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Localization of the window functions of dual and tight Gabor frames generated by the Gaussian function E. A. Kiselev, L. A. Minin, I. Ya. Novikov, S. N. Ushakov Voronezh State University, Voronezh, Russia
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     § 1. IntroductionThe family of functions
$$
\begin{equation}
g (x)=\exp\biggl(-\frac{(x-a)^2} {2} \biggr)e^{i b x}, \qquad a, b \in \mathbb{R},
\end{equation}
\tag{1.1}
$$
was introduced in quantum mechanics by Schrödinger in 1926. The case of $a$ and $b$ defined on a rectilinear lattice was considered first by von Neumann in 1929 (see the appendix to [1]) (see the appendix to [1]). He conjectured that, for real positive parameters $v_{1}$ and $v_{2}$ such that $ v_{1}v_{2}=2\pi$, the system of functions
$$
\begin{equation}
g_{k, m}(x, v_1, v_2)=\exp\biggl(-\frac{(x-v_{1}k)^{2}}{2}\biggr)e^{i v_{2}mx}, \qquad k, m \in \mathbb{Z},
\end{equation}
\tag{1.2}
$$
is complete in $L_{2}(\mathbb{R})$, and, in addition, the Gram–Schmidt orthogonalization procedure converts this system to an orthonormal basis consisting of well localized basis functions. In the case when $ v_{1} v_{2}=2\pi$ (see the appendix to [1]) the system of functions (1.2) was shown to be complete in 1971 [2], [3], and in 1975 it was verified that no expansion in this system is stable (see [4]). One method for stability provision is based on a transition to overcomplete systems (frames). In 1946 Gabor [5] proposed to use the system of functions (1.2) for digital signal processing. After Glauber’s paper [6] (1963) functions of the form (1.1) and (1.2) are commonly called coherent states. In what follows, in place of $v_1$ and $v_2$ we use two sets of parameters:
$$
\begin{equation*}
\omega_{1}, \omega_{2} >0, \quad \omega_1 \omega_2 > 2\pi, \quad\text{and}\quad \alpha_{1}, \alpha_{2}>0, \quad \alpha_1 \alpha_2 < 2\pi.
\end{equation*}
\notag
$$
According to [7], Ch. 3, § 3.4, [8] and [9], the overcomplete system of functions
$$
\begin{equation}
g_{k, m} (x,\alpha_1,\alpha_2)=\exp \biggl (- \frac{(x-k\alpha_1)^2} {2} \biggr ) e^{i m \alpha_2 x}, \qquad k,m \in \mathbb{Z},
\end{equation}
\tag{1.3}
$$
forms a Gabor frame with window function $g(x)=\exp (-x^2/2)$. The window function of the dual frame is usually found either by expanding in a Neumann series (see [7], Ch. 3, § 3.2, and [10]) or by finite-dimensional reduction (see [11] and [12]). A detailed account of Gabor frames can be found in [13] and [14]. In the case when
$$
\begin{equation}
\alpha_{1} \alpha_{2}=\frac\pi n, \qquad n \in \mathbb{N},
\end{equation}
\tag{1.4}
$$
Janssen [15]–[18] obtained explicit formulae for the window functions of dual frames and tight frames. In the present paper we consider just this case. Time-frequency localization is an important characteristic for the description of frames and bases. The better an orthogonal basis is localized, the more efficient the coefficients of expansion in this basis can be evaluated and the synthesis of the original signal is realized. In the case of frames good localization is required not only for the original frame, but also for the dual frame, because this latter is used in the evaluation of the expansion coefficients. Time-frequency localization can be described using several approaches (see Ch. 2 in [19]). One of them is based on estimates for the rate of decay of the original function, its derivatives, and the Fourier transform. The second variant calls for the calculation of the numerical characteristic known as the uncertainty constant. The uncertainty constant attains its minimum at the coherent states (1.1). Other analogues of uncertainty constants are also available. Problems involving search of well-localized functions are subsumed into the class of window design problems. The first orthogonal basis with uniformly bounded uncertainty constant is the basis of shifts and dilations of a single function proposed by Meyer [20] in 1986. Subsequently, in 1988, based on coherent states, Bourgain [21] constructed a basis in which the uncertainty constant is arbitrarily close to the optimal value; however, the corresponding orthogonalization procedure does not preserve the structure of shifts and multiple frequencies. It is known that the uncertainty constant in wavelet bases is bounded below (see Ch. 1 in [22]). In this regard we mention works by Lebedeva and Protasov [23], [24] on the refinement of the time-frequency localization of the scaling function. The decay rate of Gabor frames, tight frames and Weyl–Heisenberg bases has also been studied extensively (see [25] and [26]). Time-frequency localization is described, for the most part, by estimating the function and its Fourier transform. Strohmer and Beaver (see [27] and [28]) studied the problem of an optimal orthogonal frequency division multiplexing (OFDM) basis, in particular, with the use of the uncertainty constant. In this problem values close to the minimum ones were obtained. In the present paper, from the available explicit formulae, we obtain explicit expressions for the uncertainty constants of window functions; we also estimate their localization in its dependence on the parameter $n$, which characterizes the degree of overcompleteness of a frame, and on the relation between the parameters $\alpha_1$ and $\alpha_2$ of the time-frequency window. Numerical computations were used to construct families of Gabor frames which, together with their dual frames, have near-optimal uncertainty constant. Our findings are as follows: the localization worsens rapidly as $\alpha_1/\alpha_2$ tends to zero or $\infty$. On the other hand, the higher the overdetermination of the system of functions forming the frame, the better the window functions are localized. This reflects the specifics of coherent states, which combine the properties of a continuous spectrum (they are eigenfunctions of the annihilation operator, whose eigenvalues fill the whole complex plane) and a discrete spectrum (all eigenfunctions are normalizable). For a tight frame the localization of the window function with the same set of parameters is much better than that for the dual frame. The problem under consideration is closely related to the problem of interpolation in uniform shifts of the Gaussian function. Both the nodal interpolation function (see [29] and Ch. 7 in [30]) and the window function of the dual frame (see [16]) are constructed (provided that (1.4) holds) from the same coefficients. These coefficients also play an important role in the derivation of formulae for the uncertainty constants (see [31]). So, in the present paper we study their properties related to sign alternation and the monotonicity of decrease of the absolute value. When dealing with tight frames we use the following fact: constructing a window function is equivalent to orthogonalizing the Riesz system dual to the frame, and this procedure was implemented in [32]. § 2. Notation and preliminariesWe consider the Hilbert space $L_{2}(\mathbb{R})$ of complex-valued functions with inner product $(f,g)$ and norm $\|f\|_{L_2}$:
$$
\begin{equation*}
(f,g)=\int_{-\infty}^{\infty} f(x) g^*(x) \,dx\quad\text{and} \quad \|f\|_{L_2}^2=\int_{-\infty}^{\infty} |f(x)|^2 \,dx,
\end{equation*}
\notag
$$
where $g^*(x)$ is complex conjugation. The Fourier transform of a function $g(x)$ is defined by
$$
\begin{equation*}
\widehat{g}(\xi)=\frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} g(x) e^{-i\xi x}\,dx;
\end{equation*}
\notag
$$
the Fourier transform is a unitary linear operator on $L_{2}(\mathbb{R})$. Applying the Fourier transform to the discrete system of coherent states (1.2) we obtain the system of coherent states
$$
\begin{equation*}
\widehat{g}_{k, m}(\xi,v_{1},v_{2})=\exp\biggl(-\frac{(\xi-v_{2}m)^{2}}{2}\biggr) e^{-i v_{1}k\xi} e^{i v_{1} v_{2}km}
\end{equation*}
\notag
$$
with phase factor $e^{i v_{1} v_{2}km}$. The last equality assumes the simplest form for ${v_{1} v_{2}=2\pi n}$, $n \in \mathbb{N}$:
$$
\begin{equation}
\widehat{g}_{k, m}(\xi,v_{1},v_{2})=g_{m,-k}(\xi,v_{2},v_{1}).
