Paley – Wiener type theorem for functions with values in Banach spaces
UDC 517.5 Let $(\mathbb{X},\|.\|_{\mathbb{X}})$ denote a complex Banach space and $L(\mathbb{X})=BC (\mathbb{R}\to\mathbb{X})$ be the set of all $\mathbb{X}$-valued bounded continuous functions $f\colon\mathbb{R}\to\mathbb{X}.$For $f\in L(\mathbb{X})$ we define $\|f\|_{L(\mathbb{X})}=\sup\{ \|f(x)\|...
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Institute of Mathematics, NAS of Ukraine
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508271891185664 |
|---|---|
| author | Bang, H. H. Huy, V. N. Bang, H. H. Huy, V. N. |
| author_facet | Bang, H. H. Huy, V. N. Bang, H. H. Huy, V. N. |
| author_sort | Bang, H. H. |
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| datestamp_date | 2022-10-25T09:23:03Z |
| description | UDC 517.5
Let $(\mathbb{X},\|.\|_{\mathbb{X}})$ denote a complex Banach space and $L(\mathbb{X})=BC (\mathbb{R}\to\mathbb{X})$ be the set of all $\mathbb{X}$-valued bounded continuous functions $f\colon\mathbb{R}\to\mathbb{X}.$For $f\in L(\mathbb{X})$ we define $\|f\|_{L(\mathbb{X})}=\sup\{ \|f(x)\|_{\mathbb{X}}\colon x\in\mathbb{R}\}.$ Then $(L(\mathbb{X}),\|.\|_{L(\mathbb{X})})$ itself is a Banach space. The Beurling spectrum $\mathrm{Spec}(f)$ of a function $f\in L(\mathbb{X})$ is defined by $$\mathrm{Spec}(f)=\left\{\zeta\in\mathbb{R}\colon\forall\epsilon>0 \exists \varphi\in\mathcal{S}(\mathbb{R})\colon\mbox{supp}\,\widehat{\varphi}\subset(\zeta-\epsilon,\zeta+\epsilon),\varphi*f\nequiv 0\right\}.\right\}.$$ We obtain the following Paley-Wiener type theorem for functions with values in Banach spaces:
Let $f\in L(\mathbb{X})$ and $K$ be an arbitrary compact set in $\mathbb{R}.$ Then $\mbox{Spec}(f)\subset K$ if and only if for any $\tau > 0$ there exists a constant $C_\tau < \infty$ such that $$\|P(D)f\|_{L(\mathbb{X})}\leq C_\tau\|f\|_{L(\mathbb{X})}\sup\limits_{x\in K^{(\tau)}} |P(x)|$$for all polynomials with complex coefficients $P(x),$ where the differential operator $P(D)$ is obtained from $P(x)$ by substituting $x \to -i\,\dfrac{d}{dx},$ $\dfrac{d}{dx}$ is the usual derivative in $L(\mathbb{X})$ and $K^{(\tau)}$ is the $\tau$-neighborhood in $\mathbb{C}$ of $K.$
Moreover, Paley-Wiener type theorem for integral operators and one for some special compacts $K$ are also given. |
| doi_str_mv | 10.37863/umzh.v74i6.2382 |
| first_indexed | 2026-03-24T02:22:34Z |
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DOI: 10.37863/umzh.v74i6.2382
UDC 517.5
H. H. Bang (Inst. Math. Vietnam Acad. Sci. and Technology, Hanoi, Vietnam),
V. N. Huy (Hanoi, Univ. Sci., Vietnam Nat. Univ., and TIMAS, Thang Long Univ., Hanoi, Vietnam)
PALEY – WIENER TYPE THEOREM FOR FUNCTIONS
WITH VALUES IN BANACH SPACES*
ТЕОРЕМА ТИПУ ПЕЛI – ВIНЕРА ДЛЯ ФУНКЦIЙ
IЗ ЗНАЧЕННЯМИ У БАНАХОВИХ ПРОСТОРАХ
Let (\BbbX , \| .\| \BbbX ) denote a complex Banach space and L(\BbbX ) = BC(\BbbR \rightarrow \BbbX ) be the set of all \BbbX -valued bounded continuous
functions f : \BbbR \rightarrow \BbbX . For f \in L(\BbbX ) we define \| f\| L(\BbbX ) = \mathrm{s}\mathrm{u}\mathrm{p}\{ \| f(x)\| \BbbX : x \in \BbbR \} . Then (L(\BbbX ), \| .\| L(\BbbX )) itself is a
Banach space. The Beurling spectrum \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(f) of a function f \in L(\BbbX ) is defined by
\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(f) = \{ \zeta \in \BbbR : \forall \epsilon > 0 \exists \varphi \in \scrS (\BbbR ) : supp \widehat \varphi \subset (\zeta - \epsilon , \zeta + \epsilon ), \varphi \ast f \not \equiv 0\} .
We obtain the following Paley – Wiener type theorem for functions with values in Banach spaces:
Let f \in L(\BbbX ) and K be an arbitrary compact set in \BbbR . Then Spec(f) \subset K if and only if for any \tau > 0 there exists
a constant C\tau < \infty such that
\| P (D)f\| L(\BbbX ) \leq C\tau \| f\| L(\BbbX ) \mathrm{s}\mathrm{u}\mathrm{p}
x\in K(\tau )
| P (x)|
for all polynomials with complex coefficients P (x), where the differential operator P (D) is obtained from P (x) by
substituting x \rightarrow - i
d
dx
,
d
dx
is the usual derivative in L(\BbbX ) and K(\tau ) is the \tau -neighborhood in \BbbC of K.
Moreover, Paley – Wiener type theorem for integral operators and one for some special compacts K are also given.
Нехай (\BbbX , \| .\| \BbbX ) — комплексний простiр Банаха i L(\BbbX ) = BC(\BbbR \rightarrow \BbbX ) — множина всiх обмежених неперервних
\BbbX -значних функцiй f : \BbbR \rightarrow \BbbX . Для f \in L(\BbbX ) вводиться позначення \| f\| L(\BbbX ) = \mathrm{s}\mathrm{u}\mathrm{p}\{ \| f(x)\| \BbbX : x \in \BbbR \} . Тодi
(L(\BbbX ), \| .\| L(\BbbX )) сам є банаховим простором. Спектр Берлiнга \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(f) функцiї f \in L(\BbbX ) визначається як
\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(f) = \{ \zeta \in \BbbR : \forall \epsilon > 0 \exists \varphi \in \scrS (\BbbR ) : supp \widehat \varphi \subset (\zeta - \epsilon , \zeta + \epsilon ), \varphi \ast f \not \equiv 0\} .
