Abstract
The Poisson summation formula for Hardy spaces \(H^p\left( T_\Gamma \right) \) in tubes \(T_\Gamma \subset \mathbb {C}^n\) for \(p\in \left( 0,1\right] \) is obtained. Unlike the case of \(L^p\left( \mathbb {R}^n\right) \) spaces, the formula holds everywhere in \(T_\Gamma \) without any additional assumptions. To the best of our knowledge, the result is new even for the univariate case—Hardy spaces in the upper half-plane.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
The Poisson summation formula is one of the famous results of the Fourier analysis. The classical one (see, e.g., [19, Chapter VII, Section 2, Theorem 2.4]) says that if \(f\in L^1\left( \mathbb {R}^n\right) \), then the series \(\sum _{m\in \Lambda }f\left( x+m\right) \) converges in the norm of \(L^1\left( \mathbb {T}^n\right) \) and the resulting function in \(L^1\left( \mathbb {T}^n\right) \) has the Fourier expansion
Here, and throughout the article, \(\Lambda \) denotes the unit lattice, i.e., the additive group of points in \(\mathbb {R}^n\) having integral coordinates; \(\left( \cdot ,\cdot \right) \) is the inner product in \(\mathbb {R}^n\).
In other words, the Poisson summation formula states that two approaches to periodization of a function f give the same result. This immediately leads to the question when actually equality
holds. For example, if we have the inversion formula for f, that is:
with \(\left| f\left( x\right) \right| \le A\left( 1+\left| x\right| \right) ^{-n-\delta }\) and \(\left| \widehat{f}\left( t\right) \right| \le A\left( 1+\left| t\right| \right) ^{-n-\delta }\), \(\delta >0\), then (1) holds for any \(x\in \mathbb {R}^n\) (see [19, Chapter VII, Section 2, Corollary 2.6]). There are other results stating that (1) in the univariate case holds under relaxed requirements (see, e.g., [6, 10]). Note that having just \(f,\widehat{f}\in L^1\left( {\mathbb R}\right) \) is not enough. For example, Katznelson [11] proved that there exists a function f, such that \(f\in L^1\left( \mathbb {R}\right) \cap C_0\left( \mathbb {R}\right) \) and \(\widehat{f}\in L^1\left( \mathbb {R}\right) \), but \(f\left( 0\right) =1\), \(f\left( n\right) =0\), \(n\in \mathbb {Z}\!{\setminus }\!\left\{ 0\right\} \), \(\widehat{f}\left( n\right) =0\), \(n\in \mathbb {Z}\).
There are several generalizations of the Poisson summation formula. It has many interesting applications, such as periodization and sampling operators [1], uncertainty principle [6], problems of number theory (see, e.g., [14, 15]), and many applications in Physics, Signal Processing, Communications, Electromagnetics, etc. (see [8, 16], and references therein). In particular, the Poisson summation formula is a useful tool for dealing with various Dyadic Green Functions. There is even its generalization for manifolds with boundary [7]. New results, generalizations, and applications of the formula are still appearing (see, e.g., [3, 4, 9, 12]).
It is interesting that Riemann used the Poisson summation formula to prove one of his results on zeta-function [17]. We refer the reader to the work of Miller and Schmid [14] for an excellent review of related problems.
The Poisson summation formula itself has been obtained for several functional spaces. However, the results usually state (1) almost everywhere only.
For functions of bounded variation on \(\mathbb {R}\), the Poisson summation formula valid pointwise is obtained by Trigub [20, Lemma 2] (see also [21, Section 3.1.11]). For functions of bounded variation on \(\mathbb {R}^d\), a generalization of Trigub’s result was recently obtained by Liflyand and Stadtmüller [13]. It is interesting that the formula holds under mild restrictions, sharpness of which is also discussed in [13].
Recent work of Butzer et al. [2] is devoted to obtaining generalizations of several classical formulas and inequalities in various function spaces. The Poisson summation formula was among them. The idea is that the classical formulas and inequalities usually become invalid in not-so-perfect spaces. To get their analogues, the authors developed a sophisticated and promising technique of the so-called unified distance concept.