\end{equation}
\tag{2.1}
$$
The inner products of functions in this system can be written as
$$
\begin{equation*}
\begin{aligned} \, & (g_{k, m}(x,v_1, v_2), g_{k', m'} (x,v_1, v_2) )=\sqrt{\pi} \exp\biggl(-\frac{(k-k')^2 v_1^2}{4} \biggr) \\ &\qquad\qquad \times \exp\biggl(-\frac{(m-m')^2 v_2^2}{4} \biggr) \exp\biggl(\frac{i(k+k')(m-m') v_1 v_2}{2}\biggr). \end{aligned}
\end{equation*}
\notag
$$
If $v_1 v_2=4 \pi n$, $n \in \mathbb{N}$, then the factors with indices $k$, $k'$ and $m$, $m'$ are separated in the inner product:
$$
\begin{equation}
\begin{aligned} \, \notag &(g_{k, m}(x,v_1, v_2), g_{k', m'} (x,v_1, v_2) ) \\ &\qquad =\sqrt{\pi} \exp\biggl(-\frac{(k-k')^2 v_1^2}{4} \biggr) \exp\biggl(-\frac{(m-m')^2 v_2^2}{4} \biggr) . \end{aligned}
\end{equation}
\tag{2.2}
$$
Definition 1 (see [22], Ch. 1). The functions $\varphi_{k, m}(x) \in L_{2}(\mathbb{R})$, $k, m \in \mathbb{Z}$, form a Riesz system if there exist positive constants $A_R$ and $B_R$ such that, for any sequence of coefficients $c=\{c_{k, m}\} \in \ell_{2}$, The corresponding norm $\|c\|_{\ell_2}$ is defined by The largest $A_R$ is called the lower Riesz constant, and the smallest $B_R$ is called the upper Riesz constant. Analogues of the Riesz constants in the case of a finite set of functions are the minimum and maximum eigenvalues of the Gram matrix; their ratio $B_R/A_R$, which is equal to the condition number of the Gram matrix, measures the stability of expansion with respect to this system of functions Consider the closure of the set of linear combinations of functions in a Riesz system. To find the orthogonal projection of a function in $L_{2}(\mathbb{R})$ onto the resulting subspace one must orthogonalize the original system or construct a biorthogonal system. Definition 2 (see [22], Ch. 1). The functions $\varphi_{k, m}(x), \psi_{k, m}(x) \in L_{2}(\mathbb{R})$, where $k, m \in \mathbb{Z}$, form a biorthogonal system if where $\delta_{kk'}$ and $\delta_{m m'}$ are the Kronecker deltas. For overcomplete sets of functions (which cannot form a Riesz system because of linear dependence) one usually uses frames. Definition 3 (see [7], Ch. 3, and [14], Ch. 1). Functions $g_{k, m}(x) \in L_{2}(\mathbb{R})$, ${k,m \in \mathbb{Z}}$, form a frame if there exist positive constants $A_{F}$ and $B_{F}$ such that, for all ${f \,{\in}\, L_{2}(\mathbb{R})}$, The largest possible $A_{F}$ is called the lower bound of the frame, and the least possible $B_{F}$ is the upper bound of the frame. If $A_{F}=B_{F}$, then the frame is a tight frame. Gabor frames obey the duality principle, which relates them to Riesz systems (see [12]). Theorem 1. A family of functions $g_{k, m}(x, \alpha_1, \alpha_2)$, $k, m \in \mathbb{Z}$, is a frame in $L_2(\mathbb{R})$ if and only if the functions $g_{k, m}(x, \omega_1, \omega_2)$, $k, m \in \mathbb{Z}$, with parameters $\omega_{1}=2\pi /\alpha_2$, $\omega_2=2\pi/\alpha_1$ form a Riesz system in this space. In addition, the bounds $A_F$ and $B_F$ of the frame and the corresponding Riesz constants $A_R$ and $B_R$ are related by There are several variants of the duality principle (see [11], [12] and [15]); here we present the one most convenient for the purposes of our paper. An application of Theorem 1 to a tight frame produces the dual frame to a Riesz system of pairwise orthogonal functions. An important characteristic of a function capable of assessing its localization quality in both the time and frequency domain is the uncertainty constant. Let $g(x), x g(x) \in L_2(\mathbb{R})$, $\| g \|_{L_2} \neq 0$. The mean value $\langle g \rangle$ and radius $\Delta(g)$ of the function $g$ are defined by
$$
\begin{equation*}
\langle g \rangle=\frac{1}{\|g\|_{L_2}^2} \int_{-\infty}^{\infty} x |g(x)|^2 \,dx
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\Delta(g)=\biggl (\frac{1}{\|g\|_{L_2}^2} \int_{-\infty}^{\infty} (x-\langle g \rangle)^2 |g(x)|^2 \,dx \biggr )^{1/2}.
