Euler Gamma-function Γ(ζ):
Γ(ζ + 1) = ζΓ(ζ) , see [6] , 12.12, (19a)
Γ(ζ)Γ(1− ζ) =
π
sinπζ
, see [6] , 12.14, (19b)
Γ(ζ)Γ
(
ζ +
1
2
)
= 2
√
π 2−2ζΓ(2ζ), see [6] , 12.15. (19c)
Lemma 2. The following identities hold:
√
2
π
(
cos
π
2
ζ
)
Γ(ζ) = 2 ζ− 1
2
Γ
( ζ
2
)
Γ
(
1
2 − ζ
2
) , (20a)
√
2
π
(
sin
π
2
ζ
)
Γ(ζ) = 2 ζ− 1
2
Γ
(
1
2 + ζ
2
)
Γ
(
1− ζ
2
) . (20b)
P r o o f. From (19b) it follows that
cos
π
2
ζ =
π
Γ
(
1
2 − ζ
2
)
Γ
(
1
2 + ζ
2
) .
From (19c) it follows that
Γ(ζ) = π−
1
2 Γ
( ζ
2
)
Γ
(
1
2 + ζ
2
)
2ζ−1 .
Combining the last two formulas, we obtain (20a). Combining the last formula
with the formula
sin
π
2
ζ =
π
Γ
( ζ
2
)
Γ
(
1− ζ
2
) ,
we obtain (20b).
Lemma 3. The values κc(a), κc(1−a), κs(a), κs(1−a), which appear as the
coefficients of the linear combinations (13), are:
κc(a) = 2
1
2
−a Γ(1
2 − a
2 )
Γ(a
2 )
, κc(1− a) = 2a− 1
2
Γ(a
2 )
Γ(1
2 − a
2 )
, (21a)
κs(a) = 2
1
2
−a Γ(1− a
2 )
Γ(1
2 + a
2 )
, κs(1− a) = 2a− 1
2
Γ(1
2 + a
2 )
Γ(1− a
2 )
. (21b)
4. From the expressions (21) we can see that the products κc(a)κc(1 − a)
and κs(a)κs(1− a) do not depend on a,
κc(a)κc(1− a) = 1, κs(a)κs(1− a) = 1 0 < Re a < 1.
Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 4 481
V. Katsnelson
Theorem 1. Let a ∈ C, 0 < Re a < 1, a 6= 1
2 , and κc(a), κc(1 − a), κs(a),
κs(1− a) be the values appeared in (21).
Then:
1. The functions
E+
c (t, a) =
√
κc(1− a)t−a +
√
κc(a)ta−1, (22a)
E−
c (t, a) =
√
κc(1− a)t−a −
√
κc(a)ta−1 (22b)
of the variable t ∈ R+ are the eigenfunctions (in the broad sense) of the co-
sine transform C corresponding to the eigenvalues +1 and −1, respectively,
E+
c (t, a) = lim
R→∞
√
2
π
R∫
0
cos(tξ) E+
c (ξ, a) dξ,
E−
c (t, a) = − lim
R→∞
√
2
π
R∫
0
cos(tξ) E−
c (ξ, a) dξ.
2. The functions
E+
s (t, a) =
√
κs(1− a)t−a +
√
κs(a)ta−1, (23a)
E−
s (t, a) =
√
κs(1− a)t−a −
√
κs(a)ta−1 (23b)
of the variable t ∈ R+ are the eigenfunctions (in the broad sense) of the
sine transform S corresponding to the eigenvalues +1 and −1, respec-
tively,
E+
s (t, a) = lim
R→∞
√
2
π
R∫
0
sin(tξ) E+
s (ξ, a) dξ, (24)
E−
s (t, a) = − lim
R→∞
√
2
π
R∫
0
sin(tξ) E−
s (ξ, a) dξ. (25)
For the fixed t ∈ (0,∞), the limits exist uniformly with respect to a, from any
compact subset of the strip 0 < Re a < 1.
R e m a r k 2. In (22) and (23), the values of the square roots
√
κ(a) and√
κ(1− a) should be chosen such that their products equal 1.
