Fourier cosine and sine transforms and generalized Lipschitz classes in uniform metric
For functions $f \in L^1(\mathbb{R}_{+})$ with cosine (sine) Fourier transforms $\widehat{f}_c(\widehat{f}_s)$ in $L^1(\mathbb{R})$, we give necessary and sufficient conditions in terms of $\widehat{f}_c(\widehat{f}_s)$ for $f$ to belong to generalized Lipschitz classes $H^{\omega, m}$ and $h^{\ome...
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| author | Golubov, B. I. Volosivets, S. S. Голубов, Б. І. Волосівец, С. С. |
| author_facet | Golubov, B. I. Volosivets, S. S. Голубов, Б. І. Волосівец, С. С. |
| author_sort | Golubov, B. I. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:30:33Z |
| description | For functions $f \in L^1(\mathbb{R}_{+})$ with cosine (sine) Fourier transforms $\widehat{f}_c(\widehat{f}_s)$ in $L^1(\mathbb{R})$,
we give necessary and sufficient conditions in terms of $\widehat{f}_c(\widehat{f}_s)$ for $f$ to belong to generalized Lipschitz classes $H^{\omega, m}$ and $h^{\omega, m}$.
Conditions for the uniform convergence of the Fourier integral and for the existence of the Schwartz derivative are also obtained. |
| first_indexed | 2026-03-24T02:26:37Z |
| format | Article |
| fulltext |
UDC 517.51
B. I. Golubov (Moscow Inst. Phys. and Technol. (State Univ.), Russia),
S. S. Volosivets (Saratov State Univ., Russia)
FOURIER COSINE AND SINE TRANSFORMS
AND GENERALIZED LIPSCHITZ CLASSES IN UNIFORM METRIC*
КОСИНУС- I СИНУС-ПЕРЕТВОРЕННЯ ФУР’Є
ТА УЗАГАЛЬНЕНI КЛАСИ ЛIПШИЦЯ В РIВНОМIРНIЙ МЕТРИЦI
For functions f ∈ L1(R+) with cosine (sine) Fourier transforms f̂c (f̂s) in L1(R), we give necessary and sufficient
conditions in terms of f̂c (f̂s) for f to belong to generalized Lipschitz classes Hω,m and hω,m. Conditions for the uniform
convergence of the Fourier integral and for the existence of the Schwartz derivative are also obtained.
Для функцiй f ∈ L1(R+) iз косинус-(синус-) перетвореннями Фур’є f̂c (f̂s) у L1(R) наведено (в термiнах f̂c (f̂s))
необхiднi та достатнi умови належностi функцiй f до узагальнених класiв Лiпшиця Hω,m та hω,m. Також отримано
умови рiвномiрної збiжностi iнтеграла Фур’є та iснування похiдної Шварца.
1. Introduction. Let f : R → C be a Lebesgue integrable function over R+ = [0,+∞), i.e.,
f ∈ L1(R+). Then the Fourier cosine and sine transforms of f are defined by
f̂c(x) =
(
2
π
)1/2 ∫
R+
f(t) cosxt dt, f̂s(x) =
(
2
π
)1/2 ∫
R+
f(t) sinxt dt, x ∈ R.
If, in addition, f̂c ∈ L1(R+) (f̂s ∈ L1(R+)) and f ∈ C(R+) (f is continuous on R+), then the
inversion formula
f(t) =
(
2
π
)1/2 ∫
R+
f̂c(x) cosxt dx
f(t) =
(
2
π
)1/2 ∫
R+
f̂s(x) sinxt dx
(1.1)
takes place for all t ∈ R+. A proof is similar to that of inversion formula for
f̂(x) = (2π)−1/2
∫
R
f(t)e−ixt dt
and f ∈ L1(R) ∩ C(R) (see [1, p. 192], Chapter 5). In this case we have also limx→+∞ f(x) = 0,
that is f ∈ C0(R+). In all results connected with cosine (sine) Fourier transform we consider the
even (odd) extension fe (fo) of a function f ∈ C0(R+) onto R. For m ∈ N and f defined on R let
introduce the m-th symmetric difference ∆̇m
h f(x) =
∑m
j=0
(−1)m−j
(
m
j
)
f(x + (m − 2j)h/2). If
f ∈ C0(R+) (i.e., f ∈ C(R) and limx→±∞ f(x) = 0) and ‖f‖ = supx∈R |f(x)|, then ωm(f, δ) :=
:= sup{‖∆̇m
h f‖ : 0 ≤ h ≤ δ} is the m-th modulus of smoothness.
