Abstract
Let \(1\leq p\leq \infty \), \(f\in L_{p}({\mathbb T})\) and \(0 \notin \text {supp} \hat {f}\). Then, in this paper, we obtain the following local spectral formula for the integral operator I on \(L_{p}({\mathbb T})\), the space of 2π-periodic functions belonging to L p (−π,π): \( \lim _{n\rightarrow \infty } \|I^{n} f\|_{p,{\mathbb T}}^{1/n}= \sigma ^{-1}, \) where \(\sigma =\min \{ |k| : k \in \text {supp} \hat {f} \}, If(x)={{\int }_{0}^{x}} f(t) dt -c_{f}, x\in \mathbb {R}\) and the constant c f is chosen such that \({\int }_{0}^{2\pi } If (x) dx=0\). The local spectral formula for polynomial integral operators on \(L_{p}({\mathbb T})\) is also given.
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1 Introduction
It is well known the following spectral radius formula in the Theory of Banach algebras: Let A be a unital Banach algebra and x ∈ A. Then, the limit \( \lim _{m\to \infty } \| x^{m} \|^{1/m}\) always exists and satisfies
where \(r_{A}(x)= \sup \{|\lambda | : \lambda \in \sigma _{A}(x)\}\) is the spectral radius of x, σ A (x) is the spectrum of x which is the set of all \(\lambda \in \mathbb {C}\) such that x − λ 1 is not invertible in A and 1 is the unit in A (see [12]).
It was proved in [6] the following result:
Theorem A
Let \(1\leq p\leq \infty \) and \( D^{n} f\in L_{p}(\mathbb {R}), n=0,1,2,..\). Then, there always exists the following limit
and
where \(\hat {f}\) is the Fourier transform of f.
Note that in the traditional terminology, \(\text {supp} \hat {f}\) is called the spectrum of f. Theorem A shows that differential operator D on \(L_{p}(\mathbb {R})\) has the local spectral radius formula, and σ f is called the local spectral radius of differential operator D at f. Theorem A, which also describes behavior of the sequences of norms of the derivatives of functions based on their spectrum, has been studied and developed in various directions such as extending it to other function spaces, to more general differential operators, to n-dimensional case, as well as replacing Fourier transform by other integral operators (see e.g., [1–11, 13–16]). In [16], V.K. Tuan first gave the local spectral formula for the integral operator \(\mathcal I\) on \(L_{2}(\mathbb {R})\): Let \(f \in L_{2}(\mathbb {R})\) and \(\delta _{f}=\inf \{ |\xi | : \quad \xi \in \text {supp} \hat {f} \}>0\). Then, \(\mathcal I^{n}f\) exists and belongs to \(L_{2}(\mathbb {R})\) for all n, and
where
the improper indefinite Riemann integral, and \(\mathcal I^{n}=(\mathcal I)^{n}\).
This Tuan’s result was extended to \(L_{p}(\mathbb {R})\), \(1\leq p \leq \infty \) in [9]. We propose in this paper to find the local spectral formula for integral operators on \(L_{p}({\mathbb T})\). Section 2 provides the local spectral formula for the integral operator I on \(L_{p}({\mathbb T})\), where \(If(x)={{\int }_{0}^{x}} f(t) dt -c_{f}, x\in \mathbb {R}\) and the constant c f is chosen such that \({\int }_{0}^{2\pi } If (x) dx=0\), and Section 3 deals with the formula for polynomial integral operators on \(L_{p}({\mathbb T})\).