482 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 4
Eigenfunctions of the Cosine and Sine Transforms
R e m a r k 3. For a = 1
2 , there is only one eigenfunction
E
(
t, 1
2
)
= 2t−
1
2 .
R e m a r k 4. Since
E+
c (t, a) = E+
c (t, 1− a), E−
c (t, a) = −E−
c (t, 1− a), (26a)
E+
s (t, a) = E+
s (t, 1− a), E−
s (t, a) = −E−
s (t, 1− a), (26b)
each eigenfunction appears in the family {E±
c,s(t, a)}0<Re a<1 twice. To avoid this
redundancy, we should consider the family where only one of the points a or 1−a
appears.
5. If 0 < Re a < 1 and x(t) is any of the eigenfunctions of the form either (22),
or (23), then the integral
∫
R+
|x(t)|2 diverges. Thus, none of these eigenfunctions
belong to L2(R+). This integral diverges both at the points t = +0 and t =
+∞. However, this integral diverges variously for a with Re a = 1
2 and for a
with Re a 6= 1
2 . If Re a = 1
2 , then the integrals diverge logarithmically both at
t = +0 and t = +∞. If Re a 6= 1
2 , then the integrals diverge more strongly:
powerwisely. We try to construct the eigenfunctions of the operator C (of the
operator S) from L2 as the continuous combinations of the eigenfunctions of the
form (22) (of the form (23)). Our hope is that the singularities of the ”continuous
linear combinations” of eigenfunctions, which are in some sense an averaging of
the eigenfunctions of the family, are weaker than the singularities of individual
eigenfunctions. These continuous linear combinations should not include the
eigenfunctions of the form (22) and (23) with a : Re a 6= 1
2 . The singularities
of eigenfunctions with a : Re a 6= 1
2 are too strong and can not disappear by
averaging. Thus, we have to restrict ourselves to a’s of the form a = 1
2 +iτ, τ ∈ R.
Considering the case Re a = 1
2 in more detail, we introduce special notation
for the eigenfunctions E±
c,s(t,
1
2 + iτ)}:
e+
c (t, τ) = 1
2
√
π
E+
c (t, 1
2 + iτ), e−c (t, τ) = 1
2i
√
π
E−
c (t, 1
2 + iτ), (27a)
e+
s (t, τ) = 1
2
√
π
E+
s (t, 1
2 + iτ), e−s (t, τ) = 1
2i
√
π
E−
s (t, 1
2 + iτ). (27b)
(We include the normalizing factor 1
2
√
π
in the definition of the functions e±c,s.)
According to (21), (22), the functions e±c,s(t, τ) can be expressed as
e+
c (t, τ) = 1
2
√
π
(
t−
1
2
−iτ c(τ) + t−
1
2
+iτ c(−τ)
)
, (28a)
e−c (t, τ) = 1
2i
√
π
(
t−
1
2
−iτ c(τ)− t−
1
2
+iτ c(−τ)
)
, (28b)
Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 4 483
V. Katsnelson
e+
s (t, τ) = 1
2
√
π
(
t−
1
2
−iτ s(τ) + t−
1
2
+iτ s(−τ)
)
, (29a)
e−s (t, τ) = 1
2i
√
π
(
t−
1
2
−iτ s(τ)− t−
1
2
+iτ s(−τ)
)
, (29b)
where c(τ), s(τ) are the ”phase factors”
c(τ) = 2i τ
2 exp
{
i arg Γ(1
4 + i τ
2 )
}
, −∞ < τ < ∞, (30a)
s(τ) = 2i τ
2 exp
{
i arg Γ(3
4 + i τ
2 )
}
, −∞ < τ < ∞. (30b)
In (30), exp{i arg Γ(ζ)} = Γ(ζ)
|Γ(ζ)| .
Since c(τ) = c(−τ), s(τ) = s(−τ) for real τ , the values of the functions e+
c (t, τ),
e−c (t, τ), e+
s (t, τ), e−s (t, τ) are real for t ∈ (0,∞), τ ∈ (0,∞).
R e m a r k 5. The parameter τ , which enumerates the families {e±c (t, τ},
{e±s (t, τ}, runs over the interval (0,∞). There is no need to consider negative τ .