\end{equation*}
\notag
$$
One defines similarly the mean value $\langle \widehat{g} \rangle$ and radius $\Delta(\widehat{g})$ of the Fourier transform $\widehat{g}$ in the case when $\xi \widehat{g}(\xi) \in L_2(\mathbb{R})$. Namely.
$$
\begin{equation*}
\langle \widehat{g} \rangle=\frac{1}{\|\widehat{g}\|_{L_2}^2} \int_{-\infty}^{\infty} \xi |\widehat{g}(\xi)|^2 \,d\xi
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\Delta(\widehat{g})=\biggl (\frac{1}{\|\widehat{g}\|_{L_2}^2} \int_{-\infty}^{\infty}(\xi-\langle \widehat{g} \rangle)^2 |\widehat{g}(\xi)|^2\, d\xi \biggr )^{1/2}.
\end{equation*}
\notag
$$
We will use the Jacobi theta function of the third kind (see Ch. 21 in [34]), for which we need its representation as a series
$$
\begin{equation*}
\vartheta_3(x,q)=\sum_{k=-\infty}^\infty q^{k^2} e^{2ikx}, \qquad |q|<1,
\end{equation*}
\notag
$$
and as a Jacobi product
$$
\begin{equation*}
\vartheta_{3}(x,q)=\prod_{k=1}^\infty(1-q^{2k})(1+2q^{2k-1}\cos{2x}+q^{4k-2}).
\end{equation*}
\notag
$$
The last equality implies that $\vartheta_{3}(x,q)>0$ for all $ x \in \mathbb{R}$, $q \in (-1,1)$. The theta function can be also defined on the complex plane, but here we consider only the real argument. We let $c_{k} (\omega)$ denote the coefficients of the Fourier series
$$
\begin{equation}
\sum_{k=-\infty}^{\infty} c_{k}(\omega) e^{ikx} =\frac{1}{\vartheta_{3}(x/2, \exp (-\omega^2/4))}.
\end{equation}
\tag{2.4}
$$
According to Maz’ya and Schmidt (see [29] and [30], Ch. 7, § 7.3, Lemma 7.8),
$$
\begin{equation}
c_k (\omega)=\frac{1}{K(\omega)} \exp \biggl(\frac{k^{2} \omega^2 }{4 } \biggr) \sum_{r=|k|}^{\infty}(-1)^{r} \exp \biggl(-\frac{(r+0.5)^{2} \omega^{2}}{4} \biggr),
\end{equation}
\tag{2.5}
$$
where
$$
\begin{equation*}
K(\omega)=\sum_{r=-\infty}^{\infty}(4r+1)\exp \biggl(-\frac{(2r+0.5)^{2} \omega^{2}}{4 } \biggr).
\end{equation*}
\notag
$$
These coefficients are related to the problem of interpolation in the system of integer shifts of the Gaussian function, because (2.4) is equivalent to
$$
\begin{equation}
\sum_{k=-\infty}^{\infty} c_{k}(\omega) \exp\biggl(-\frac{\omega^2 (m-k)^2}{4}\biggr)= \delta_{m 0} , \qquad m \in \mathbb{Z}.
\end{equation}
\tag{2.6}
$$
From (2.5) it follows that
$$
\begin{equation}
c_k (\omega) \in \mathbb{R}\quad\text{and} \quad c_k (\omega)=c_{-k} (\omega) \quad \forall k \in \mathbb{N}.
\end{equation}
\tag{2.7}
$$
Note that formula (2.5) for the coefficients of the Fourier series (2.4) was essentially proved by Whittaker in 1903 (see [34], Ch. 21, Example 14). In addition, the coefficients $ c_{k}(\omega)$ can be found numerically using the discrete Fourier transform. For an account of this procedure, see [35]. According to [11] and [13], in the case when $\alpha_1 \alpha_2 <2 \pi$ the Gabor frame (1.3) corresponds to the dual Riesz system
$$
\begin{equation}
\varphi_{k,m}(x,\omega_1,\omega_2)=\exp\biggl(-\frac{(x-\omega_{1}k)^{2}}{2}\biggr) e^{i\omega_{2}mx}, \qquad k, m \in \mathbb{Z},
\end{equation}
\tag{2.8}
$$
with parameters
$$
\begin{equation}
\omega_1=\frac{2 \pi} {\alpha_2}\quad\text{and} \quad \omega_2=\frac{2 \pi} {\alpha_1},\qquad \omega_1 \omega_2 >2 \pi.