Denote by Φ the set of all continuous and increasing on R+ functions ω such that ω(0) = 0 and
ω(2t) ≤ Cω(t), t ∈ R+. If ω ∈ Φ and
∫ δ
0
t−1ω(t) dt = O(ω(δ)), then ω belongs to the Bari class
*The work of the first author is supported by the Russian Foundation for Basic Research under Grant № 11-01-00321 and
by the project “Contemporary problems of analysis and mathematical physics” fulfilled by the Moscow Institute of Physics
and Technologies (State University). The work of the second author is supported by the Russian Foundation for Basic
Research under Grant № 10-01-00270a and by a grant of the President of Russian Federation, project NSh-4383.2010.1.
c© B. I. GOLUBOV, S. S. VOLOSIVETS, 2012
616 ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 5
FOURIER COSINE AND SINE TRANSFORMS AND GENERALIZED LIPSCHITZ CLASSES . . . 617
B; if ω ∈ Φ and δm
∫ ∞
δ
t−m−1ω(t) dt = O(ω(δ)), m > 0, then ω belongs to the Bari – Stechkin
class Bm (see [2]). If ω ∈ Φ and ω(λδ) ≤ Cλmω(δ) for all λ ≥ 1, δ > 0, then ω ∈ Nm. It is well
known that ωm(f, δ) ∈ Nm (see [3], Chapter 3). By definition, Hω,m = {f ∈ C0(R) : ωm(f, t) ≤
≤ Cω(t), t ∈ R+} and hω,m = {f ∈ C0(R) : ωm(f, t) = o(ω(t)), t → 0} for ω ∈ Φ. The class
Hω,1(hω,1) with ω(t) = tα, 0 < α ≤ 1, will be denoted by Lip(α) (lip(α)). There is a different
notation for the class Hω,2 (hω,2) with ω(t) = tα, 0 < α ≤ 2. In the paper [4] it was denoted
by Zyg(α) (zyg(α)). F. Moricz [4] established several theorems connecting the behaviour of f̂ and
classes Lip(α), Zyg(α), lip(α), zyg(α). The main content of these results is represented in the
following theorem.
Theorem A. (i) If f ∈ L1(R) ∩ C(R) and for some α ∈ (0,m], m = 1, 2, we have∫
|t|<y
|tmf̂(t)| dt = O(ym−α) for all y > 0, (1.2)
then f̂ ∈ L1(R) and f ∈ Lip(α) for m = 1 and f ∈ Zyg(α) for m = 2.
(ii) If f, f̂ ∈ L1(R), f ∈ Lip(α) for some α ∈ (0, 1], m = 1, or f ∈ Zyg(α) for some α ∈ (0, 2],
m = 2, and tmf̂(t) ≥ 0 for all t ∈ R, then (1.2) holds.
(iii) Both statements (i) and (ii) are valid for 0 < α < m, m = 1, 2, if the right-hand side of
(1.2) is replaced by o(ym−α), y → 0, and the condition f ∈ Lip(α) or f ∈ Zyg(α) is replaced by
f ∈ lip(α) or f ∈ zyg(α) correspondingly.
In the paper [5] Theorem A was generalized to arbitrary m ∈ N and ω belonging to the class B
or Bm. Such theorems in the case of trigonometric series are known as Boas-type results. Interesting
survey of earlier results may be found in [6]. R. P. Boas, L. Leindler, J. Nemeth and S. Tikhonov [7, 8]
considered the cases of cosine and sine series separately, while F. Moricz [9 – 11] and second author
[12] studied such conditions in terms of complex Fourier coefficients (about papers of L. Leindler
and J. Nemeth see Introduction and references in [7]). Let an, bn are cosine and sine coefficients
of f ∈ L1
2π and ωβ(f, δ) is a modulus of continuity of order β > 0. Using our notations, we can
formulate S. Tikhonov’s results from [7] as follows.
Theorem B. Let ω ∈ Φ and β > 0, f ∈ C2π is even, an ≥ 0 for all n ∈ Z+.
(A) If β 6= 2l − 1, l ∈ N, and ω ∈ B, then the conditions ωβ(f, 1/n) = O(ω(1/n)), n ∈ N, and∑n
k=1
kβak = O(nβω(1/n)) are equivalent.
(B) If β = 2l − 1, l ∈ N, and ω ∈ B, then the condition ωβ(f, 1/n) = O(ω(1/n)) is equivalent
to
n∑
k=1
kβ+1ak = O(nβ+1ω(1/n)), n ∈ N,
and
n∑
k=1
kβak sin kx = O(nβω(1/n)), n ∈ N,
uniformly in x ∈ [0, 2π].
(C) If ω ∈ Bβ, then the conditions ωβ(f, 1/n) = O(ω(1/n)), n ∈ N, and
∑∞
k=n
ak =
= O(ω(1/n)), n ∈ N, are equivalent.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 5
618 B. I. GOLUBOV, S. S. VOLOSIVETS
Parts (A) and (B) of Theorem B are valid for odd functions f, but exceptional values of β are
2l, l ∈ N (see [7]). V. Fülöp [13] obtained analogs of the Theorem A for cosine and sine Fourier
transforms.
By definition, a function f has the Schwartz derivative of order m ∈ N in the point x and this
derivative equals to A if there exists limh→0 h
−m∆̇m
h f(x) = A. In [5] the following result is proved.
Theorem C. Let f ∈ L1(R) ∩ C(R), m ∈ N and∫
|t|>y
|f̂(t)| dt = o(y−m), y → +∞.