(See Remark 4).
6. Let us introduce four integral transforms T+
c , T−c , T+
s , T−s .
For φ(t) ∈ L1(R+) and t > 0, let us define
(T+
c φ)(t) =
∫
R+
e+
c (t, τ)φ(τ) dτ, (T−c φ)(t) =
∫
R+
e−c (t, τ)φ(τ) dτ, (31a)
(T+
s φ)(t) =
∫
R+
e+
s (t, τ)φ(τ) dτ, (T−s φ)(t) =
∫
R+
e−s (t, τ)φ(τ) dτ. (31b)
Lemma 4. If φ(τ) ∈ L1(R+), and x(t) = (Tφ)(t), where T is any of the
above–introduced four transformations T±c,s, then the function x(t) is continuous
on the interval (0,∞) and the estimate
|x(t)| ≤ 1√
π
‖φ‖
L1(R+)
· t− 1
2 , 0 < t < ∞, (32)
holds.
P r o o f. Let e(t, τ) be any of the four above–introduced functions e+
c (t, τ),
e−c (t, τ), e+
s (t, τ), e−s (t, τ). The function e(t, τ) is continuous with respect to t at
each t > 0, τ > 0 and satisfies the estimate
|e(t, τ | ≤ 1√
π
t−
1
2 , 0 < t < ∞, 0 < τ < ∞. (33)
Now Lemma 4 is a consequence of the standard results of the Lebesgue integration
theory.
484 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 4
Eigenfunctions of the Cosine and Sine Transforms
Theorem 2. Let φ(τ) be a function satisfying the condition
∞∫
0
|φ(τ)|eπ
2
τ dτ < ∞, (34)
and
x+
c (t) = (T+
c φ)(t), x−c (t) = (T−c φ)(t), (35)
x+
s (t) = (T+
s φ)(t), x−s (t) = (T−c φ)(t). (36)
Then the functions x+
c (t), x−c (t) are the eigenfunctions (in the broad sense) of
the cosine transform C, and the functions x+
s (t), x−s (t) are the eigenfunctions (in
the broad sense) of the sine transform S, i.e.,
x+
c (t) = lim
R→∞
√
2
π
R∫
0
cos(tξ) x+
c (ξ) dξ, (37a)
x−c (t) = − lim
R→∞
√
2
π
R∫
0
cos(tξ) x−c (ξ) dξ (37b)
and
x+
s (t) = lim
R→∞
√
2
π
R∫
0
sin(tξ) x+
s (ξ) dξ, (38a)
x−s (t) = − lim
R→∞
√
2
π
R∫
0
sin(tξ) x−s (ξ) dξ (38b)
for every t ∈ (0,∞). In particular, in (37), (38) the limits exist.
P r o o f. According to Theorem 1 and (27),
e+
c (t, τ) = lim
R→∞
√
2
π
R∫
0
cos(tξ)e+
c (ξ, τ) dξ for every t, τ.
Multiplying by φ(τ) and integrating with respect to τ , we obtain
x+
c (t) =
√
2
π
∞∫
0
(
lim
R→∞
R∫
0
cos(tξ)e+
c (ξ, τ) dξ
)
φ(τ) dτ.
Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 4 485
V. Katsnelson
From (17) we obtain the estimate
∣∣∣∣
R∫
0
cos(tξ)e+
c (ξ, τ) dξ
∣∣∣∣ ≤ Ct−
1
2 e
π
2
τ , ∀ R < ∞, τ ∈ R+, t ∈ R+,
where the value C < ∞ does not depend on R, τ, t. This estimate and condition
(34) for the function φ(t) allow us to apply the Lebesgue dominated convergence
theorem
∞∫
0
(
lim
R→∞
R∫
0
cos(tξ)e+
c (ξ, τ) dξ
)
φ(τ) dτ
= lim
R→∞
∞∫
0
( R∫
0
cos(tξ)e+
c (ξ, τ) dξ
)
φ(τ) dτ. (39)
Thus,
x+
c (t) = lim
R→∞
√
2
π
∞∫
0
( R∫
0
cos(tξ)e+
c (ξ, τ) dξ
)
φ(τ) dτ.