2 Local Spectral Formula for Integral Operator on \(L_{p}({\mathbb T})\)
Recall first some notations and results needed in the sequel. Denote by \(\mathcal {S}(\mathbb {R})\), the Schwartz space of rapidly decreasing functions and by \(\mathcal {S}^{\prime }(\mathbb {R})\) the dual space of \(\mathcal {S}(\mathbb {R})\), the space of tempered distributions on \(\mathbb {R}\), it is the set of all functions \( f: \mathcal {S}(\mathbb {R}) \to \mathbb {C}\) that are linear and continuous. One usually denotes by < f,φ > the value in \(\mathbb {C}\) that the distribution \(f \in \mathcal {S}^{\prime }(\mathbb {R})\) assigns to \(\varphi \in \mathcal {S}(\mathbb {R})\). Let \(f\in L_{1}(\mathbb {R})\) and \(\hat {f}=\mathcal {F}f\) be its Fourier transform
The Fourier transform of a tempered generalized function f can be defined via the formula
Let \(\mathbb {T}=[-\pi , \pi ]\), \(1\leq p\leq \infty \), K be an arbitrary set in \(\mathbb {R}\) and 𝜖 > 0. Denote by \(K_{\epsilon } :=\{ \xi \in \mathbb {R}: \quad \exists x\in K : |x-\xi |< \epsilon \}\) and \(L_{p}({\mathbb T})\) the set of all 2π-periodic functions f on \(\mathbb {R}\) such that the norm
is finite. Then, \(L_{p}({\mathbb T}) \subset \mathcal {S}^{\prime }(\mathbb {R})\). Therefore, for each function \(f \in L_{p}({\mathbb T})\) then \(f, \hat {f}\) are the tempered distributions which are defined as follows
Moreover,
We have the following Young inequality for periodic functions: Let \(1 \leq p\leq \infty \), \(f\in L_{p}({\mathbb T})\), \(g\in L_{1}(\mathbb {R})\). Then, \(f*g \in L_{p}({\mathbb T})\) and
where the convolution f ∗ g is defined as
The following notion of primitives of a tempered generalized function was given in [9]: Let \(f\in \mathcal {S}^{\prime }(\mathbb {R})\). The tempered generalized function I f is termed a primitive of f if D(I f) = f, that is,
It was proved in [9] that for each \(f\in \mathcal {S}^{\prime }(\mathbb {R})\), the set
is always not empty and if G 1,G 2 are two primitives of f then
It is known the following result in [17]:
Lemma 1
Let \(1\leq p \leq \infty \) and \(f\in L_{p}(\mathbb {T})\). Then, the Fourier series of f converges to f in \(\mathcal {S}^{\prime }(\mathbb {R})\):
(The functional series \({\sum }_{k \in \mathbb {Z}} f_{k}(x) \) is called convergent to f in \(\mathcal {S}^{\prime }(\mathbb {R})\) if the functional sequence \(S_{n}(x)={\sum }_{|k|\leq n} f_{k}(x)\) converges to f in \(\mathcal {S}^{\prime }(\mathbb {R})\)).
Let \(f \in L_{p}({\mathbb T})\). It follows from Lemma 1 that \(\text {supp} \hat {f}\subset \mathbb {Z}\) and \(\text {supp} \hat {f}=\{k \in \mathbb {Z}: \quad {\int }_{0}^{2\pi }f(t)e^{-ikt}dt \ne 0\}\). Hence, \({\int }_{0}^{2\pi } f(t) dt=0\) if and only if \(0\notin \text {supp} \hat {f} \).
Lemma 2
Let \(1\leq p\leq \infty \) and \(f\in L_{p}({\mathbb T})\) . Then, there exists in \(L_{p}({\mathbb T})\) a primitive of f if and only if \({\int }_{0}^{2\pi } f(t) dt=0\).
Proof
Sufficiency. We define a primitive of f as follows
where the constant c f is chosen such that \({\int }_{0}^{2\pi } If (x) dx=0,\) i.e,
Then, I f is well defined on \(\mathbb {R}\) and D(I f) = f.
Further, using \(f\in L_{p}({\mathbb T})\) and \({\int }_{0}^{2\pi } If (x) dx=0\), we have for all \(x \in \mathbb {R}\)
So, \(If \in L_{p}({\mathbb T})\). Therefore, it follows from Lemma 1 that \(\text {supp} \widehat {I f} \subset \mathbb {Z}\) and the Fourier series of I f converges to I f in \(\mathcal {S}^{\prime }(\mathbb {R})\):
It follows from \({\int }_{0}^{2\pi } If (x) dx=0\) that
and then \(0 \not \in \text {supp} \widehat {I f}\).
Necessity. Assume that there exists a primitive \(G\in L_{p}({\mathbb T})\) of f. Using (1), we obtain I f = G + c, where I f is defined in (2). Therefore, \(If\in L_{p}({\mathbb T})\). Hence,
The proof is complete. □
For illustration, \(I(\sin x)=-\cos x\), \(I(\cos x)=\sin x,I(e^{ikx})= \frac {1}{ik}e^{ikx}, k\in \mathbb {Z} \symbol {92}\{0\}\).
Lemma 3
Let \(f\in L_{p}({\mathbb T})\). There exists at most one primitive G of f satisfying \(0 \notin \text {supp} \hat {G} \).