Then the Schwartz derivative of order m exists at the point x and equals to A if and only if the
principal value of the integral (2π)−1/2
∫
R
(it)mf̂(t)eitx dt exists and equals to A.
It is known the following theorem of R. Paley [15] (see also [16, p. 277], Ch. 4).
Theorem D. Let the Fourier series a0/2+
∑∞
n=1
(an cosnx+bn sinnx) of a function f ∈ C2π
has non-negative coefficients an, bn. Then this series converges uniformly on R.
F. Moricz [11] proved a similar result.
Theorem E. Let the Fourier series
∑
k∈Z
f̂(k)eikx of a function f ∈ C2π is such that
kf̂(k) ≥ 0, k ∈ Z. Then this series converges uniformly on R.
The aim of present paper is to obtain the sufficient conditions in order that functions to belong
to the class Hω,m or hω,m in terms of cosine and sine Fourier transforms. These conditions are
necessary for functions with non-negative cosine and sine transforms. Also we obtain analogs of
Theorems C and D (see Theorems 3 and 4). Theorem 5 is a generalization of Theorems 4, 5 and 8
from the paper [13].
2. Auxiliary results. For f ∈ L1(R+) let us consider the Fejer operator
σt(f)c(x) =
(
2
π
)1/2 t∫
0
(
1− |u|
t
)
f̂c(u) cosxu du, x ∈ R+,
and de La Vallee Poussin operator vt(f)c = 2σ2t(f) − σt(f). Similarly we define σt(f)s(x) and
vt(f)s(x). By definition σt(f)c(x) and vt(f)c(x) are even while σt(f)s(x) and vt(f)s(x) are odd.
Let us remind that an entire function f(z) has exponential type t ≥ 0 (f ∈ Et) if for each ε > 0
there exists A = A(ε) > 0 such that |f(z)| ≤ Ae(t+ε)|z| for all z ∈ C. By UC(R) (BUC(R))
we denote the space of uniformly continuous (bounded uniformly continuous) functions on R. For a
function f ∈ BUC(R) we set At(f) = inf{‖f − g‖∞ : g ∈ BUC(R) ∩ Et}, t ∈ R+.
It is clear that C0(R) ⊂ BUC(R). Lemma 1 connects the direct approximation theorems for
At(f) and properties of vt(f)c (vt(f)s) (see [14], Ch. 5, § 5.1 and Ch. 8, § 8.6).
Lemma 1. If f ∈ BUC(R), m ∈ N, t > 0 and f is even (odd), then
‖f − vt(f)c‖∞ ≤ C1At(f) ≤ C2ωm (f, 1/t)
(‖f − vt(f)s‖∞ ≤ C1At(f) ≤ C2ωm (f, 1/t)).
A function γ(t) will be called almost increasing (almost decreasing) if there exists a constant
k := k(γ) ≥ 1, such that kγ(t) ≥ γ(u) (kγ(u) ≥ γ(t)) for 0 ≤ u ≤ t.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 5
FOURIER COSINE AND SINE TRANSFORMS AND GENERALIZED LIPSCHITZ CLASSES . . . 619
Lemma 2 [2]. (i) Let ω ∈ Φ. Then ω ∈ Bk, k ∈ N, if and only if there exists α ∈ (0, k) such
that tα−kω(t) is almost decreasing.
(ii) Let ω ∈ Φ. Then ω ∈ B if and only if there exists α ∈ (0, 1) such that t−αω(t) is almost
increasing.
Lemma 3. Let F ∈ L1(R+) is differentiable on R+ and F ′ = f ∈ L1(R+). Then tF̂c(t) =
= −f̂s(t) and tF̂s(t)− (2/π)1/2F (0) = f̂c(t) on R+.
Proof. We have F (x) = F (0) +
∫ x
0
f(t) dt, x ∈ R+. Since f ∈ L1(R+), there exists
limx→+∞ F (x) = F (0) +
∫ ∞
0
f(t) dt. But F ∈ L1(R+) implies limx→+∞ F (x) = 0. Using inte-
gration by parts, we obtain
f̂s(t) =
(
2
π
)1/2 ∫
R+
f(u) sin tu du =
(
2
π
)1/2
F (u) sin tu |∞0 −
∫
R+
t cos tuF (u) du
= −tF̂c(t).
Second identity is proved in a similar way.
Lemma 3 is proved.
Lemma 4 [5]. (i) If ω ∈ Bm, m ∈ N, g(t) is a non-negative measurable function and
∞∫
y
g(t)dt = O (ω (1/y)) , y > 0, (2.1)
then ymg(t) ∈ L1
loc(R+) and
y∫
0
tmg(t) dt = O (ymω (1/y)) , y > 0. (2.2)
(ii) If ω ∈ B, g(t) is a non-negative measurable function and tmg(t) ∈ L1
loc(R+), then (2.2)
implies (2.1).