On the other hand, using estimate (33) for e+
c (ξ, τ), we can justify the change of
order of integration in the series integral which appears on the right–hand side
of the above equality. For any finite R,
∞∫
0
( R∫
0
cos(tξ)e+
c (ξ, τ) dξ
)
φ(τ) dτ
=
R∫
0
cos(tξ)
( ∞∫
0
e+
c (ξ, τ)φ(τ) dτ
)
dξ =
R∫
0
cos(tξ)x+
c (ξ) dξ.
Finally, we obtain the equality x+
c (t) = limR→∞
R∫
0
cos(tξ)x+
c (ξ) dξ, i.e., equality
(37a) for the function x+
c . Equality (37b) for the function x−c and equalities (38)
for the functions x+
s , x+
s can be obtained analogously.
R e m a r k 6. In Theorem 2 we assume that the function φ satisfies condition
(34). Assuming only that
∞∫
0
|φ(τ)| dτ < ∞, we can not justify equality (39). To
486 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 4
Eigenfunctions of the Cosine and Sine Transforms
apply the Lebesgue dominated convergence theorem, we need the estimate
sup
R∈(0,∞)
τ∈(−∞,∞)
∣∣∣∣
R∫
0
(cos ξ) · ξ−1
2+iτ dξ
∣∣∣∣ < ∞.
We are, however, able to establish (17), but this estimate is not strong enough.
The question of whether equalities (37), (38) hold under the assumption
∞∫
0
|φ(τ)| dτ < ∞ remains open.
7. Our considerations in the context of the L2-theory on the operators
C and S are based on the L2-theory for the Melline transform. (See the article
”Melline Transform” on page 192 of [7, Vol. 6] and references there.) The Melline
transform M is defined by
(Mf)(ζ) =
∞∫
0
f(t)tζ−1 dt.
If the function f(t) ∈ L2(R+) is compactly supported in the open interval (0,∞),
then the function Φ(ζ) = (Mf)(ζ) of the variable ζ is defined in the whole
complex ζ-plane and is holomorphic. The function f(t) can be recovered from
the function Φ = Mf by the formula
f(t) = 1
2πi
∫
Re ζ=c
Φ(ζ) t−ζ dζ,
where c is an arbitrary real number. Moreover, the Parseval equality
∞∫
0
|f(t)|2dt =
1
2π
∫
Re ζ= 1
2
|Φ(ζ)|2 |dζ|
holds (from which we recognize the significance of the vertical line Re ζ = 1
2).
Thus the Melline transform M generates the linear operator defined on the set
of all compactly supported functions f from L2(R+) which maps this set iso-
metrically into the space L2
(
Re ζ = 1
2
)
of the functions defined on the vertical
line Re ζ = 1
2 and which are square-integrable. Since the set of all compactly
supported functions f is dense in L2(R+), this operator can be extended to an
isometrical operator defined on the whole L2(R+) which maps L2(R+) isomet-
rically into L2
(
Re ζ = 1
2
)
. We will continue to denote this extended operator
by M.
Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 4 487
V. Katsnelson
It turns out that the operator M maps the space L2(R+) onto the whole space
L2
(
Re ζ = 1
2
)
. The inverse operator M−1 is defined everywhere on L2
(
Re ζ = 1
2
)
.
If Φ ∈ L2
(
Re ζ = 1
2
)
, then the function
f(t) = (M−1Φ)(t)
is defined as an L2(R+)-function and can be expressed as
f(t) = 1
2π
∫ ∞
−∞
Φ
(
1
2 + iτ
)
t−
1
2
−iτdτ, 0 < t < ∞. (40a)
Furthermore, the function
Φ
(
1
2 + iτ
)
= (Mf)(1
2 + iτ
)
can be expressed as
Φ
(
1
2 + iτ
)
=
∞∫
0
f(t) t−
1
2
+iτdt, −∞ < τ < ∞. (40b)
The pair of formulas (40a) and (40b) together with the Parseval equality
∞∫
0
|f(t)|2dt = 1
2π
∞∫
−∞
∣∣Φ(
1
2 + iτ
)∣∣2 dτ (40c)
make up the most import part of the L2-theory of the Melline transform.