Отримано таку теорему типу Пелi – Вiнера для функцiй iз значеннями у просторах Банаха:
Нехай f \in L(\BbbX ) i K — довiльна компактна множина в \BbbR . У цьому випадку Spec(f) \subset K тодi й лише тодi,
коли для будь-якого \tau > 0 iснує стала C\tau < \infty така, що
\| P (D)f\| L(\BbbX ) \leq C\tau \| f\| L(\BbbX ) \mathrm{s}\mathrm{u}\mathrm{p}
x\in K(\tau )
| P (x)|
для всiх полiномiв з комплексними коефiцiєнтами P (x), де диференцiальний оператор P (D) отримано з P (x)
замiною x \rightarrow - i
d
dx
,
d
dx
— звичайна похiдна у L(\BbbX ) i K(\tau ) — \tau -окiл для K у \BbbC .
Також наведено теорему типу Пелi – Вiнера для iнтегральних операторiв та деяких спецiальних компактiв K.
1. Introduction. The relation between properties of functions and their spectrum (the support of
their Fourier transform) are interested for many mathematicians. The Paley – Wiener theorem is one
of the well-known results belonging to this direction. The initial Paley – Wiener theorem was proved
for L2-functions, was extended to generalized functions by L. Schwartz and has many generalizations
* This paper was supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED)
under grant number 101.02-2018.300.
c\bigcirc H. H. BANG, V. N. HUY, 2022
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 6 731
732 H. H. BANG, V. N. HUY
(see, for example, [1 – 8, 10 – 14, 16, 17]. In this paper we provide Paley – Wiener type theorem for
functions with values in Banach space.
Let f \in L1(\BbbR ) and \widehat f = \scrF f be the Fourier transform of f
\widehat f(\zeta ) = 1\surd
2\pi
+\infty \int
- \infty
e - ix\zeta f(x) dx,
and \u f = \scrF - 1f denote its inverse Fourier transform
\u f(\zeta ) =
1\surd
2\pi
+\infty \int
- \infty
eix\zeta f(x) dx.
Let (\BbbX , \| .\| \BbbX ) denote a complex Banach space and L(\BbbX ) = BC(\BbbR \rightarrow \BbbX ) be the set of all \BbbX -valued
bounded continuous functions f : \BbbR \rightarrow \BbbX . For a given function f \in L(\BbbX ) we define \| f\| L(\BbbX ) =
= \mathrm{s}\mathrm{u}\mathrm{p}\{ \| f(x)\| \BbbX : x \in \BbbR \} . Then (L(\BbbX ), \| .\| L(\BbbX )) itself is a Banach space. We define the derivative
Df of f \in L(\BbbX ), as usual,
Df(\zeta ) = \mathrm{l}\mathrm{i}\mathrm{m}
x\rightarrow 0
f(\zeta + x) - f(\zeta )
x
.
Given f \in L(\BbbX ). It was shown in [15] that for every \lambda \in \BbbC \setminus i\BbbR the equation \lambda u(x) - Du(x) = f(x)
has a unique solution uf,\lambda \in L(\BbbX ). That means the operator \lambda - D is invertible and (\lambda - D) - 1f =
= uf,\lambda . More exactly,
(\lambda - D) - 1f(\zeta ) =
\left\{
\int \infty
0
e - \lambda xf(\zeta + x) dx, if Re\lambda > 0,
-
\int \infty
0
e - \lambda xf(\zeta + x) dx, if Re\lambda < 0.
Hence, the spectrum of the differential operator \lambda - D is i\BbbR and
+\infty \int
- \infty
\varphi (x)(\lambda - D) - nf(x) dx =
+\infty \int
- \infty
\bigl[
(\lambda - D) - n\varphi (x)
\bigr]
f(x) dx
for any f \in L(\BbbX ) and \varphi \in \scrS (\BbbR ), where \scrS (\BbbR ) is the Schwartz space. The convolution \varphi \ast f of f
with a Schwartz function \varphi is defined as follows:
\varphi \ast f(\zeta ) =
+\infty \int
- \infty
\varphi (\zeta - x)f(x) dx.
The Beurling spectrum \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(f) of a function f \in L(\BbbX ) is defined by
\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(f) = \{ \zeta \in \BbbR : \forall \epsilon > 0 \exists \varphi \in \scrS (\BbbR ) : \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p} \widehat \varphi \subset (\zeta - \epsilon , \zeta + \epsilon ), \varphi \ast f \not \equiv 0\} .
Note that \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(f) is always a closed subset of \BbbR . Let K \subset \BbbR and \tau > 0. We put
K(\tau ) := \{ \zeta \in \BbbC : \exists x \in K : | x - \zeta | < \tau \} ,
which is the \tau -neighborhood in \BbbC of K and K\tau := \{ \zeta \in \BbbR : \exists x \in K : | x - \zeta | < \tau \} , \BbbZ + =
= \{ 0, 1, 2, . . .\} .
Let P (x) be a polynomial. The differential operator P (D) is obtained from P (x) by substituting
x\rightarrow - i d
dx
.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 6
PALEY – WIENER TYPE THEOREM FOR FUNCTIONS WITH VALUES IN BANACH SPACES 733
2. Paley – Wiener type theorem for differential operators. 2.1. Paley – Wiener type theorem
for any compact \bfitK .
Theorem 2.1. Let f \in L(\BbbX ) and K be an arbitrary compact set in \BbbR . Then \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(f) \subset K if
and only if for any \tau > 0 there exists a constant C\tau ,K <\infty independent of f such that
\| P (D)f\| L(\BbbX ) \leq C\tau ,K\| f\| L(\BbbX ) \mathrm{s}\mathrm{u}\mathrm{p}
x\in K(\tau )
| P (x)| (1)
for all polynomials with complex coefficients P (x).
To obtain the theorem, we need the following results.
Lemma 2.1 (Young inequality for Banach spaces). Let f \in L(\BbbX ) and \varphi \in \scrS (\BbbR ). Then \varphi \ast f \in
\in L(\BbbX ) and
\| \varphi \ast f\| L(\BbbX ) \leq \| f\| L(\BbbX )\| \varphi \| L1 .
Proof. We see that
\| \varphi \ast f\| L(\BbbX ) = \mathrm{s}\mathrm{u}\mathrm{p}
s\in \BbbR
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
+\infty \int
- \infty
\varphi (s - t)f(t) dt
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\BbbX
\leq \mathrm{s}\mathrm{u}\mathrm{p}
s\in \BbbR
+\infty \int
- \infty
\| \varphi (s - t)f(t)\| \BbbX dt \leq
\leq \| f\| L(\BbbX ) \mathrm{s}\mathrm{u}\mathrm{p}
s\in \BbbR
+\infty \int
- \infty
| \varphi (s - t)| dt = \| f\| L(\BbbX )\| \varphi \| L1 ,
which completes the proof.
Lemma 2.2 [15]. Let f \in L(\BbbX ) and \varphi , \psi \in \scrS (\BbbR ). Assume that \widehat \varphi = 0 on \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(f) and\widehat \psi = (2\pi ) - 1/2 on \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(f). Then \varphi \ast f = 0 and \psi \ast f = f.