\end{equation}
\tag{2.9}
$$
According to Janssen [16], under condition (1.4) (which corresponds to $\omega_1 \omega_2=4\pi n$) the window function of the frame dual to (1.3) is
$$
\begin{equation}
\widetilde {g} (x, \omega_1, \omega_2 )=\frac{1} {2n \sqrt {\pi}} \sum_{k,m=-\infty}^{\infty} c_{k} (\omega_1) c_{m} (\omega_2) g_{k, m} (x, \omega_1, \omega_2).
\end{equation}
\tag{2.10}
$$
Correspondingly, $\widetilde {\varphi} (x,\omega_1, \omega_2)=2n \widetilde {g} (x, \omega_1, \omega_2 )$ is the window function of the system biorthogonal to (2.8). We give a sketch of the proof, which reveals the link with the interpolation problem. From (2.2) it follows that
$$
\begin{equation*}
\begin{aligned} \, &(\widetilde {\varphi} (x,\omega_1, \omega_2), \varphi_{k', m'} (x, \omega_{1}, \omega_{2}) ) \\ &\qquad =\sum_{k, m} c_k (\omega_1) c_m (\omega_2) \exp\biggl(-\frac{(k-k')^2 \omega_1^2}{4} \biggr) \exp\biggl(-\frac{(m-m')^2 \omega_2^2}{4} \biggr) \\ &\qquad =\sum_{k} c_k (\omega_1) \exp\biggl(-\frac{(k-k')^2 \omega_1^2}{4} \biggr) \sum_{m} c_m (\omega_2) \exp\biggl(-\frac{(m-m')^2 \omega_2^2}{4} \biggr). \end{aligned}
\end{equation*}
\notag
$$
To complete the proof it remains to note that
$$
\begin{equation*}
(\widetilde {\varphi}(x,\omega_1, \omega_2),g_{k', m'} (x, \omega_{1}, \omega_{2}) )= \delta_{k' 0}\delta_{m' 0}
\end{equation*}
\notag
$$
by (2.6). In numerical practice it is more convenient to represent the window function of the dual frame in the form
$$
\begin{equation}
\widetilde{g} (x, \omega_1, \omega_2)=\frac{\sum_{k=-\infty}^{\infty} c_{k} (\omega_1) \exp (- (x-k\omega_1)^2/2 )} {2 n \sqrt {\pi}\, \vartheta_{3}(\omega_2 x/2, \exp(-\omega_2^2/4)) },
\end{equation}
\tag{2.11}
$$
where the parameters $\alpha_1$ and $\alpha_2$ of the frame are related to $\omega_1$ and $\omega_2$ by (2.9). Applying the orthogonalization procedure (which was was proposed in [32] for $\omega_1 \omega_2=4\pi n$, $n \in \mathbb{N}$) to the Riesz system (2.8) we find that a tight Gabor frame is the dual frame to the resulting orthonormal system. Let us briefly describe this procedure. We let $c_{k}^{\bot}(\omega)$ denote the coefficients of the Fourier series
$$
\begin{equation}
\sum_{k=-\infty}^{\infty} c_{k}^{\bot}(\omega) e^{ikx}=\frac{1}{\sqrt{\vartheta_{3}(x/2,\exp(-\omega^{2}/4)})}.
\end{equation}
\tag{2.12}
$$
Consider the function
$$
\begin{equation}
g^{\bot} (x,\omega_{1},\omega_{2})=\frac{\sum_{k=-\infty}^{\infty}c_{k}^{\bot} (\omega_{1})\exp\bigl(-(x-\omega_{1}k)^{2}/2\bigr)} {\sqrt {\sqrt{\pi}\, \vartheta_{3}(\omega_{2}x/2,\exp{(-\omega_{2}^{2}/4)})}}.
\end{equation}
\tag{2.13}
$$
The system of functions
$$
\begin{equation*}
\varphi_{k,m}^{\bot}(x) =g^{\bot}(x-\omega_{1}k) e^{i\omega_{2}mx}, \qquad k, m \in \mathbb{Z},
\end{equation*}
\notag
$$
is orthonormal in $L_{2}(\mathbb{R})$. Correspondingly, $g^{\bot} (x,\omega_{1}, \omega_{2})$ is the window function of the tight Gabor frame with parameters $\alpha_1=2\pi/\omega_{2}$ and $\alpha_2=2\pi/\omega_{1}$. The above relation is slightly simpler and convenient for calculations than the one in [17].