We claim that for Hardy spaces \(H^p\left( T_\Gamma \right) \), \(p\in \left( 0,1\right] \), the classical formula (1) holds everywhere. No additional assumptions are needed. Moreover, the periodization of f is analytic in \(T_\Gamma \). The exact statement is Theorem 1 below. This answers a question asked by Roald M. Trigub (Donetsk National University, Ukraine) several years ago. In his opinion, the inversion formula for the Fourier transform obtained in [22] should result in something like the Poisson summation formula.
2 Definitions and Main Results
Let us recall some notation related to Hardy spaces in tubes, which are a natural multivariate generalization of the upper half-plane in \(\mathbb {C}\).
Let B be an open set in \({\mathbb R}^n\), \(n\in {\mathbb N}\). Following [19, Chapter III], the tube with base B is
The Hardy space \(H^p\left( T_B\right) \), \(p\in \left( 0,\infty \right] \), consists of functions f holomorphic in \(T_B\), such that
is finite. We will also use the following notation:
This notation is convenient to use in both contexts: \(f_\delta (z)=f(z+i\delta )\), \(z\in {\mathbb C}^n\), and \(f(z)=f(x+iy)=f_y(x)\), \(x\in {\mathbb R}^n\). Thus, \(\Vert f\Vert _{H^p(T_B)}=\sup _{y\in B}\Vert f_y\Vert _p\), where \(\Vert \cdot \Vert _p\) is a standard norm (or pre-norm) in \(L^p({\mathbb R}^n)\).
Since the case of an arbitrary tube \(T_B\) is too complicated, the attention is usually restricted to the case, where B is chosen to be an open cone.
A non-empty open set \(\Gamma \subset {\mathbb R}^n\) is called an open cone if \(0\notin \Gamma \) and whenever \(x,y\in \Gamma \) and \(\alpha ,\beta >0\), the linear combination \(\alpha x+\beta y\in \Gamma \). In particular, \(\Gamma \) is a convex set. The closure of an open cone is called a closed cone.
For any open cone \(\Gamma \), the set
is closed. If \(\Gamma ^*\) has non-empty interior, then it is a closed cone, and \(\Gamma \) is called a regular cone. The closed cone \(\Gamma ^*\) is called the cone dual to \(\Gamma \).
Let us start with an easy but very useful observation, which follows from the proof of [19, Chapter III, Section 2, Lemma 2.12].
Lemma 1
([22, Lemma 1]) Let \(\Gamma \) be an open cone in \({\mathbb R}^n\), \(p\in \left( 0,\infty \right] \), and \(q\in \left[ p,\infty \right] \). If \(f\in H^p\left( T_\Gamma \right) \), then for any \(\delta \in \Gamma \), we have \(f_\delta \in H^q\left( T_\Gamma \right) \) and
where \(\Omega _m\) is the volume of the unit ball in \(R^m\), i.e., \(\Omega _m=\pi ^{m/2}/\Gamma \left( m/2+1\right) \), and \(D_{\delta ,\Gamma }=\mathop {\mathrm{dist}}\left( \delta ,{\mathbb R}^n\!{\setminus }\!\Gamma \right) \).
The Fourier transform of a function \(f\in L^1\left( {\mathbb R}^n\right) \) is defined by
It is extended to the case of a function from \(L^2\left( {\mathbb R}^n\right) \) in the usual way.
Lemma 1 and [22, Theorem1] justify the following definition of the Fourier transform of a function from \(H^p\left( T_\Gamma \right) \) and \(p\in \left( 0,1\right] \).
Definition 1
The Fourier transform of a function \(f\in H^p\left( T_\Gamma \right) \), \(p\in \left( 0,1\right] \), is defined by
It is easy to see that the right-hand side of (2) is independent of \(\delta \). Let us also note that for \(p=1\), our \(\widehat{f}\) coincides with the classical Fourier transform of the limit function \(F\left( x\right) :=\lim _{\zeta \rightarrow 0,\,\zeta \in \Gamma }f_\zeta \left( x\right) \), \(x\in {\mathbb R}^n\).