Lemma 5 [5]. (i) If ω ∈ Bm, m ∈ N, g(x) is a non-negative, measurable function on R+
satisfying (2.1) and
∞∫
y
g(t) dt = o(ω(y−1)), y → +∞, (2.3)
then tmg(t) ∈ L1
loc(R+) and
y∫
0
tmg(t) dt = o(ymω(y−1)), y → +∞. (2.4)
(ii) If ω ∈ B, g : R+ → R+ is a measurable function such that tmg(t) ∈ L1
loc(R+) and (2.4)
holds, then (2.3) also holds.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 5
620 B. I. GOLUBOV, S. S. VOLOSIVETS
3. Main results.
Theorem 1. (i) If f ∈ L1(R+) ∩ C0(R+), m ∈ N, ω ∈ B and
y∫
0
tm|f̂c(t)| dt = O(ymω(1/y)) for all y > 0, (3.1)
or
y∫
0
tm|f̂s(t)| dt = O(ymω(1/y)) for all y > 0, (3.2)
then f̂c ∈ L1(R) (or f̂s ∈ L1(R)) and fe ∈ Hω,m (or fo ∈ Hω,m).
(ii) If m ∈ N be even, fe ∈ L1(R) ∩Hω,m and f̂c(t) keeps its sign on R+, then (3.1) holds. If
m ∈ N be odd, ω ∈ Bm, fe ∈ L1(R) ∩Hω,m and f̂c(t) keeps its sign on R+, then (3.1) holds.
(iii) If m ∈ N be odd, ω ∈ Φ, fo ∈ L1(R) ∩ Hω,m and f̂s(t) keeps its sign on R+, then (3.2)
holds. If m ∈ N be even and fo ∈ L1(R) ∩Hω,m and f̂s(t) keeps its sign on R+, then (3.2) holds.
Proof. (i) By Lemma 4(i) the integral
∫ ∞
y
|f̂c(t)| dt is finite for all y > 0 and it is well known
that f̂c ∈ C0(R+). Therefore, f̂c ∈ L1(R+). Further,
∆̇m
h cosxt = Re ∆̇m
h e
ixt = Re
[
eixt
(
2i sin
ht
2
)m]
, m ∈ N, h > 0.
For even m we have ∆̇m
h cosxt = (−1)m/2 cosxt(2 sinht/2)m and for odd m we see that
∆̇m
h cosxt = (−1)(m+1)/2 sinxt(2 sinht/2)m. Similar formulas are valid for ∆̇m
h sinxt. By the
inversion formula (1.1) we find that
∆̇m
h fe(x) =
(
2
π
)1/2
(−1)m/2
∫
R+
f̂c(t) cosxt
(
2 sin
ht
2
)m
dt, m is even,
(
2
π
)1/2
(−1)(m+1)/2
∫
R+
f̂c(t) sinxt
(
2 sin
ht
2
)m
dt, m is odd,
(3.3)
and
∆̇m
h fo(x) =
(
2
π
)1/2
(−1)m/2
∫
R+
f̂s(t) sinxt
(
2 sin
ht
2
)m
dt, m is even,
(
2
π
)1/2
(−1)(m+1)/2
∫
R+
f̂s(t) cosxt
(
2 sin
ht
2
)m
dt, m is odd.
(3.4)
Thus, in all cases ∆̇m
h fe(x)
(
∆̇m
h fo(x)
)
is either even or odd function of x. From (3.3) we deduce
|∆̇m
h fe(x)| ≤
(
2
π
)1/2 1/h∫
0
+
∞∫
1/h
|f̂c(t)| ∣∣∣∣2 sin
ht
2
∣∣∣∣m dt =:
(
2
π
)1/2
(Ih + Jh)
for h > 0. By (3.1) and inequality | sin t | ≤ t, t ∈ R+, we have
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 5
FOURIER COSINE AND SINE TRANSFORMS AND GENERALIZED LIPSCHITZ CLASSES . . . 621
|Ih| ≤
1/h∫
0
hmtm|f̂c(t)| dt ≤ C1h
mh−mω(h) = C1ω(h). (3.5)
On the other hand, by Lemma 4(ii) we see that
|Jh| ≤ 2m
∞∫
1/h
|f̂(t)| dt ≤ C2ω(h). (3.6)
Combining (3.1) and (3.2) yields fe ∈ Hω,m. For f̂s and fo the proof is similar.
(ii) Let f̂c(t) ≥ 0 for t ≥ 0 and m is even. Then from the condition fe ∈ Hω,m and inequality
sin t ≥ 2t/π, t ∈ [0, π/2], we obtain
C3ω(h) ≥ |∆̇m
h f(0)| =
(
2
π
)1/2 ∫
R+
f̂c(t)
(
2 sin
ht
2
)m
dt ≥ C4
1/h∫
0
f̂c(t)(ht)
m dt
or
∫ 1/h
0
tmf̂c(t) dt ≤ C5h
−mω(h), that is equivalent to (3.1).
If f̂c(t) ≥ 0 for t ≥ 0 and m is odd, then by Lemma 1 we have
fe(0)− vt(fe)(0) =
(
2
π
)1/2 2t∫
t
(u
t
− 1
)
f̂c(u) du+
∞∫
2t
f̂c(u) du
≤ C6ω
(
1
t
)
,
whence
∞∫
t
f̂c(u) du ≤ C7ω
(
2
t
)
≤ C8ω
1
t
.