8. Developing the L2-theory of the cosine and sine transforms, we first of all
prove
Lemma 5. Let φ(t) ∈ L1(R+) ∩ L2(R+). Then
∫
R+
|(Tφ)(t)|2dt =
∫
R+
|φ(τ)|2dτ , (41)
where T is any of the above–introduced (see (31)) four transformations T±c,s.
P r o o f. The proof is based on the Parseval equality for the Melline
transform. We present the transformations T±c,s as the inverse Melline transforms.
Given a function φ(τ) defined for τ ∈ (0,∞), we introduce the functions
Φ+
c
(
1
2 + iτ
)
=
√
π c(τ) φ(|τ |), (42a)
Φ−c
(
1
2 + iτ
)
=1
i sign (τ)
√
π c(τ) φ(|τ |), (42b)
488 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 4
Eigenfunctions of the Cosine and Sine Transforms
and
Φ+
s
(
1
2 + iτ
)
=
√
π s(τ) φ(|τ |), (43a)
Φ−s
(
1
2 + iτ
)
=1
i sign (τ)
√
π s(τ) φ(|τ |), (43b)
which are defined for τ ∈ (−∞,∞). Here c(τ), s(τ) are the ”phase factors”
introduced in (30). It is clear that
∣∣Φ(
1
2 + iτ
)∣∣ =
√
πφ(|τ)|, thus
∞∫
−∞
∣∣Φ(
1
2 + iτ
)∣∣2 dτ = 2π
∞∫
0
|φ(τ)|2dτ,
where Φ is any of the four functions Φ+
c , Φ−c , Φ+
s , Φ−s . Comparing (31a), (28a)
and (42a), we can see that the function (T+
c φ)(t) can be interpreted as the inverse
Melline transform of the function Φ+
c ,
(T+
c φ)(t) = 1
2π
∞∫
−∞
t−
1
2
−iτΦ+
c
(
1
2 + iτ
)
dτ. (44a)
The Parseval equality transform, as applied to the inverse Melline transform of
the function ϕ+
c (τ), yields
∞∫
0
|(T+
c φ)(t)|2dt = 1
2π
∞∫
−∞
∣∣Φ+
c
(
1
2 + iτ
)∣∣2dτ =
∞∫
0
|φ(τ)|2dτ.
This is equality (41) for the transform T+
c .
The functions T−c φ, T+
s φ, T−s φ can also be interpreted as the inverse Melline
transforms:
(T−c φ)(t) = 1
2π
∞∫
−∞
t−
1
2
−iτΦ−c
(
1
2 + iτ
)
dτ (44b)
and
(T+
s φ)(t) = 1
2π
∞∫
−∞
t−
1
2
−iτΦ+
s
(
1
2 + iτ
)
dτ, (45a)
(T−s φ)(t) = 1
2π
∞∫
−∞
t−
1
2
−iτΦ−s
(
1
2 + iτ
)
dτ. (45b)
The Parseval equalities, as applied to the inverse Melline transform of the func-
tions Φ−c , Φ+
s and Φ−s , yield equalities (41) for the transforms T−c , T+
s and T−s ,
respectively.
Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 4 489
V. Katsnelson
9. According to Lemma 5, the operators T+
c , T−c , T+
s , T−s are linear operators
each of which is defined on the linear manifold L1(R+) ∩ L2(R+) of the Hilbert
space L2(R+) and which maps this linear manifold into L2(R+) isometrically.
Since the set L1(R+)∩L2(R+) is dense in L2(R+), each of these operators can be
extended to an operator defined on the whole space L2(R+), which maps L2(R+)
into L2(R+) isometrically. We will continue to write T+
c , T−c , T+
s and T−s for the
extended operators.
We now consider the operators T+
c , T−c , T+
s , T−s as the operators defined
on all of L2(R+), mapping L2(R+) into L2(R+) isometrically and acting on the
functions φ(t) ∈ L1(R+) ∩ L2(R+) according to (31).