Proof
Assume that G 1,G 2 are primitives of f satisfying \(0 \notin \text {supp} \widehat {G}_{j}, j=1,2\). Then, it follows from (1) that G 1 = G 2 + c, where c is a constant. Hence, \(\widehat {G}_{1} - \widehat {G}_{2}=c \sqrt {2\pi }\delta _{0}\), where δ 0 is Dirac function at 0. Then, it follows from \(0 \not \in \widehat {G}_{1} - \widehat {G}_{2}\) and suppδ 0 = {0} that c = 0, and then G 1 = G 2. □
Formula (2) gives an operator denoted by I on \(L^{*}_{p}({\mathbb T}):= \{ f\in L_{p}({\mathbb T}) : 0 \notin \text {supp} \hat {f} \}\). Combining Lemmas 2, 3, we have
Theorem 1
Let \(1\leq p\leq \infty \) and \(f\in L^{*}_{p}({\mathbb T})\) . Then, \((I^{n} f)_{n=0}^{\infty } \subset L_{p}({\mathbb T})\), D m(I m + n f) = I n f and \(0 \notin \text {supp} \widehat {I^{n} f} \) for all \(m,n \in \mathbb {Z}_{+}\), where I 0 f := f.
Remark 1
Let \(1\leq p\leq \infty \) and \(f\in L_{p}({\mathbb T})\). By Lemma 2, the differential equation D h = f has a solution h in \(L_{p}({\mathbb T})\) if and only if \({\int }_{0}^{2\pi } f(t) dt=0\). Moreover, if \({\int }_{0}^{2\pi } f(t) dt=0\) then \(h=If+c, c \in {\mathbb C}\) are all solutions of D h = f. Clearly, \(L^{*}_{p}({\mathbb T})\) is a closed subspace of \(L_{p}({\mathbb T})\), and the linear integral operator I acts invariantly and continuously on \(L^{*}_{p}({\mathbb T})\). For each \(f\in L^{*}_{p}({\mathbb T})\), there exists exactly one solution of D h = f, which is I f. The continuity of the operator I is obtained from the following inequality \(\|I f\|_{p,{\mathbb T}} \leq \frac {\pi }{2} \| f\|_{p,{\mathbb T}}\), which is implied from the Bohr inequality \(\| g\|_{p,{\mathbb T}} \leq \frac {\pi }{2} \| Dg\|_{p,{\mathbb T}}\) (see [8]) by substituting g → I f and using D(I f) = f.
Now, we give the local spectral formula for the integral operator I on \(L_{p}({\mathbb T})\).
Theorem 2
Let \(1\leq p\leq \infty \) and \(f\in L^{*}_{p}({\mathbb T})\). Then, we have the following limit
where \(\sigma =\min \{ |k| : \quad k \in \text {supp} \hat {f} \}\).
Proof
Without loss of generality, we may assume that
Hence, for each 𝜖 > 0, there exists a function \( \varphi \in C_{0}^{\infty }(\mathbb {R}), \text {supp}\varphi \subset (\sigma -\epsilon , \sigma +\epsilon )\) such that \( <\hat {f}, {\varphi } >\ne 0\). Put
Then, taking \(\hat {f}=(i\xi )^{n} \widehat {I^{n}f}\) into account, we get
Denote
By virtue of (4) and Hölder inequality, one has
where 1/p + 1/q = 1. Since I n f is periodic, it follows that \(\|I^{n} f\|_{p,j}=\|I^{n} f\|_{p,{\mathbb T}}, \forall j \in \mathbb {Z}\). Then using (5), we get
This helps us to deduce
From the definition of φ n , we obtain
where B(σ,𝜖) := (σ − 𝜖,σ + 𝜖), and
Thus,
where
Taking account of (7), we get
if \(1\leq q<\infty \), and
This helps us to obtain
which implies
By virtue of this and (6), one has
Letting 𝜖 → 0, we obtain
Finally, we show
Indeed, since D n I n f = f, \(\hat f= (ix)^{n}\widehat {I^{n} f}\). Therefore,
Then, it follows from \(0\notin \text {supp} \widehat {I^{n}f}\) that
For any 𝜖 ∈ (0,σ/2), we choose a function \(h \in C^{\infty }(\mathbb {R})\) satisfying the following conditions \( h(\xi )=1 \quad \text {for } \xi \in (-\infty , -(\sigma -\epsilon )] \cup [ \sigma -\epsilon , +\infty ),\) and h(ξ) = 0 for ξ ∈ (−(σ − 2𝜖),σ − 2𝜖). Then, h(ξ)/(i ξ)n is well defined and it follows from \(\hat {f} =(i\xi )^{n}\widehat {I^{n} f}\) that \(h(\xi ) \hat f= (i\xi )^{n}\widehat {I^{n} f}\). Hence, \(\hat f h(\xi )/(i\xi )^{n} = \widehat {I^{n} f}\). So, for n ≥ 3 :
So, we have for n ≥ 3
Hence,
We put for n ≥ 3
Then, for n ≥ 3, we get
Therefore, invoke
to deduce that
In view of (11) and (12), one has
and then (10) by letting 𝜖 → 0. Combining (9) and (10), we conclude
The proof is complete. □
Let \(1\leq p \leq \infty , f \in L_{p}^{*}({\mathbb T})\) and \(m \in \mathbb {N}\). Denote by T m the set of all trigonometric polynomials of degree < m, and
the error of approximation of f by elements from T m .