Using condition ω ∈ Bm and Lemma 4(i), we obtain (3.1).
(iii) If f̂s(t) ≥ 0 for t ≥ 0 and m is odd, then the proof is similar to that of the item (ii) for even
m. Let f̂s(t) ≥ 0 for t ≥ 0, m is even and f ∈ Hω,m. Then for t > 0 by (3.4) we have
C9ω(t) ≥ |∆̇m
t f(x)| =
(
2
π
)1/2 ∣∣∣∣∣∣∣
∫
R+
f̂s(u) sinxu
(
2 sin
tu
2
)m
du
∣∣∣∣∣∣∣ .
Integrating previous inequality by x ∈ [0, t], we obtain∣∣∣∣∣∣∣
t∫
0
∫
R+
f̂s(u) sinxu
(
2 sin
tu
2
)m
du dx
∣∣∣∣∣∣∣ ≤ C9
t∫
0
ω(t) du = C9tω(t)
or
C10
1/t∫
0
u−1f̂s(u)(tu)m+2 du ≤
1/t∫
0
f̂s(u)u−1(1− cos tu)
(
2 sin
tu
2
)m
du =
=
1/t∫
0
∫ t
0
sinxu dxf̂s(u)
(
2 sin
tu
2
)m
du ≤ C9tω(t).
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 5
622 B. I. GOLUBOV, S. S. VOLOSIVETS
From last inequality in the form
∫ 1/t
0
f̂s(u)um+1 du = O(t−m−1ω(t)), t > 0, the condition ω ∈ B
and Lemma 4(ii) we deduce that
∫ ∞
y
f̂s(t) dt = O(ω(1/y)), y > 0. Using ω ∈ Bm and Lemma 4(i),
we obtain (3.2).
Theorem 1 is proved.
Remark 1. In parts (ii) and (iii) of Theorem 1 one may assume non-negativity or non-positivity
of Re f̂c, Im f̂c, Re f̂s, Im f̂s instead of f̂c and f̂s. Theorem 1 is a generalization of Theorems 1, 2, 6
and 7 from [13] and a non-periodic analog of theorem B and its sine counterpart (see Theorems 3.1
and 3.2 in [7]).
Corollary 1. Let f ∈ L1(R+) ∩ C0(R+), f̂c(t) keeps its sign on R+, m ∈ N, ω ∈ Bm ∩ B.
Then the following three conditions are equivalent:
1) fe ∈ Hω,m;
2) (3.1), and
3)
∞∫
y
f̂c(t) dt = O(ω(1/y)), y > 0. (3.7)
Analogous proposition is valid for f̂s and f0.
Theorem 2. (i) If m ∈ N is odd, ω ∈ B ∩Nm, f ∈ L1(R+) ∩ C0(R+) and f̂c(t) ≥ 0 on R+,
then fe ∈ Hω,m if and only if
y∫
0
tm+1f̂c(t) dt = O(ym+1ω(1/y)), y > 0, (3.8)
and
y∫
0
tmf̂c(t) sinxt dt = O(ymω(1/y)), y > 0, (3.9)
uniformly in x ∈ R+.
(ii) If m ∈ N is even, ω ∈ B∩Nm, f ∈ L1(R+)∩C0(R+) and f̂s(t) ≥ 0 on R+, then fo ∈ Hω,m
if and only if
y∫
0
tm+1f̂s(t) dt = O(ym+1ω(1/y)), y > 0, (3.10)
and
y∫
0
tmf̂s(t) sinxt dt = O(ymω(1/y)), y > 0, (3.11)
uniformly in x ∈ R+.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 5
FOURIER COSINE AND SINE TRANSFORMS AND GENERALIZED LIPSCHITZ CLASSES . . . 623
Proof. (i) By Lemma 4(ii), (3.8) implies (3.7). Using (3.3), we have for h > 0
|∆̇m
h f(x)| ≤
(
2
π
)1/2
∣∣∣∣∣∣∣
1/h∫
0
f̂c(t) sinxt
(
2 sin
th
2
)m
dt
∣∣∣∣∣∣∣+
∞∫
1/h
f̂c(t) dt
=:
=:
(
2
π
)1/2
(Ih(x) + Jh(x))
and Jh(x) = O(ω(h)), h > 0, by (3.7). From Taylor’s formula we obtain 2 sin th/2 = th +
+ α(th)(th)3, where |α(t)| ≤ C, t ∈ R, whence
Ih(x) ≤ C1
∣∣∣∣∣∣∣
1/h∫
0
f̂c(t) sinxt(th)m dt
∣∣∣∣∣∣∣+
+C1
∣∣∣∣∣∣∣
1/h∫
0
m∑
j=1
(
m
j
)
(th)m−j(α(th))j(th)3j f̂c(t) sinxt dt
∣∣∣∣∣∣∣ =: I
(1)
h (x) + I
(2)
h (x).