It was proved in [9] the following radial spectral formula.
Lemma 2.3. Let f \in L(\BbbX ) and P (x) be a polynomial. Assume that \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(f) is compact. Then
there always exists the following limit:
\mathrm{l}\mathrm{i}\mathrm{m}
m\rightarrow \infty
\| Pm(D)f\| 1/mL(\BbbX )
and
\mathrm{l}\mathrm{i}\mathrm{m}
m\rightarrow \infty
\| Pm(D)f\| 1/mL(\BbbX ) = \mathrm{s}\mathrm{u}\mathrm{p} \{ | P (\zeta )| : \zeta \in \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(f)\} .
Proof of Theorem 2.1. Necessity. We choose a function \vargamma \in C\infty
0 (\BbbR ) such that \vargamma (\zeta ) = (2\pi ) - 1/2
if \zeta \in K\tau /4 and \vargamma (\zeta ) = 0 if \zeta /\in K\tau /2. Then it follows from \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(f) \subset K and Lemma 2.2 that
f = \scrF - 1(\vargamma ) \ast f and Dnf =
\bigl(
Dn\scrF - 1(\vargamma )
\bigr)
\ast f = \scrF - 1 (\vargamma (\zeta )(i\zeta )n) \ast f for all n \in \BbbZ +. Combining
these and the definition of the differential operator P (D), we obtain
P (D)f = \scrF - 1(\vargamma (\zeta )P (\zeta )) \ast f.
Therefore, by Lemma 2.1, we have
\| P (D)f\| L(\BbbX ) \leq \| f\| L(\BbbX )
\bigm\| \bigm\| \scrF - 1 (\vargamma (\zeta )P (\zeta ))
\bigm\| \bigm\|
L1 =
= \| f\| L(\BbbX ) \| \scrF (\vargamma (\zeta )P (\zeta ))\| L1 = \| f\| L(\BbbX )\| \Psi \| L1 ,
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 6
734 H. H. BANG, V. N. HUY
where
\Psi (x) := (\scrF (\vargamma (\zeta )P (\zeta ))) (x).
Hence, from\int
\BbbR
| \Psi (x)| dx \leq
\biggl(
\mathrm{s}\mathrm{u}\mathrm{p}
x\in \BbbR
|
\bigl(
1 + x2
\bigr)
\Psi (x)|
\biggr) \left( \int
\BbbR
dx
1 + x2
\right) = \pi \mathrm{s}\mathrm{u}\mathrm{p}
x\in \BbbR
\bigm| \bigm| \bigl( 1 + x2
\bigr)
\Psi (x)
\bigm| \bigm| ,
we obtain
\| P (D)f\| L(\BbbX ) \leq \pi \| f\| L(\BbbX ) \mathrm{s}\mathrm{u}\mathrm{p}
x\in \BbbR
\bigm| \bigm| \bigl( 1 + x2
\bigr)
\Psi (x)
\bigm| \bigm| . (2)
For \beta \in \{ 0, 1, 2\} we get the following estimate:
\mathrm{s}\mathrm{u}\mathrm{p}
x\in \BbbR
\bigm| \bigm| \bigm| x\beta \Psi (x)
\bigm| \bigm| \bigm| = (2\pi ) - 1/2 \mathrm{s}\mathrm{u}\mathrm{p}
x\in \BbbR
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\int
\BbbR
e - ix\zeta D\beta (\vargamma (\zeta )P (\zeta )) d\zeta
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| =
= (2\pi ) - 1/2 \mathrm{s}\mathrm{u}\mathrm{p}
x\in \BbbR
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\int
\zeta \in K\tau /2
e - ix\zeta D\beta (\vargamma (\zeta )P (\zeta )) d\zeta
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq (2\pi ) - 1/2
\int
\zeta \in K\tau /2
\bigm| \bigm| \bigm| D\beta (\vargamma (\zeta )P (\zeta ))
\bigm| \bigm| \bigm| d\zeta .
Then it follows from Leibniz’s rule that
\mathrm{s}\mathrm{u}\mathrm{p}
x\in \BbbR
\bigm| \bigm| \bigm| x\beta \Psi (x)
\bigm| \bigm| \bigm| \leq (2\pi ) - 1/2
\int
\zeta \in K\tau /2
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\sum
\gamma \leq \beta
\beta !
\gamma !(\beta - \gamma )!
D\gamma \vargamma (\zeta )D\beta - \gamma P (\zeta )
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| d\zeta \leq
\leq (2\pi ) - 1/2
\sum
\gamma \leq \beta
\left( \beta !
\gamma !(\beta - \gamma )!
\mathrm{s}\mathrm{u}\mathrm{p}
x\in K\tau /2
\bigm| \bigm| \bigm| D\beta - \gamma P (x)
\bigm| \bigm| \bigm| \int
\zeta \in K\tau /2
| D\gamma \vargamma (\zeta )| d\zeta
\right) \leq
\leq (2\pi ) - 1/2\mathrm{m}\mathrm{a}\mathrm{x}
\theta \leq 2
\mathrm{s}\mathrm{u}\mathrm{p}
x\in K\tau /2
\bigm| \bigm| \bigm| D\theta P (x)
\bigm| \bigm| \bigm| \sum
\gamma \leq \beta
\left( \beta !
\gamma !(\beta - \gamma )!
\int
\zeta \in K\tau /2
| D\gamma \vargamma (\zeta )| d\zeta
\right) . (3)
For each x \in K\tau /2, we consider \gamma x = \{ z \in \BbbC : | z - x| = \tau /2\} as a simple closed curve oriented
counterclockwise. Because P (x) is a holomorphic function and by Cauchy’s integral formula for
derivatives, we obtain, for n = 0, 1, 2, . . . ,
DnP (x) =
n!
2\pi i
\int
\gamma x
P (z)dz
(z - x)n+1
.
Hence,
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 6
PALEY – WIENER TYPE THEOREM FOR FUNCTIONS WITH VALUES IN BANACH SPACES 735
| DnP (x)| \leq
n! \mathrm{s}\mathrm{u}\mathrm{p}z\in \gamma x | P (z)|
(\tau /2)n
.
Since \gamma x \subset K(\tau ) and above inequalities are true for all x \in K\tau /2, we deduce
\mathrm{s}\mathrm{u}\mathrm{p}
x\in K\tau /2
| DnP (x)| \leq
n! \mathrm{s}\mathrm{u}\mathrm{p}z\in K(\tau ) | P (z)|
(\tau /2)n
for n = 0, 1, 2, . . . . So,
\mathrm{s}\mathrm{u}\mathrm{p}
x\in K\tau /2
| DnP (x)| \leq A\tau \mathrm{s}\mathrm{u}\mathrm{p}
x\in K(\tau )
| P (x)| (4)
for n = 0, 1, 2, where A\tau = 2
\bigl(
1 + (2/\tau )2
\bigr)
is independent of P (x). By using (3), (4), we have
\mathrm{s}\mathrm{u}\mathrm{p}
x\in \BbbR
\bigm| \bigm| \bigm| x\beta \Psi (x)
\bigm| \bigm| \bigm| \leq (2\pi ) - 1/2
\sum
\gamma \leq \beta
\left( \beta !