Furthermore, if \(f\in H^p\left( T_\Gamma \right) \) for some \(p\in \left( 0,1\right] \), then the following inversion formula holds true (see [22])
Therefore, for any \(p\in \left( 0,1\right] \), the space \(f\in H^p\left( T_\Gamma \right) \) contains non-zero functions if and only if the cone \(\Gamma \) is regular (in fact, this is true for \(p\in \left( 0,\infty \right) \), since \(f\in H^p\) implies \(\left( f\right) ^{\left[ p\right] +1}\in H^s\) with \(s=p/\left( \left[ p\right] +1\right) \in \left( 0,1\right] \), where \(\left[ p\right] \) denotes the integral part of p). This is why we investigate only the case of a regular cone.
The existence of the inversion formula (3) suggested that the Poisson summation formula should hold in \(H^p\left( T_\Gamma \right) \) and \(p\in \left( 0,1\right] \), without any additional assumptions. Indeed, the following result holds.
Theorem 1
Let \(\Gamma \) be a regular cone in \(\mathbb {R}^n\), \(n\in \mathbb {N}\), and let \(f\in H^p\left( T_\Gamma \right) \) for some \(p\in \left( 0,1\right] \). Then
Moreover, both the sides of (4) are holomorphic in \(T_\Gamma \).
Note that Theorem 1 holds not just because of the inversion formula for the Fourier transform. The most important part is that the Fourier transform defined by (2) has a well-controlled growth (see estimate (13) below), which implies absolute and uniform convergence of the series in the right-hand side of (4). This growth estimate and the inversion formula (3) make \(H^p\left( T_\Gamma \right) \) spaces so special when \(p\in \left( 0,1\right] \), and allows us to obtain various inequalities, convenient conditions for Fourier multipliers, etc. (see [18, 22, 23]).
3 Proof
To prove Theorem 1, we need the following lemma, which could be of independent interest.
Lemma 2
Let \(\Gamma \) be a regular cone in \(\mathbb {R}^n\), \(n\in \mathbb {N}\), and let \(f\in H^p\left( T_\Gamma \right) \) for some \(p\in \left( 0,1\right] \). Then, for any \(\delta \in \Gamma \), the series
converges absolutely and uniformly on \(x\in Q^n\), where \(Q^n=\left[ -\frac{1}{2},\frac{1}{2}\right) ^n\)—the fundamental cube. Moreover, for any \(x\in Q^n\), we have
where \(D_{\delta ,\Gamma }=\mathop {\mathrm{dist}}\left( \delta ,{\mathbb R}^n\!{\setminus }\!\Gamma \right) ,\Omega _m=\pi ^{m/2}/\Gamma (m/2+1)\)
Proof
Take an arbitrary \(\alpha \in \left( 0,1\right) \), such that the closure of the (n-dimensional) ball centered at \(\delta \) and of radius \(\alpha \), \(\overline{B_{\mathbb {R}^n}\left( \delta ,\alpha \right) }\subset \Gamma \). Since f is analytic in \(T_\Gamma \), \(\left| f\right| \) is subharmonic there. Therefore, taking arbitrary \(m=\left( m_1,\dots ,m_n\right) \in \Lambda \) and \(x\in Q^n\) and considering that the closure of the (2n-dimesional) ball \(\overline{B_{\mathbb {C}^n}\left( x+m+i\delta ,\alpha \right) }\subset T_\Gamma \), we arrive at
where \(\mu \) is the Lebesgue (volume) measure in \(\mathbb {C}^n\) treated as \(\mathbb {R}^{2n}\).
Since \(\alpha \in \left( 0,1\right) \), it is easy to see that \(B_{\mathbb {C}^n}\left( x+m+i\delta ,\alpha \right) \subset E_{m,\delta ,\alpha }\), where
Now, the last estimate implies
Furthermore,
Since
Note that the last inequality follows from Lemma 1.
Thus, from (6), (7), and (8), we easily conclude that any partial sum of the series \(\sum _{m\in \Lambda }\left| f\left( x+m+i\delta \right) \right| \) is bounded above by
Hence, the series \(\sum _{m\in \Lambda }f\left( x+m+i\delta \right) \) converges absolutely for any \(x\in Q^n\) and
Finally, if we pass to the limit as \(\alpha \rightarrow \min \left\{ D_{\delta ,\Gamma },1\right\} -\) (from the left), we get (5).