It is clear that
I
(1)
h (x) ≤ C1h
m
∣∣∣∣∣∣∣
1/h∫
0
f̂c(t)t
m sinxt dt
∣∣∣∣∣∣∣ = O(ω(h)), h > 0,
uniformly in x ∈ R+ according to (3.9). On the other hand,
I
(2)
h (x) ≤ C2
m∑
j=1
hm+2j
1/h∫
0
tm+2j f̂c(t) dt. (3.12)
Since Nm ⊂ Bm+2j by Lemma 2 for all 1 ≤ j ≤ m, each term from the right-hand side of (3.12)
is O(ω(h)) according to (3.7) and Lemma 4(i). Thus, Ih(x) = O(ω(h)), h > 0, and |∆̇m
h f(x)| =
= O(ω(h)), h > 0.
Conversely, it is easy to see that Hω,m ⊂ Hω,m+1 by definition and Nm ⊂ Bm+1 by Lemma 2.
Hence, under conditions of theorem we have f ∈ Hω,m+1 with ω ∈ Bm+1. Since m + 1 is even,
by Theorem 1(ii) we obtain (3.8). Using above notations, we have Ih(x) ≤ Jh(x) + C3|∆̇m
h f(x)|
and I(1)h (x) ≤ C4(I
(2)
h (x) + Jh(x) + |∆̇m
h f(x)|. By Lemma 4(ii) and condition ω ∈ B, (3.8) implies
(3.7). Finally, ω ∈ Nm ⊂ Bm+2j and (3.7) implies I(2)h (x) = O(ω(h)), h > 0, as above. Thus,
I
(1)
h (x) = O(ω(h)), h > 0, unformly in x ∈ R+, that is equivalent to (3.9).
(ii) The proof is similar to that of (i).
Theorem 2 is proved.
Corollary 2. (i) If m ∈ N is odd, ω(t) = tm, f ∈ L1(R+) ∩ C0(R+) and f̂c(t) ≥ 0 on R+,
then fe ∈ Hω,m if and only if
y∫
0
tm+1f̂c(t) dt = O(y), y > 0, and
y∫
0
tmf̂c(t) sinxt dt = O(1), y > 0,
uniformly in x ∈ R+.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 5
624 B. I. GOLUBOV, S. S. VOLOSIVETS
(ii) Similar assertion is valid for f̂s, fo and even m ∈ N.
Remark 2. Theorem 2 is an analog of Theorems 3.1 and 3.2, part (B), in [7] (see the item (B)
in Theorem B). Corollary 2 is an extensoin of Theorem 3 in [13], where the necessary and sufficent
condition for f ∈ Lip(1) in terms of f̂c is given.
Theorem 3. (i) Let f ∈ L1(R+) ∩ C0(R+), m ∈ N and
∞∫
y
|f̂c(t)| dt = o(y−m), y → +∞. (3.13)
Then the Schwartz derivative of f of order m exists in the point x > 0 and equals to A(x) if and
only if the integral (2/π)1/2
∫
R+
tmf̂c(t) cos(xt+mπ/2) dt converges and equals to A(x).
(ii) Similar assertion is valid for f̂s(t).
Proof. By (3.3) we have
∆̇m
h f(x) =
(
2
π
)1/2 1/h∫
0
+
∞∫
1/h
f̂c(t) cos
(
xt+m
π
2
)(
2 sin
ht
2
)m
dt =
=:
(
2
π
)1/2
(Ah(x) +Bh(x)).
According to (3.13) we have Bh(x) = o(hm), h→ 0. Using identity 2 sin th/2 = th+ α(th)(th)3,
where α(t) = O(1), t ∈ R (see the proof of Theorem 2), we write
Ah(x) =
1/h∫
0
f̂c(t)(ht)
m cos
(
xt+m
π
2
)
dt+
+
m∑
j=1
(
m
j
) 1/h∫
0
f̂c(t) cos
(
xt+m
π
2
)
(ht)m+2j(α(ht))j dt =: A
(1)
h (x) +A
(2)
h (x).
Since
∫ ∞
y
|f̂c(t)| dt = o(ω(1/y)), y → +∞, for ω(t) = tm and tm ∈ Nm ⊂ Bm+2j for all
1 ≤ j ≤ m, by Lemma 5(i) we obtain
A
(2)
h (x) = O
m∑
j=1
hm+2j
1/h∫
0
|f̂c(t)|tm+2j dt
= o(hm+2jh−m−2jhm) = o(hm), h→ 0.
Therefore, the existence of the limit
B(x) := lim
h→0
h−mA
(1)
h (x) =
(
2
π
)1/2 ∫
R+
f̂c(t)t
m cos
(
xt+m
π
2
)
dt
is equivalent to the existence of limh→0 h
−m∆̇m
h f(x) =: A(x) and in the last case B(x) = A(x).
(ii) The proof of this item is similar to that of (i).
Theorem 3 is proved.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 5
FOURIER COSINE AND SINE TRANSFORMS AND GENERALIZED LIPSCHITZ CLASSES . . . 625
Remark 3. Theorem 3 is an analog of Theorem C.