\gamma !(\beta - \gamma )!
A\tau \mathrm{s}\mathrm{u}\mathrm{p}
x\in K(\tau )
| P (x)|
\int
\zeta \in K\tau /2
| D\gamma \vargamma (\zeta )| d\zeta
\right) \leq
\leq 4(2\pi ) - 1/2A\tau A\tau ,K \mathrm{s}\mathrm{u}\mathrm{p}
x\in K(\tau )
| P (x)| (5)
for all \beta = 0, 1, 2, where
A\tau ,K := \mathrm{m}\mathrm{a}\mathrm{x}
\gamma \leq 2
\int
\zeta \in K\tau /2
| D\gamma \vargamma (\zeta )| d\zeta .
Then it follows from (5) that
\mathrm{s}\mathrm{u}\mathrm{p}
x\in \BbbR
\bigm| \bigm| \bigl( 1 + x2
\bigr)
\Psi (x)
\bigm| \bigm| \leq 8(2\pi ) - 1/2A\tau A\tau ,K \mathrm{s}\mathrm{u}\mathrm{p}
x\in K(\tau )
| P (x)| . (6)
From (2) and (6) we obtain (1).
Sufficiency. Assume (1) is true, we need to prove \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(f) \subset K. Indeed, assume the contrary
that there exists \varrho \in \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(f) and \varrho \not \in K. We construct the polynomial G(x) = t - (x - \varrho )2, where
t = \mathrm{s}\mathrm{u}\mathrm{p}x\in K(x - \varrho )2. Then applying (1) for P (x) = Gm(x), we get, for all m \in \BbbZ +,
\| Gm(D)f\| L(\BbbX ) \leq C\tau \| f\| L(\BbbX ) \mathrm{s}\mathrm{u}\mathrm{p}
x\in K(\tau )
| Gm(x)| ,
which gives
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
m\rightarrow \infty
\Bigl(
\| Gm(D)f\| L(\BbbX )
\Bigr) 1/m
\leq \mathrm{s}\mathrm{u}\mathrm{p}
x\in K(\tau )
| G(x)| .
Letting \tau \rightarrow 0, we obtain
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
m\rightarrow \infty
\Bigl(
\| Gm(D)f\| L(\BbbX )
\Bigr) 1/m
\leq \mathrm{s}\mathrm{u}\mathrm{p}
x\in K
| G(x)| . (7)
Then it follows from Lemma 2.3 that
| G(\varrho )| \leq \mathrm{s}\mathrm{u}\mathrm{p}
x\in K
| G(x)|
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 6
736 H. H. BANG, V. N. HUY
and then
t = | G(\varrho )| \leq \mathrm{s}\mathrm{u}\mathrm{p}
x\in K
\bigl(
t - (x - \varrho )2
\bigr)
.
This is a contradiction. So, \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(f) \subset K.
Theorem 2.1 is proved.
It follows from Lemma 2.3 that if f \in L(\BbbX ) and \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(f) \subset K, then for any \tau > 0 there exists
a constant CP,\tau ,f <\infty (CP,\tau ,f depends on P, \tau and f ) such that
\| Pm(D)f\| L(\BbbX ) \leq CP,\tau ,f\| f\| L(\BbbX ) \mathrm{s}\mathrm{u}\mathrm{p}
x\in K(\tau )
| Pm(x)| \forall m \in \BbbN ,
while by Theorem 2.1 we have the stronger result that for any \tau > 0 there exists a constant C\tau <\infty
(independent of P, m, f ) such that
\| Pm(D)f\| L(\BbbX ) \leq C\tau \| f\| L(\BbbX ) \mathrm{s}\mathrm{u}\mathrm{p}
x\in K(\tau )
| Pm(x)| .
2.2. Paley – Wiener type theorem for sets generated by polynomials. Let P (x) be a polynomial
with complex coefficients. We put, for r > 0,
Q(P )r := \{ x \in \BbbR : | P (x)| \leq r\}
and Q(P )r is called the set generated by P (x) with respect to r. Note that if deg(P ) \geq 1, then
the Q(P )r is compact. Moreover, if a, b \in \BbbR , a \leq b, \alpha > 0, then [a, a + \alpha ] \cup [b, b + \alpha ] are sets
generated by polynomials.
Theorem 2.2. Let f \in L(\BbbX ), r > 0 and P (x) be a polynomial. Then \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(f) \subset Q(P )r if and
only if for any \tau > 0 there exists a constant C\tau ,r,P <\infty independent of f such that
\| Pm(D)f\| L(\BbbX ) \leq C\tau ,r,P \| f\| L(\BbbX )(r + \tau )m (8)
for all m \in \BbbZ +.
Proof. Necessity is follows from Theorem 2.1.
Sufficiency. Assume the contrary that there exists \sigma \in \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(f) and \sigma \not \in Q(P )r. Combining
\sigma /\in Q(P )r and Q(P )r = \{ x \in \BbbR : | P (x)| \leq r\} , we have
| P (\sigma )| > r.
By using (8), we obtain
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
m\rightarrow \infty
\Bigl(
\| Pm(D)f\| L(\BbbX )
\Bigr) 1/m
\leq r + \tau . (9)
Applying Lemma 2.3, we have
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
m\rightarrow \infty
\Bigl(
\| Pm(D)f\| L(\BbbX )
\Bigr) 1/m
\geq | P (\sigma )| . (10)
From (9) and (10), we get | P (\sigma )| \leq r + \tau . Letting \tau \rightarrow 0, we obtain | P (\sigma )| \leq r. This is a
contradiction. So, \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(f) \subset Q(P )r.
Theorem 2.2 is proved.
By Theorem 2.2 we get the following corollary.
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PALEY – WIENER TYPE THEOREM FOR FUNCTIONS WITH VALUES IN BANACH SPACES 737
Corollary 2.1. Let r > 0 and f \in L(\BbbX ). Then \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(f) \subset [ - r, r] if and only if for any \tau > 0
there exists a constant C\tau <\infty such that
\| Dmf\| L(\BbbX ) \leq C\tau (r + \tau )m\| f\| L(\BbbX )
for all m \in \BbbZ +.