Theorem 3
Let \(1\leq p \leq \infty , f \in L_{p}^{*}({\mathbb T})\) and \(m \in \mathbb {N}\). Then,
Proof
Let \(\sigma \in \text {supp} \hat {f}, |\sigma | \geq m\). Then, for each 𝜖 ∈ (0,1) there exists a function \( \varphi \in C_{0}^{\infty }(\mathbb {R}), \text {supp}\varphi \subset (\sigma -\epsilon , \sigma +\epsilon )\) such that \( <\hat {f}, {\varphi } >\ne 0\). Put
Note that for all trigonometric polynomials \(P(x)={\sum }_{j=-(m-1)}^{m-1} a_{j} e^{i j x} \in T_{m}\) we have
Then, it follows from \( | <\hat {f}, {\varphi } > | = | <I^{n} f, \varphi _{n} >|\) that
Denote
Using (13) and Hölder inequality, one has
where 1/p + 1/q = 1. Since I n f(x) − P(x) is periodic, it follows that \(\|I^{n} f-P\|_{p,j}=\|I^{n} f -P\|_{p,{\mathbb T}}, \forall j \in \mathbb {Z}\). Then using (14), we get
for all trigonometric polynomials P(x) ∈ T m . Therefore,
This helps us to deduce
As in the proof of Theorem 2, we have
Hence, by (15), we obtain
So,
Moreover, from Theorem 2, we have
where
Note that Q(x) ∈ T m and
So,
Then, it follows from \(E_{m}(I^{n} f) \leq \|I^{n} (f -Q ) \|_{p,{\mathbb T}}\) and (17) that
Combining (16) and (18), we obtain
The proof is complete. □
Corollary 1
Let \(1\leq p \leq \infty , f \in L_{p}^{*}({\mathbb T})\) and \(m \in \mathbb {N}\). Then,
3 Local Spectral Formula for Polynomial Integral Operators
Let \(P(x)={\sum }_{k=0}^{m} a_{k} x^{k}\) be a polynomial of degree m. The integral operator P(I) is obtained from P(x) by substituting x →−i I: \(P(I)f= {\sum }_{k=0}^{m} a_{k} (-i)^{k} I^{k} f\).
Arguing similarly, we obtain the following local spectral formula for polynomial integral operators.
Theorem 4
Let \(1\leq p\leq \infty \) and \(f\in L^{*}_{p}({\mathbb T})\). Then, we have the following estimate
Moreover, if \(\text {supp}\hat {f}\) is a compact set or P(0) = 0, then
Theorem 5
Let \(1\leq p \leq \infty , f \in L_{p}^{*}({\mathbb T})\) , \(m \in \mathbb {N}\) and \(\text {supp}\hat {f}\) be compact or P(0) = 0. Then,
where \(E_m (f) =\inf _{P\in T_m} {\|f- P\|_{p,{\mathbb T}}}\) is the error of approximation of f by elements from T m and T m is the set of all trigonometric polynomials of degree < m.
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Acknowledgements
The research of V. N. Huy is funded by the Vietnam National University, Hanoi (VNU) under project number QG.16.08. A part of this work was done when V. N. Huy is working at the Vietnam Institute for Advanced Study in Mathematics (VIASM); the author would like to thank the VIASM for providing a fruitful research environment and working condition. The authors would like to thank the referees for useful remarks and comments.
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Bang, H.H., Huy, V.N. Local Spectral Formula for Integral Operators on \(L_{p}({\mathbb T})\) . Vietnam J. Math. 45, 737–746 (2017). https://doi.org/10.1007/s10013-017-0242-2
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DOI: https://doi.org/10.1007/s10013-017-0242-2