Theorem 4. Let f ∈ L1(R+) ∩ UC(R+), f̂s(t) ≥ 0 (f̂c(t) ≥ 0) on R+. If F (x) =
=
∫ x
0
f(t) dt ∈ L1(R+), then
f(x) =
(
2
π
)1/2
lim
y→∞
y∫
0
f̂s(t) sinxt dt
f(x) =
(
2
π
)1/2
lim
y→∞
y∫
0
f̂c(t) cosxt dt
uniformly in x ∈ R+.
Proof. If f ∈ L1(R) is even, then F (x) =
∫ x
0
f(t) dt is odd on R and vice versa. As it is noted
in [5], for f ∈ L1(R)∩UC(R) we have |∆̇2
hF (x)| = o(h), h→ 0, i.e., F ∈ hω,2 for ω(t) = t. Now
we consider odd f (f ≡ fo) and even F. By Theorem 8 in [13] or Theorem 5 below we have
y∫
0
t2|F̂c(t)| dt = o(y2y−1) = o(y), y → +∞, (3.14)
and by Lemma 5
∞∫
y
|F̂c(t)| dt = o(y−1), y → +∞, (3.15)
since ω(t) = t ∈ B2. Using the fact that F̂c(t) ∈ C0(R+) and (3.15), we obtain F̂c(t) ∈ L1(R+)
and by inversion formula (1.1)
F (x+ h)− F (x) = −
(
2
π
)1/2 1/h∫
0
+
∞∫
1/h
F̂c(t)(cosxt− cos(x+ h)t) dt =:
=: −
(
2
π
)1/2
(Ah(x) +Bh(x)).
By virtue of (3.15) we have Bh(x) = o(h), h → 0, uniformly in x ∈ R+. On the other hand, using
identity cosxt− cos(x+ h)t = cosxt(1− cosht) + sinxt sinht, we see that
Ah(x) =
1/h∫
0
F̂c(t)2 sin2
(
ht
2
)
cosxt dt+
1/h∫
0
F̂c(t) sinxt sinht dt =: A
(1)
h (x) +A
(2)
h (x).
By (3.14) and inequality | sin t| ≤ t, t ≥ 0, we obtain
|A(1)
h (x)| ≤ h2
1/h∫
0
|F̂c(t)|t2 dt = o(h), h→ 0,
uniformly in x ∈ R+, while
A
(2)
h (x) = h
1/h∫
0
F̂c(t)t sinxt dt+
1/h∫
0
F̂c(t)α
3(ht)(ht)3 dt =: A
(3)
h (x) +A
(4)
h (x)
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 5
626 B. I. GOLUBOV, S. S. VOLOSIVETS
(see the proof of Theorem 2). From (3.15) and condition ω ∈ B3 for ω(t) = t due to Lemma 5(i) we
have
|A(4)
h (x)| = O
h3 1/h∫
0
t3|F̂c(t)| dt
= o(h3h−3h) = o(h), h→ 0,
also uniformly in x ∈ R+. Thus, by Lemma 3
F (x+ h)− F (x)
h
= −
(
2
π
)1/2 1/h∫
0
F̂c(t)t sinxt dt+ o(1) =
=
(
2
π
)1/2 1/h∫
0
f̂s(t) sinxt dt+ o(1), h→ 0.
Similar relation holds for (F (x)− F (x− h))/h and tending h to zero yields
f(x) =
(
2
π
)1/2 ∞∫
0
f̂s(t) sinxt dt
uniformly in x ∈ R+. The proof of the second statement of Theorem 4 is similar to that of the first
one.
Theorem 4 is proved.
Remark 4. Theorem 4 is a non-periodic analog of Theorem D of R. Paley [15].
Theorem 5. (i) If f ∈ L1(R+) ∩ C0(R+), m ∈ N, ω ∈ B and
y∫
0
tm|f̂c(t)| dt = o(ymω(1/y)), y → +∞, (3.16)
or
y∫
0
tm|f̂s(t)| dt = o(ymω(1/y)), y → +∞, (3.17)
and (3.1) or (3.2) respectively hold for all y > 0, then f̂c ∈ L1(R+) (or f̂s ∈ L1(R+)) and fe ∈ hω,m
(or fo ∈ hω,m).
(ii) If m ∈ N and fe (or fo) satisfy conditions of Theorem 1 (ii) (or Theorem 1 (iii)), then
fe ∈ hω,m implies (3.16) (or fe ∈ hω,m implies (3.17)).
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 5
FOURIER COSINE AND SINE TRANSFORMS AND GENERALIZED LIPSCHITZ CLASSES . . . 627
Proof. (i) By condition of Theorem for every ε > 0 there exists y0(ε), such that(
2
π
)1/2 y∫
0
tm|f̂c(t)| dt < εymω(1/y) for all y > y0.
If Ih and Jh are defined in the proof of Theorem 1, then similarly to (3.5) we have |Ih| ≤
≤ εhmh−mω(h) = εω(h) for 0 < h < y−10 . On the other hand, by Lemma 5 (ii) we have |Jh| =
= o(ω(h)), h→ 0. Thus, |∆̇m
h f(x)| = O(Ih + Jh) = o(ω(h)) and fe ∈ hω,m (fo ∈ hω,m).