In general, for a, b \in \BbbR , a < b, then [a, b] is the set generated by the polynomial P (x) =
= x - a+ b
2
with respect to
b - a
2
. Therefore, \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(f) \subset [a, b] if and only if for any \tau > 0 there
exists a constant C\tau <\infty such that\bigm\| \bigm\| \bigm\| \bigm\| \biggl( x - a+ b
2
\biggr) m
(D)f
\bigm\| \bigm\| \bigm\| \bigm\|
L(\BbbX )
\leq C\tau
\biggl(
b - a
2
+ \tau
\biggr) m
\| f\| L(\BbbX )
for all m \in \BbbZ +.
Moreover, for a, b \in \BbbR , a < b, \alpha > 0, then [a, a + \alpha ] \cup [b, b + \alpha ] is the set generated by the
polynomial Q(x) = x2 - (a + b + \alpha )x + ab +
(a+ b)\alpha
2
with respect to r =
(b - a)\alpha
2
. Hence,
\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c} (f) \subset [a, a+ \alpha ] \cup [b, b+ \alpha ] if and only if for any \tau > 0 there exists a constant C\tau <\infty such
that
\| Qm(D)f\| L(\BbbX ) \leq C\tau
\biggl(
(b - a)\alpha
2
+ \tau
\biggr) m
\| f\| L(\BbbX )
for all m \in \BbbZ +. Consequently, for 0 < a < b, \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(f) \subset [a, b] \cup [ - b, - a] if and only if for any
\tau > 0 there exists a constant C\tau <\infty such that\bigm\| \bigm\| \bigm\| \bigm\| \biggl( x2 - a2 + b2
2
\biggr) m
(D)f
\bigm\| \bigm\| \bigm\| \bigm\|
L(\BbbX )
\leq C\tau
\biggl(
b2 - a2
2
+ \tau
\biggr) m
\| f\| L(\BbbX )
for all m \in \BbbZ +.
3. Paley – Wiener type theorem for integral operators. 3.1. Paley – Wiener type theorem
for any compact \bfitK . We define I\lambda = (\lambda - D) - 1, where \lambda \in \BbbC \setminus i\BbbR . The integral operator P (I\lambda ) is
obtained from P (x) by substituting x\rightarrow I\lambda . We have the following result for P (I\lambda ).
Theorem 3.1. Let K be a compact set in \BbbR , f \in L(\BbbX ) and \lambda \in \BbbC \setminus i\BbbR . Then \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(f) \subset K if
and only if for any \tau > 0 there exists C\tau ,K > 0 independent of f such that
\| P (I\lambda )f\| L(\BbbX ) \leq C\tau ,K \mathrm{s}\mathrm{u}\mathrm{p}
x\in K(\tau )
| P (1/(\lambda - ix))| \| f\| L(\BbbX ) (11)
for all polynomials with complex coefficients P (x).
Proof. Necessity. Assume that \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(f) \subset K. Now, we choose a function \phi \in C\infty (\BbbR ) such that
\phi (x) = (2\pi ) - 1/2 if x \in K\tau /4 and \phi (x) = 0 if x /\in K\tau /2. Since \lambda \in \BbbC \setminus i\BbbR , there is a small enough
positive number \tau such that \lambda - ix \not = 0 for all x \in K(\tau ), so, the following function is well defined:
\Phi = \scrF - 1 (\phi (x)P (1/(\lambda - ix))) .
By using Lemma 2.2, we have f = \u \phi \ast f, and then
(\lambda - D)
\Bigl(
(I\lambda \u \phi ) \ast f
\Bigr)
= \lambda
\Bigl(
(I\lambda \u \phi ) \ast f
\Bigr)
- D
\Bigl(
(I\lambda \u \phi ) \ast f
\Bigr)
=
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738 H. H. BANG, V. N. HUY
= \lambda
\Bigl(
(I\lambda \u \phi ) \ast f
\Bigr)
- (DI\lambda \u \phi ) \ast f =
\Bigl(
(\lambda - D)(I\lambda \u \phi )
\Bigr)
\ast f = \u \phi \ast f = f.
Hence, I\lambda f =
\Bigl(
I\lambda \u \phi
\Bigr)
\ast f. Similarly, Ik\lambda f =
\Bigl(
Ik\lambda
\u \phi
\Bigr)
\ast f \forall k \in \BbbZ +. So,
P (I\lambda )f =
\Bigl(
P (I\lambda )\u \phi
\Bigr)
\ast f. (12)
From Ik\lambda
\u \phi = \scrF - 1
\bigl(
\phi (x)/(\lambda - ix)k
\bigr)
, we conclude P (I\lambda )\u \phi = \scrF - 1 (\phi (x)P (1/(\lambda - ix))) = \Phi .
Therefore, applying (12), we get P (I\lambda )f = \Phi \ast f. Hence, it follows from Lemma 2.1 that
\| P (I\lambda )f\| L(\BbbX ) \leq \| f\| L(\BbbX )\| \Phi \| L1 . (13)
For each x \in K\tau /2, we consider \gamma = \{ z \in \BbbC : | z - x| = \tau /2\} as a simple closed curve on K(\tau )
oriented counterclockwise. Because Q(x) := P (1/(\lambda - ix)) is a holomorphic function on K(\tau ) and
by Cauchy’s integral formula for derivatives, we obtain
DnQ(x) =
n!
2\pi i
\int
\gamma
Q(z)dz
(z - x)n+1
, n = 0, 1, 2, . . . .
Consequently,
| DnQ(x)| \leq
n! \mathrm{s}\mathrm{u}\mathrm{p}z\in \gamma | Q(z)|
(\tau /2)n
, n = 0, 1, 2, . . . .
Therefore, since \gamma \subset K(\tau ),
\mathrm{s}\mathrm{u}\mathrm{p}
x\in K\tau /2
| DnQ(x)| \leq
n! \mathrm{s}\mathrm{u}\mathrm{p}z\in K(\tau ) | Q(z)|
(\tau /2)n
, n = 0, 1, 2, . . . .
Then it follows from
\surd
2\pi \mathrm{s}\mathrm{u}\mathrm{p}
x\in \BbbR
\bigm| \bigm| \bigl( 1 + x2
\bigr)
\Phi (x)
\bigm| \bigm| \leq
\leq
\int
x\in K\tau /2
\bigl( \bigm| \bigm| D2 (\phi (x)P (1/(\lambda - ix)))
\bigm| \bigm| + | \phi (x)P (1/(\lambda - ix))|
\bigr)
dx
that
\mathrm{s}\mathrm{u}\mathrm{p}
x\in \BbbR
\bigm| \bigm| \bigl( 1 + x2
\bigr)
\Phi (x)
\bigm| \bigm| \leq C(\tau ,K) \mathrm{s}\mathrm{u}\mathrm{p}
x\in K(\tau )
| P (1/(\lambda - ix))| , (14)
where C(\tau ,K) does not depend on P (x). Further, we have
\| \Phi \| L1 \leq \pi C(\tau ,K) \mathrm{s}\mathrm{u}\mathrm{p}
x\in \BbbR
\bigm| \bigm| \bigl( 1 + x2
\bigr)
\Phi (x)
\bigm| \bigm| . (15)
Combining (13) – (15), we obtain
\| P (I\lambda )f\| L(\BbbX ) \leq \pi C(\tau ,K) \mathrm{s}\mathrm{u}\mathrm{p}
x\in K\tau
| P (1/(\lambda - ix))| \| f\| L(\BbbX ).