Theorem 3.
1. The range of values of the operator T+
c is the eigensubspace C+1 of the
operator C;
2. The range of values of the operator T−c is the eigensubspace C−1 of the
operator C;
3. The range of values of the operator T+
s is the eigensubspace S+1 of the
operator S;
4. The range of values of the operator T−s is the eigensubspace S−1 of the
operator S.
R e m a r k 7. Since the operators T+
c , T−c , T+
s , T−s act isometrically from
L2(R+) into L2(R+), the equalities
(T+
c )∗T+
c = I, T+
c (T+
c )∗ = P+
c , CT+
c = T+
c ; (46a)
(T−c )∗T−c = I, T−c (T−c )∗ = P−c , CT−c = −T−c (46b)
and
(T+
s )∗T+
s = I, T+
s (T+
s )∗ = P+
s , ST+
s = T+
s ; (47a)
(T−s )∗T−s = I, T−s (T−s )∗ = P−s , ST−s = −T−s (47b)
hold, where P+
c , P−c , P+
s and P−s are orthogonal projectors from L2(R)+ onto
the eigensubspaces C+1, C−1,S+1 and S−1, respectively, and (T+
c )∗, (T−c )∗, (T+
s )∗,
(T−s )∗ are the operators Hermitian-conjugated to the operators (T+
c ), (T−c ), T+
s ), (T−s )
with respect to the standard scalar product in the Hilbert space L2(R+).
In particular, the operators (T+
c )∗, (T−c )∗, (T+
s )∗ and (T−s )∗ are generalized
inverses ? of the operators T+
c , T−c , T+
s and T−s , respectively.
? In the sense of Moore–Penrose, for example.
490 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 4
Eigenfunctions of the Cosine and Sine Transforms
It is worth mentioning that
(
(T+
c )∗ x
)
(τ) =
∫
R+
e+
c (t, τ)x(t) dt,
(
(T−c )∗x
)
(τ) =
∫
R+
e−c (t, τ)x(t) dt, (48a)
(
(T+
s )∗x
)
(τ) =
∫
R+
e+
s (t, τ) x(t) dt,
(
(T−s )∗x
)
(τ) =
∫
R+
e−s (t, τ) x(t) dt. (48b)
Theorem 3 is a consequence of the following
Lemma 6. Let a function x(t) belong to L2(R+), and x̂c(t) and x̂s(t) be the
cosine and sine Fourier transforms of the function x:
x̂c(t) =
√
2
π
∞∫
0
x(s) cos(ts) ds, (49a)
x̂s(t) =
√
2
π
∞∫
0
x(s) sin(ts) ds. (49b)
Let Φx(ζ), Φx̂c(ζ) and Φx̂s(ζ) be the Melline transforms of the functions x, x̂c and
x̂s, respectively. (All three functions x, x̂c, x̂s belong to L2(0,∞), so their Melline
transforms exist and they are the L2 functions on the vertical line Re ζ = 1
2 .)
Then for ζ : Re ζ = 1
2 , the equalities
Φx̂c(ζ) = Φx(1− ζ) · 2 ζ− 1
2
Γ
( ζ
2
)
Γ
(
1
2 − ζ
2
) , (50a)
Φx̂s(ζ) = Φx(1− ζ) · 2 ζ− 1
2
Γ
(
1
2 + ζ
2
)
Γ
(
1− ζ
2
) (50b)
hold.
P r o o f. It is enough to prove Eqs. (50) assuming that the functions
x(t), x̂c(t), x̂s(t) are continuous and belong to L2(R+) ∩ L1(R+): the set of these
functions x is dense in L2(R) and all three transforms, cosine, sine and Melline
transforms, act continuously from L2 to L2. Under these extra assumptions on
the functions x(t), x̂c(t), x̂s(t), the Melline transforms Φx(ζ), Φx̂c(ζ), Φx̂c(ζ) are
defined everywhere on the vertical line Re ζ = 1
2 and are continuous functions.
For such x, Eqs. (50) will be established for every ζ : Re ζ = 1
2 .
Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 4 491
V. Katsnelson
We fix ζ : Re ζ = 1
2 . The Melline transform Φx̂c(ζ) is
Φx̂c(ζ) = lim
R→∞
R∫
0
x̂c(t)tζ−1dt.
Substituting (49a) for x̂c(t) into the last formula, we obtain
Φx̂c(ζ) = lim
R→∞
R∫
0
(√
2
π
∞∫
0
x(s) cos(ts) ds
)
tζ−1 dt. (51)
For fixed finite R, we change the order of integration
R∫
0
( ∞∫
0
x(s) cos(ts) ds
)
tζ−1 dt =
∞∫
0
x(s)
( R∫
0
cos(ts) tζ−1 dt
)
ds .
The change of order of integration is justified by Fubini’s theorem. Changing the
variable ts = τ , we get
R∫
0
cos(ts) tζ−1 dt = s−ζ
Rs∫
0
cos(τ) τ ζ−1 dτ .
Thus
R∫
0
(√
2
π
∞∫
0
x(s) cos(ts) ds
)
tζ−1 dt
=
∞∫
0
x(s)s−ζ
(√
2
π
Rs∫
0
cos(τ) τ ζ−1 dτ
)
ds . (52)
According to Lemma 1, for every s > 0,
lim
R→∞
Rs∫
0
cos(τ) τ ζ−1 dτ =
(
cos
π
2
ζ
)
Γ(ζ) .
The value
ρ∫
0
(cos τ)τ ζ−1 dτ , considered as a function of ρ, vanishes at ρ = 0, is a
continuous function of ρ, and has a finite limit as ρ →∞. Therefore there exists
492 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 4
Eigenfunctions of the Cosine and Sine Transforms
a finite M(ζ) < ∞ such that the estimate holds
∣∣∣
ρ∫
0
(cos τ)τ ζ−1 dτ
∣∣∣ ≤ M(ζ), where
the value M(ζ) does not depend on ρ. In other words,
∣∣∣∣∣
Rs∫
0
cos(τ) τ ζ−1 dτ
∣∣∣∣∣ ≤ M(ζ) < ∞ ∀ s,R : 0 ≤ s < ∞, 0 ≤ R < ∞ .
By the Lebesgue dominated convergence theorem,
lim
R→∞
∞∫
0
x(s)s−ζ
(√
2
π
Rs∫
0
cos(τ) τ ζ−1 dτ
)
ds
=
∞∫
0
x(s)s−ζ
(√
2
π
∞∫
0
cos(τ) τ ζ−1 dτ
)
ds . (53)
Taking into account equalities (51), (53) and using (16a) and (20a), we reduce
the last equality to the form
Φx̂c(ζ) =
∞∫
0
x(s) s−ζ ds · 2 ζ− 1
2
Γ
( ζ
2
)
Γ
(
1
2 − ζ
2
) .
To obtain (50a) from the previous equality, we need only to consider that
∞∫
0
x(s) s−ζ ds = Φx(1− ζ) .
Equality (50b) can be proved in a similar way.
R e m a r k 8. Equalities (50) can be presented in the form
Φx̂c
(
1
2 + iτ
)
= Φx
(
1
2 − iτ
) · c 2(τ), (54a)
Φx̂s
(
1
2 + iτ
)
= Φx
(
1
2 − iτ
) · s 2(τ), (54b)
where c(τ) and s(τ) were introduced in (30).