(ii) Let m be even and f̂c(t) ≥ 0 on R+. If f ∈ hω,m, then
εω(h) ≥ |∆̇m
h f(0)| ≥ C1
1/h∫
0
f̂c(t)(ht)
m dt, 0 < h < h0(ε),
whence
∫ 1/h
0
|tmf̂c(t)| dt = o(h−mω(h)), h→ 0, and (3.16) is proved.
Let m be odd, f̂c(t) ≥ 0 on R+ and ω ∈ Bm. Similarly to the proof of Theorem 1 (ii) we find that∫ ∞
2t
f̂c(u) du < εω(1/t) for t > t0(ε) and
∫ ∞
t
f̂c(u) du = o(ω(1/t)), t → +∞. Using condition
ω ∈ Bm and Lemma 5 (i), we obtain (3.16).
The case of odd m and f̂s ≥ 0 is similar to the case of even m and f̂c ≥ 0. Finally, if m
is even, ω ∈ B and f̂s(t) ≥ 0 on R+, then similarly to the proof of Theorem 1 (iii) we have∫ 1/t
0
u−1f̂s(u)(tu)m+2 du ≤ εtω(1/t) for t > t0(ε) and by Lemma 5 (ii) we deduce that
∞∫
y
f̂s(t) dt = o(ω(1/y)), y → +∞. (3.18)
Using ω ∈ Bm and Lemma 5 (ii), we obtain (3.17).
Theorem 5 is proved.
Remark 5. Theorem 5 is a generalization of Theorems 4, 5 and 8 from [13].
Corollary 3. Let f ∈ L1(R+) ∩ C0(R+), f̂c(t) keeps its sign on R+, m ∈ N, ω ∈ Bm ∩ B.
Then three conditions f ∈ hω,m, (3.16) and (3.18) are equivalent. Similar assertion is valid for f̂s.
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ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 5
628 B. I. GOLUBOV, S. S. VOLOSIVETS
10. Moricz F. Higher order Lipschitz classes of functions and absolutely convergent Fourier series // Acta math. hung. –
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Received 17.11.11
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 5
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| id | umjimathkievua-article-2602 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:26:37Z |
| publishDate | 2012 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/98/c4aec8c8404ad4adfb06958340f91a98.pdf |
| spelling | umjimathkievua-article-26022020-03-18T19:30:33Z Fourier cosine and sine transforms and generalized Lipschitz classes in uniform metric Косинус- i синус-перетворення Фур’є та узагальненi класи Лiпшиця в рiвномiрнiй метрицi Golubov, B. I. Volosivets, S. S. Голубов, Б. І. Волосівец, С. С. For functions $f \in L^1(\mathbb{R}_{+})$ with cosine (sine) Fourier transforms $\widehat{f}_c(\widehat{f}_s)$ in $L^1(\mathbb{R})$, we give necessary and sufficient conditions in terms of $\widehat{f}_c(\widehat{f}_s)$ for $f$ to belong to generalized Lipschitz classes $H^{\omega, m}$ and $h^{\omega, m}$. Conditions for the uniform convergence of the Fourier integral and for the existence of the Schwartz derivative are also obtained. Для функцiй $f \in L^1(\mathbb{R}_{+})$ iз косинус-(синус-) перетвореннями Фур’є $\widehat{f}_c(\widehat{f}_s)$ у $L^1(\mathbb{R})$ наведено (в термiнах $\widehat{f}_c(\widehat{f}_s)$ необхiднi та достатнi умови належностi функцiй $f$ до узагальнених класiв Лiпшиця $H^{\omega, m}$ та $h^{\omega, m}$. Також отримано умови рiвномiрної збiжностi iнтеграла Фур’є та iснування похiдної Шварца. Institute of Mathematics, NAS of Ukraine 2012-05-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2602 Ukrains’kyi Matematychnyi Zhurnal; Vol. 64 No. 5 (2012); 616-627 Український математичний журнал; Том 64 № 5 (2012); 616-627 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2602/1963 https://umj.imath.kiev.ua/index.php/umj/article/view/2602/1964 Copyright (c) 2012 Golubov B. I.; Volosivets S. S. |
| spellingShingle | Golubov, B. I. Volosivets, S. S. Голубов, Б. І. Волосівец, С. С. Fourier cosine and sine transforms and generalized Lipschitz classes in uniform metric |
| title | Fourier cosine and sine transforms and generalized Lipschitz classes in uniform metric |
| title_alt | Косинус- i синус-перетворення Фур’є та узагальненi класи Лiпшиця в рiвномiрнiй метрицi |
| title_full | Fourier cosine and sine transforms and generalized Lipschitz classes in uniform metric |
| title_fullStr | Fourier cosine and sine transforms and generalized Lipschitz classes in uniform metric |
| title_full_unstemmed | Fourier cosine and sine transforms and generalized Lipschitz classes in uniform metric |
| title_short | Fourier cosine and sine transforms and generalized Lipschitz classes in uniform metric |
| title_sort | fourier cosine and sine transforms and generalized lipschitz classes in uniform metric |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2602 |
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