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PALEY – WIENER TYPE THEOREM FOR FUNCTIONS WITH VALUES IN BANACH SPACES 739
Sufficiency. Assume the contrary that there exists \sigma \in \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(f) and \sigma \not \in K. Let t = \mathrm{s}\mathrm{u}\mathrm{p}x\in K(\sigma -
- x)2, we define the following polynomial:
Q(x) = t+ (x - \lambda + i\sigma )2.
Clearly,
\mathrm{s}\mathrm{u}\mathrm{p}
x\in K
| Q(\lambda - ix)| = \mathrm{s}\mathrm{u}\mathrm{p}
x\in K
\bigm| \bigm| t - (\sigma - x)2
\bigm| \bigm| < t = | Q(\lambda - i\sigma )| .
Put
H =
\biggl\{
z = a+ bi : | a| < \mathrm{R}\mathrm{e}\lambda , | b| < 2(| \sigma | + \mathrm{s}\mathrm{u}\mathrm{p}
x\in K
| x| )
\biggr\}
,
H1 =
\biggl\{
z = a+ bi : | a| < \mathrm{R}\mathrm{e}\lambda
2
, | b| < | \sigma | + \mathrm{s}\mathrm{u}\mathrm{p}
x\in K
| x|
\biggr\}
and R(x) = Q
\biggl(
1
x
\biggr)
. Then it follows from \lambda \in \BbbC \setminus i\BbbR that R(x) is a holomorphic function on
H and
| R(1/(\lambda - i\sigma ))| > \mathrm{s}\mathrm{u}\mathrm{p}
x\in K
| R(1/(\lambda - ix))| . (16)
Because R(x) is a holomorphic function in the complex domain H, there exists a sequence of
polynomials \{ Pn\} such that Pn converges uniformly to R(x) on H1. Combining this with (16), we
can choose an integer j0 such that
| Pj0(1/(\lambda - i\sigma ))| > \mathrm{s}\mathrm{u}\mathrm{p}
x\in K
| Pj0(1/(\lambda - ix))| . (17)
For a small enough positive number \tau we have Pj0(1/(\lambda - ix)) \not = 0 for all x \in (\sigma - \tau , \sigma + \tau ).
From the definition of Beurling spectrum, there exists \varphi \in C\infty
0 (\BbbR ), \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\varphi \subset (\sigma - \tau , \sigma + \tau ) such
that \u \varphi \ast f \not \equiv 0. Put
\varphi m = \scrF - 1
\bigl(
\varphi (x)/Pm
j0 (1/(\lambda - ix))
\bigr)
.
Then \varphi m is well defined, \varphi m \in \scrS (\BbbR ) and Pm
j0
(I\lambda )\varphi m = \u \varphi . Clearly, \varphi m \ast
\bigl(
Ik\lambda f
\bigr)
=
\bigl(
Ik\lambda \varphi m
\bigr)
\ast f for
all k \in \BbbZ +. That gives \varphi m \ast
\Bigl(
Pm
j0
(I\lambda )f
\Bigr)
=
\Bigl(
Pm
j0
(I\lambda )\varphi m
\Bigr)
\ast f, and then \varphi m \ast Pm
j0
(I\lambda )f = \u \varphi \ast f.
So, by Lemma 2.1, we get
0 < \| \u \varphi \ast f\| L(\BbbX ) =
\bigm\| \bigm\| \varphi m \ast Pm
j0 (I\lambda )f
\bigm\| \bigm\|
L(\BbbX ) \leq
\bigm\| \bigm\| Pm
j0 (I\lambda )f
\bigm\| \bigm\|
L(\BbbX ) \| \varphi m\| L1 .
Consequently,
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
m\rightarrow \infty
\bigm\| \bigm\| Pm
j0 (I\lambda )f
\bigm\| \bigm\| 1/m
L(\BbbX ) \geq 1/ \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
m\rightarrow \infty
\| \varphi m\| 1/m
L1 . (18)
From
\mathrm{s}\mathrm{u}\mathrm{p}
x\in \BbbR
\bigm| \bigm| (1 + x2)\varphi m(x)
\bigm| \bigm| \leq
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 6
740 H. H. BANG, V. N. HUY
\leq 1\surd
2\pi
\sigma +\tau \int
\sigma - \tau
\bigl( \bigm| \bigm| D2
\bigl(
\varphi (x)/Pm
j0 (1/(\lambda - ix))
\bigr) \bigm| \bigm| + \bigm| \bigm| \varphi (x)/Pm
j0 (1/(\lambda - ix))
\bigm| \bigm| \bigr) dx
we can deduce that
\mathrm{s}\mathrm{u}\mathrm{p}
x\in \BbbR
\bigm| \bigm| \bigl( 1 + x2
\bigr)
\varphi m(x)
\bigm| \bigm| \leq C1m
2 \mathrm{s}\mathrm{u}\mathrm{p}
x\in (\sigma - \tau ,\sigma +\tau )
\bigm| \bigm| \bigm| 1/Pm+2
j0
(1/(\lambda - ix))
\bigm| \bigm| \bigm| (19)
for some C1 independent of m. Then it follows from \| \varphi m\| L1 \leq \pi \mathrm{s}\mathrm{u}\mathrm{p}x\in \BbbR
\bigm| \bigm| \bigl( 1 + x2
\bigr)
\varphi m(x)
\bigm| \bigm| that
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
m\rightarrow \infty
\| \varphi m\| 1/m
L1 \leq \mathrm{s}\mathrm{u}\mathrm{p}
x\in (\sigma - \tau ,\sigma +\tau )
| 1/Pj0(1/(\lambda - ix))| . (20)
Relations (18) and (20) imply
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
m\rightarrow \infty
\bigm\| \bigm\| Pm
j0 (I\lambda )f
\bigm\| \bigm\| 1/m
L(\BbbX ) \geq \mathrm{i}\mathrm{n}\mathrm{f}
x\in (\sigma - \tau ,\sigma +\tau )
| Pj0(1/(\lambda - ix))| .
Letting \tau \rightarrow 0, we get
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
m\rightarrow \infty
\bigm\| \bigm\| Pm
j0 (I\lambda )f
\bigm\| \bigm\| 1/m
L(\BbbX ) \geq | Pj0(1/(\lambda - i\sigma ))| . (21)
Combining this with (17), we have
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
m\rightarrow \infty
\bigm\| \bigm\| Pm
j0 (I\lambda )f
\bigm\| \bigm\| 1/m
L(\BbbX ) > \mathrm{s}\mathrm{u}\mathrm{p}
x\in K
| Pj0(1/(\lambda - ix))| .