To prove that the series \(\sum _{m\in \Lambda }\left| f\left( x+m+i\delta \right) \right| \) converges uniformly on \(x\in Q^n\), let us estimate its remainder. For \(N\in \mathbb {N}\), from (6) and (7), we have
The last equality follows from the Tonelli’s theorem (with summation treated as integration with respect to the discrete measure).
Estimate (8) justifies the application of the Lebesgue dominated convergence theorem:
Taking \(\limsup _{N\rightarrow \infty }\) in (9) and considering (10), we conclude that
Thus, the series \(\sum _{m\in \Lambda }\left| f\left( x+m+i\delta \right) \right| \) converges uniformly on \(x\in Q^n\), whence the series \(\sum _{m\in \Lambda }f\left( x+m+i\delta \right) \) also converges uniformly on \(x\in Q^n\). \(\square \)
Proof of Theorem 1
In our standard notation, \(z=x+iy\), \(x\in \mathbb {R}^n\), \(y\in \Gamma \), and \(\widehat{f}\left( t\right) =\widehat{f_y}\left( t\right) e^{2\pi \left( y,t\right) }\). Therefore, (4) can be rewritten in the following form:
Considering the periodicity of both the sides of (11), it is sufficient to prove this formula for \(x\in Q^n\).
Let us fix an arbitrary \(y\in \Gamma \). As we already know, \(f_y\in L^1\left( \mathbb {R}^n\right) \). Hence, [19, Chapter VII, Section 2, Theorem 2.4] implies that the series \(\sum _{m\in \Lambda }f_y\left( x+m\right) \) converges in \(L^1\left( Q_n\right) =L^1\left( \mathbb {T}^n\right) \). Its sum has the Fourier series:
Since \(\widehat{f_y}\left( m\right) =e^{-2\pi \left( y,m\right) }\widehat{f}\left( m\right) \), \(\mathop {\mathrm{supp}}\widehat{f}\subset \Gamma ^*\), and
(see [22, Section 1] and [5, Lemma 4]), we obtain that for any \(x\in \mathbb {R}^n\), \(m\in \Gamma ^*\),
Since clearly
we conclude that the series (12) converges uniformly and absolutely on \(x\in Q^n\). Thus, the function \(\sum _{m\in \Lambda }\widehat{f_y}\left( m\right) e^{2\pi i\left( x,m\right) }\) is continuous on \(x\in Q^n\) (or \(\mathbb {T}^n\)).
Applying [19, Chapter VII, Section 1, Corollary 1.8] to \(F\left( x\right) =\sum _{m\in \Lambda }f_y\left( x+m\right) \), we obtain that
As we already noticed, the right-hand side of (14) is continuous on \(Q^n\). The left-hand side is continuous in view of Lemma 2. Hence, (14) holds for every \(x\in Q^n\), which completes the proof of (4).