P r o o f. [Proof of Theorem 3] Let xc(t) be defined by (49a). The equality
Cx = x, i.e., the equality xc(t) = x(t) for the functions xc(t), x(t), is equivalent
to the equality
Φx̂c
(
1
2 + iτ
)
= Φx
(
1
2 + iτ
)
Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 4 493
V. Katsnelson
for their Melline transforms. According to Lemma 6, (54a), the last equality can
be reduced ? to the form
Φx
(
1
2 − iτ
) · c(τ) = Φx
(
1
2 + iτ
) · c(−τ), −∞ < τ < ∞. (55a)
Analogously, the equalities Cx = −x, Sx = x and Sx = −x for the functions
x(t) are equivalent to the equalities
Φx
(
1
2 − iτ
) · c(τ) = −Φx
(
1
2 + iτ
) · c(−τ), −∞ < τ < ∞, (55b)
and
Φx
(
1
2 − iτ
) · s(τ) = Φx
(
1
2 + iτ
) · s(−τ), −∞ < τ < ∞, (56a)
Φx
(
1
2 − iτ
) · s(τ) = −Φx
(
1
2 + iτ
) · s(−τ), −∞ < τ < ∞. (56b)
Thus each of the equalities Cx = x, Cx = −x, Sx = x, Sx = −x for the
function x(t), 0 < t < ∞, is equivalent to the symmetry condition for its Melline
transform Φx
(
1
2 + iτ
)
, −∞ < τ < ∞. These symmetry conditions, which appear
as conditions (55), (56), can be presented in the form
Φx
(
1
2 + iτ
)
=
√
π c(τ) φ(|τ |), −∞ < τ < ∞,
Φx
(
1
2 + iτ
)
= 1
i sign (τ)
√
π c(τ) φ(|τ |), −∞ < τ < ∞,
and
Φx
(
1
2 + iτ
)
=
√
π s(τ) φ(|τ |), −∞ < τ < ∞,
Φx
(
1
2 + iτ
)
= 1
i sign (τ)
√
π s(τ) φ(|τ |), −∞ < τ < ∞,
where φ(τ) is the function defined for 0 < τ < ∞. Comparing these expressions
for the function Φx
(
1
2 + iτ
)
with the expressions (28), (29) for the eigenfunctions
e+
c (t, τ), e−c (t, τ), e+
s (t, τ), e−s (t, τ), we can see that in each of the four cases the
inversion formula
x(t) = 1
2π
∞∫
−∞
t−
1
2+iτΦx
(
1
2 + iτ
)
dτ
for the Melline transform can be presented in terms of the function φ(τ) as
x(t) =
∞∫
0
e+
c (t, τ) φ(τ) dτ, x(t) =
∞∫
0
e−c (t, τ) φ(τ) dτ, (57a)
x(t) =
∞∫
0
e+
s (t, τ) φ(τ) dτ, x(t) =
∞∫
0
e−s (t, τ) φ(τ) dτ, (57b)
? Remember that c−1(τ) = c(−τ).
494 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 4
Eigenfunctions of the Cosine and Sine Transforms
respectively. Now the symmetries (55), (56) of the function Φx
(
1
2 +iτ
)
are hidden
in the structure of the functions e+
c , e−c , e+
s , e−−.
Thus, the equalities Cx = x, Cx = −x and Sx = x, Sx = −x for the functions
x are equivalent to the representability of x in one of the four forms (57), i.e.,
in the form x = T+
c φ,x = T−c φ, x = T+
s φ and x = T−s φ, respectively (with
φ ∈ L2(R+)).
Acknowledgements. I thank Armin Rahn for his careful reading of the
manuscript and his help in improving English in this paper.
References
[1] N. Wiener, The Fourier Integral and Certain of its Applications. Cambridge Univ.
Press, Cambridge, 1933.
[2] G.H. Hardy and E.C. Titchmarsh, Self-Reciprocal Functions. — Quart. J. Math.,
Oxford, Ser. 1 (1930), 196–231.
[3] G.H. Hardy and E.C. Titchmarsh, Formulae Connecting Different Classes of Self-
Reciprocal Functions. — Proc. London Math. Soc., Ser. 2 33 (1931), 225–232.
[4] E.C. Titchmarsh, Introduction to the Theory of Fourier Integrals. Clarendon Press,
Oxford, 1937. Third Edition. Chelsea Publishing Co., New York, 1986.
[5] J.L. Burchnall, Symbolic Relations Assotiated with Fourier Transforms. — Quart.
J. Math. 3 (1932), 213–223.
[6] E.T. Whittaker and G.N. Watson, A Course of Modern Analysis. Fourth Edition.
Cambridge Univ. Press, Cambridge, 1927.
[7] Encyclopaedia of Mathematics. Kluver Acad. Publ., Dordracht–Boston–London,
1995.
Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 4 495
|