This is contrary to (11).
Theorem 3.1 is proved.
3.2. Paley – Wiener type theorem for sets generated by polynomial type. Let P (x) be a poly-
nomial with complex coefficients, r > 0 and \lambda \in \BbbC \setminus i\BbbR . We put
(P )r,\lambda := \{ x \in \BbbR : | P (1/(\lambda - ix))| \leq r\}
and (P )r,\lambda is called the set generated by the polynomial of type (P (x), r, \lambda ). Note that if | P (0)| > r,
then (P )r,\lambda is compact, and if (P )r,\lambda is compact, then | P (0)| \geq r. However, | P (0)| = r does not
guarantee compactness of (P )r,\lambda or not. For example, we put P (x) = 1 + x4, P1(x) = 1 + x2 and
\lambda = r = 1. Then (P )1,1 is compact and (P1)1,1 is not compact. Indeed,\bigm| \bigm| \bigm| \bigm| P \biggl( 1
\lambda - ix
\biggr) \bigm| \bigm| \bigm| \bigm| =
\bigm| \bigm| \bigm| \bigm| \bigm| 1 +
\biggl(
1
1 - ix
\biggr) 4
\bigm| \bigm| \bigm| \bigm| \bigm| =
=
\bigm| \bigm| \bigm| \bigm| \bigm| 1 +
\biggl(
1 + ix
1 + x2
\biggr) 4
\bigm| \bigm| \bigm| \bigm| \bigm| =
\bigm| \bigm| \bigm| \bigm| \bigm| 1 + x4 - 6x2 + 1
(1 + x2)4
+
i
\bigl(
4x - 4x3
\bigr)
(1 + x2)4
\bigm| \bigm| \bigm| \bigm| \bigm| ,\bigm| \bigm| \bigm| \bigm| P1
\biggl(
1
\lambda - ix
\biggr) \bigm| \bigm| \bigm| \bigm| =
\bigm| \bigm| \bigm| \bigm| \bigm| 1 +
\biggl(
1
1 - ix
\biggr) 2
\bigm| \bigm| \bigm| \bigm| \bigm| =
=
\bigm| \bigm| \bigm| \bigm| \bigm| 1 +
\biggl(
1 + ix
1 + x2
\biggr) 2
\bigm| \bigm| \bigm| \bigm| \bigm| =
\bigm| \bigm| \bigm| \bigm| 1 - x2 - 1
(1 + x2)2
+
2xi
(1 + x2)2
\bigm| \bigm| \bigm| \bigm| ,
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PALEY – WIENER TYPE THEOREM FOR FUNCTIONS WITH VALUES IN BANACH SPACES 741
and then \bigm| \bigm| \bigm| \bigm| P \biggl( 1
\lambda - ix
\biggr) \bigm| \bigm| \bigm| \bigm| \geq \biggl( 1 + x4 - 6x2 + 1
(1 + x2)4
\biggr) 1/2
> 1,
\bigm| \bigm| \bigm| \bigm| P1
\biggl(
1
\lambda - ix
\biggr) \bigm| \bigm| \bigm| \bigm| = \biggl( 1 - 2x2 - 3
(1 + x2)2
\biggr) 1/2
< 1
for all x \in ( - \infty , - 6) \cup (6,+\infty ).
Therefore, (P )1,1 is compact but (P1)1,1 is not compact.
Moreover, if a, b \in \BbbR , a < b, then [a, b] is a set generated by polynomial type. To see this we
put c = (a+b)/2, d = (b - a)/2, \lambda = 1+ ic and we choose two numbers \kappa , r \in [1,+\infty ) satisfying
(2\kappa - 1)/
\bigl(
\kappa 2 - r2
\bigr)
= 1 + d2. Put P (x) = \kappa - x. Clearly,
| P (1/(\lambda - ix))| =
\bigm| \bigm| \bigm| \bigm| \kappa - 1
\lambda - ix
\bigm| \bigm| \bigm| \bigm| = \bigm| \bigm| \bigm| \bigm| \kappa - 1
1 + i(c - x)
\bigm| \bigm| \bigm| \bigm| = \bigm| \bigm| \bigm| \bigm| \kappa - 1 - i(c - x)
1 + (c - x)2
\bigm| \bigm| \bigm| \bigm| =
=
\Biggl( \biggl(
\kappa - 1
1 + (c - x)2
\biggr) 2
+
\biggl(
(c - x)
1 + (c - x)2
\biggr) 2
\Biggr) 1/2
=
\biggl(
\kappa 2 - 2\kappa - 1
1 + (c - x)2
\biggr) 1/2
.
Hence,
\{ x \in \BbbR : | P (1/(\lambda - ix))| \leq r\} =
\biggl\{
x \in \BbbR : \kappa 2 - 2\kappa - 1
1 + (c - x)2
\leq r2
\biggr\}
=
=
\bigl\{
x \in \BbbR : 1 + (c - x)2 \leq (2\kappa - 1)/
\bigl(
\kappa 2 - r2
\bigr) \bigr\}
=
=
\bigl\{
x \in \BbbR : 1 + (c - x)2 \leq 1 + d2
\bigr\}
.
Consequently, (P )r,\lambda = [a, b].
Theorem 3.2. Let f \in L(\BbbX ), r > 0, \lambda \in \BbbC \setminus i\BbbR , P (x) be a polynomial and (P )r,\lambda be compact.
Then \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(f) \subset (P )r,\lambda if and only if for any \tau > 0 there exists a constant C\tau ,r,\lambda ,P <\infty independent
of f such that
\| Pm(I\lambda )f\| L(\BbbX ) \leq C\tau ,r,\lambda ,P \| f\| L(\BbbX )(r + \tau )m (22)
for all m \in \BbbZ +.
Proof. Necessity is follows from Theorem 3.1.
Sufficiency. Assume the contrary that there exists \sigma \in \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(f) and \sigma \not \in (P )r,\lambda . Hence,
| P (1/(\lambda - i\sigma ))| > r. According to (22), we obtain
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
m\rightarrow \infty
\bigl(
\| Pm(I\lambda )f\| L(\BbbX )
\bigr) 1/m \leq r + \tau . (23)
Applying the proof of inequality (21), we have
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
m\rightarrow \infty
\Bigl(
\| Pm(I\lambda )f\| L(\BbbX )
\Bigr) 1/m
\geq | P (1/(\lambda - i\sigma ))| .
Combining this with (23), we deduce | P (1/(\lambda - i\sigma ))| \leq r+ \tau . Letting \tau \rightarrow 0, we obtain | P (1/(\lambda -
- i\sigma ))| \leq r. This is a contradiction. So, \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(f) \subset (P )r,\lambda .