Finally, using estimate (13) and the fact that \(\mathop {\mathrm{supp}}\widehat{f}\subset \Gamma ^*\), it is easy to see that the series in the right-hand side of (4) converges locally uniformly in \(T_\Gamma \). This implies that both the sides of (4) are holomorphic in \(T_\Gamma \) functions. \(\square \)
References
Benedetto, J.J., Zimmermann, G.: Sampling multipliers and the Poisson summation formula. Dedicated to the memory of Richard J. Duffin. J. Fourier Anal. Appl. 3(5), 505–523 (1997)
Butzer, P.L., Stens, R.L., Schmeisser, G.: Basic relations valid for the Bernstein spaces \(B^2_\sigma \) and their extensions to larger function spaces via a unified distance concept. Function spaces X, pp. 41–55, Banach Center Publications, vol. 102. Polish Academy of Sciences. Institute of Mathematics, Warsaw (2014)
Deshmukh, P.R., Gudadhe, A.S.: Poisson summation formula associated with the fractional Laplace transform. J. Sci. Arts 23(2), 151–158 (2013)
Faifman, D.: A characterization of Fourier transform by Poisson summation formula. C. R. Math. Acad. Sci. Paris 348(7–8), 407–410 (2010)
Fefferman, C., Stein, E.M.: \(H^p\) spaces of several variables. Acta Math. 129(3–4), 137–193 (1972)
Gröchenig, K.: An uncertainty principle related to the Poisson summation formula. Stud. Math. 121(1), 87–104 (1996)
Guillemin, V., Melrose, R.: The Poisson summation formula for manifolds with boundary. Adv. Math. 32(3), 204–232 (1979)
Hautot, A.: New applications of Poisson’s summation formula. J. Phys. A 8, 853–862 (1975)
Johnson-McDaniel, N.K.: A dimensionally continued Poisson summation formula. J. Fourier Anal. Appl. 18(2), 367–385 (2012)
Kahane, J.-P., Lemarié-Rieusset, P.-G.: Remarques sur la formule sommatoire de Poisson (French). [Remarks on the Poisson summation formula]. Stud. Math. 109(3), 303–316 (1994)
Katznelson, Y.: Une remarque concernant la formule de Poisson (French). Stud. Math. 29, 107–108 (1967)
Lev, N., Olevskii, A.: Quasicrystals and Poisson’s summation formula. Invent. Math. 200(2), 585–606 (2015)
Liflyand, E., Stadtmüller, U.: Poisson summation for functions of bounded variation on \(\mathbb{R}^d\). Acta Sci. Math. (Szeged) 80(3–4), 491–498 (2014)
Miller, S.D., Schmid, W.: Summation formulas, from Poisson and Voronoi to the present. In: Noncommutative Harmonic Analysis, pp. 419–440. Progress in Mathematics, vol. 220. Birkhäuser, Boston (2004)
Mordell, L.J.: Poisson’s summation formula in several variables and some applications to the theory of numbers. Math. Proc. Camb. Philos. Soc. 25(04), 412–420 (1929)
Panaretos, A.H., Anastassiu, H.T., Kaklamani, D.I.: A note on the Poisson summation formula and its application to electromagnetic problems involving cylindrical coordinates. Turk. J. Electr. Eng. 10(2), 377–384 (2002)
Riemann, B.: Über die Anzahl der Primzahlen unter einer gegebenen Grösse. Mon. Not. Berlin Akad. (Nov. 1859), pp. 671–680 (1859)
Soljanik, A.A.: On the order of approximation to functions of \(H^p\) (R) (\(0 < p \le 1\)) by certain means of Fourier integrals. Anal. Math. 12(1), 59–75 (1986)
Stein, E.M., Weiss, G.: Introduction to Fourier Analysis on Euclidean Spaces. Princeton University Press, Princeton (1971)
Trigub, R.M.: A generalization of the Euler–Maclaurin formula (Russian) Mat. Zamet. 61(2), 312–316 (1997). Translation in Mathematical Notes 61(1–2), 253–257 (1997)
Trigub, R.M., Bellinsky, E.S.: Fourier Analysis and Approximation of Functions. Kluwer Academic Publishers, Dordrecht (2004)
Tovstolis, A.V.: Fourier multipliers in Hardy spaces in tube domains over open cones and their applications. Methods Funct. Anal. Topol. 4(1), 68–89 (1998)
Tovstolis, A.V.: Fourier multipliers in Hardy spaces in tubes over open cones. Comput. Methods Funct. Theory 14(4), 681–719 (2014)
Acknowledgments
The authors thank the anonymous referees for their valuable suggestions helping to improve the article. In particular, they led to including references to several interesting results on Poisson summation formula we had missed. The authors also thank Dr. John Paul Ward (University of Central Florida) for useful interactions during the presentation of the proof at the Analysis seminar at the University of Central Florida.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Dmitry Khavinson.
Rights and permissions
About this article
Cite this article
Li, X., Tovstolis, A.V. Poisson Summation Formula in Hardy Spaces \(H^p(T_\Gamma )\), \(p\in (0,1]\) . Comput. Methods Funct. Theory 16, 689–697 (2016). https://doi.org/10.1007/s40315-016-0170-2
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40315-016-0170-2