Theorem 3.2 is proved.
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742 H. H. BANG, V. N. HUY
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Received 21.02.20
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 6
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| id | umjimathkievua-article-2382 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:22:34Z |
| publishDate | 2022 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/35/1fc7a09c1eff6e78a951654fadfdf635.pdf |
| spelling | umjimathkievua-article-23822022-10-25T09:23:03Z Paley – Wiener type theorem for functions with values in Banach spaces Paley – Wiener type theorem for functions with values in Banach spaces Bang, H. H. Huy, V. N. Bang, H. H. Huy, V. N. Real Paley-Wiener theorem, Beurling spectrum, Generalized functions, Banach spaces UDC 517.5 Let $(\mathbb{X},\|.\|_{\mathbb{X}})$ denote a complex Banach space and $L(\mathbb{X})=BC (\mathbb{R}\to\mathbb{X})$ be the set of all $\mathbb{X}$-valued bounded continuous functions $f\colon\mathbb{R}\to\mathbb{X}.$For $f\in L(\mathbb{X})$ we define $\|f\|_{L(\mathbb{X})}=\sup\{ \|f(x)\|_{\mathbb{X}}\colon x\in\mathbb{R}\}.$ Then $(L(\mathbb{X}),\|.\|_{L(\mathbb{X})})$ itself is a Banach space. The Beurling spectrum $\mathrm{Spec}(f)$ of a function $f\in L(\mathbb{X})$ is defined by $$\mathrm{Spec}(f)=\left\{\zeta\in\mathbb{R}\colon\forall\epsilon&gt;0 \exists \varphi\in\mathcal{S}(\mathbb{R})\colon\mbox{supp}\,\widehat{\varphi}\subset(\zeta-\epsilon,\zeta+\epsilon),\varphi*f\nequiv 0\right\}.\right\}.$$ We obtain the following Paley-Wiener type theorem for functions with values in Banach spaces: Let $f\in L(\mathbb{X})$ and $K$ be an arbitrary compact set in $\mathbb{R}.$ Then $\mbox{Spec}(f)\subset K$ if and only if for any $\tau &gt; 0$ there exists a constant $C_\tau &lt; \infty$ such that $$\|P(D)f\|_{L(\mathbb{X})}\leq C_\tau\|f\|_{L(\mathbb{X})}\sup\limits_{x\in K^{(\tau)}} |P(x)|$$for all polynomials with complex coefficients $P(x),$ where the differential operator $P(D)$ is obtained from $P(x)$ by substituting $x \to -i\,\dfrac{d}{dx},$ $\dfrac{d}{dx}$ is the usual derivative in $L(\mathbb{X})$ and $K^{(\tau)}$ is the $\tau$-neighborhood in $\mathbb{C}$ of $K.$ Moreover, Paley-Wiener type theorem for integral operators and one for some special compacts $K$ are also given. УДК 517.5Теорема типу Пелi – Вiнера для функцiй iз значеннями у банахових просторахНехай $(\mathbb{X}, \|.\|_{\mathbb{X}})$ — комплексний простiр Банаха i $L(\mathbb{X})=BC (\mathbb{R} \to \mathbb{X})$ — множина всiх обмежених неперервних $\mathbb{X}$ -значних функцiй$f: \mathbb{R} \to \mathbb{X}$. Для $f\in L(\mathbb{X})$ вводиться позначення$\|f\|_{L(\mathbb{X})}=\sup\{ \|f(x)\|_{\mathbb{X}}\colon x\in\mathbb{R}\}.$ Тодi $(L(\mathbb{X}),\|.\|_{L(\mathbb{X})})$ сам є банаховим простором. Спектр Берлiнга $\mathrm{Spec}(f)$ функцiї $f\in L(\mathbb{X})$ визначається як$$\mathrm{Spec}(f)=\left\{\zeta\in\mathbb{R}\colon\forall\epsilon&gt;0\ \exists\varphi\in\mathcal{S}(\mathbb{R})\colon\mbox{supp}\,\widehat{\varphi}\subset(\zeta-\epsilon,\zeta+\epsilon),\varphi*f\not\equiv 0\right\}.$$Отримано таку теорему типу Пелi – Вiнера для функцiй iз значеннями у просторах Банаха:Нехай $f\in L(\mathbb{X})$ i $K$ — довiльна компактна множина в $\mathbb{R}.$ У цьому випадку $\mbox{Spec}(f)\subset K$ тодi й лише тодi, коли для будь-якого $\tau &gt; 0$ iснує стала $C_\tau &lt; \infty$ така, що$$\|P(D)f\|_{L(\mathbb{X})}\leq C_\tau\|f\|_{L(\mathbb{X})}\sup\limits_{x\in K^{(\tau)}} |P(x)|$$для всiх полiномiв з комплексними коефiцiєнтами $P(x),$, де диференцiальний оператор $P(D)$ отримано з $P(x)$ замiною $x \to -i\,\dfrac{d}{dx},$ $\dfrac{d}{dx}$ — звичайна похiдна у $L(\mathbb{X})$ i $K^{(\tau)}$ — $\tau$ -окiл для $K$ у $\mathbb{C}$.Також наведено теорему типу Пелi – Вiнера для iнтегральних операторiв та деяких спецiальних компактiв $K$. Institute of Mathematics, NAS of Ukraine 2022-07-07 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2382 10.37863/umzh.v74i6.2382 Ukrains’kyi Matematychnyi Zhurnal; Vol. 74 No. 6 (2022); 731 - 742 Український математичний журнал; Том 74 № 6 (2022); 731 - 742 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2382/9268 Copyright (c) 2022 Ha Huy Bang, Vu Nhat Huy |
| spellingShingle | Bang, H. H. Huy, V. N. Bang, H. H. Huy, V. N. Paley – Wiener type theorem for functions with values in Banach spaces |
| title | Paley – Wiener type theorem for functions with values in Banach spaces |
| title_alt | Paley – Wiener type theorem for functions with values in Banach spaces |
| title_full | Paley – Wiener type theorem for functions with values in Banach spaces |
| title_fullStr | Paley – Wiener type theorem for functions with values in Banach spaces |
| title_full_unstemmed | Paley – Wiener type theorem for functions with values in Banach spaces |
| title_short | Paley – Wiener type theorem for functions with values in Banach spaces |
| title_sort | paley – wiener type theorem for functions with values in banach spaces |
| topic_facet | Real Paley-Wiener theorem Beurling spectrum Generalized functions Banach spaces |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2382 |
| work_keys_str_mv | AT banghh paleywienertypetheoremforfunctionswithvaluesinbanachspaces AT huyvn paleywienertypetheoremforfunctionswithvaluesinbanachspaces AT banghh paleywienertypetheoremforfunctionswithvaluesinbanachspaces AT huyvn paleywienertypetheoremforfunctionswithvaluesinbanachspaces |