§ 3. The resultsFirst we establish some properties of the coefficients $ c_{k}(\omega)$ defined by (2.4). Recall that $c_k (\omega)=c_{-k} (\omega)$ by (2.7). Theorem 2. Let $\omega >0$ and $q=\exp (-\omega^2/4)$. Then and Corollary 1. If $k \geqslant 0$ satisfies then the sequence $|c_k (\omega)|$ is strictly monotone decreasing. Corollary 3. A sufficient condition for $|c_k (\omega)|$ to decrease monotonically starting from $k=0$ is that $q <(\sqrt{5}-1)/2$, which corresponds to $ \omega > 1.38739 \dots$ . Note that, with the normalization of the Fourier transform used in our paper, the minimum value of the uncertainty constant is $1/2$, and this minimum is attained precisely at functions of the form (1.1). Theorem 3. The uncertainty constant for the window function (2.10) of the dual frame is given by Theorem 4. The squared uncertainty constant for the window function (2.13) of a tight frame is given by § 4. The proofs4.1. Proof of Theorem 2We need the following result involving the Jacobi theta function of the first kind (see [34], Ch. 21)
$$
\begin{equation*}
\vartheta_1(x,q)=2\sum_{k=0}^\infty (-1)^k q^{(k+1/2)^2} \sin((2k+1)x), \qquad |q|<1.
\end{equation*}
\notag
$$
Proof. We transform the series $K(\omega)$ as follows:
$$
\begin{equation*}
\begin{aligned} \, K(\omega) &=\sum_{r=-\infty}^{\infty} (4r+1) \exp \biggl(-\frac{(4r+1)^{2} \omega^{2}}{16} \biggr) \\ &=\sum_{r=0}^{\infty} (4r+1) \exp \biggl(-\frac{(4r+1)^{2} \omega^{2}}{16} \biggr) - \sum_{r=1 }^{\infty} (4r-1) \exp \biggl(-\frac{(4r-1)^{2} \omega^{2}}{16} \biggr). \end{aligned}
\end{equation*}
\notag
$$
The preexponential factors in the (first) series $\sum_{r=-\infty}^{\infty}$ are odd integers which leave a remainder of 1 after division by 4; for the second series, the preexponential factors leave a remainder of 3 after division by 4. We can combine them into a single series with respect to all odd numbers:
$$
\begin{equation}
K (\omega )=\sum_{m=0}^{\infty} (-1)^{m} (2m+1) \exp \biggl(-\frac{(2m+1)^{2} \omega^{2}}{16} \biggr).
\end{equation}
\tag{4.1}
$$
We have
$$
\begin{equation*}
\lim_{m \to \infty} \sqrt[m]{(2m+1) \exp \biggl(-\frac{(2 m+1)^{2} \omega^{2}}{16} \biggr)}= \lim_{m \to \infty} \exp \biggl(-\frac{(2m+1)^{2} \omega^{2}}{16 m} \biggr)=0,
\end{equation*}
\notag
$$
and by the Cauchy test this series converges absolutely for all $\omega > 0$. It remains to note that the series (4.1) is obtained by differentiating $\frac{1}{2}\vartheta_1(x,\exp(-\omega^2/4))$ with respect to $x$ at $x=0$. This proves Lemma 1. According to [34], Ch. 21,
$$
\begin{equation*}
\vartheta'_1 (0, q)=2q^{1/4} G^3, \qquad G=\prod_{n=1}^{\infty}(1-q^{2n}).
\end{equation*}
\notag
$$
Hence
$$
\begin{equation*}
K (\omega)=\exp \biggl(-\frac{\omega^2}{16}\biggr) \prod_{n=1}^{\infty} \biggl\{1-\exp \biggl(-\frac{n \omega^2 }{2}\biggr)\biggr\}^3.
\end{equation*}
\notag
$$
It follows that $K (\omega)>0$, because $0<\exp(-\omega^2/2)<1$ for $\omega \neq 0$. Hence the sign of the coefficients $c_k (\omega)$ depends on the sign of the sum of the series
$$
\begin{equation*}
\sum_{r=|k|}^{\infty}(-1)^{r} \exp \biggl(-\frac{(r+0.5)^{2} \omega^2 }{4} \biggr).
\end{equation*}
\notag
$$
The sign of the sum of this series is equal to that of its second term, because this series is alternating and since the absolute values of its terms are monotone decreasing. Hence which proves (3.1). For a nonnegative integer $k$ consider the series
$$
\begin{equation*}
X_{k}=\biggl| \sum_{r=k}^{\infty}(-1)^{r} q^{(r+0.5)^{2}} \biggr|, \quad\text{where } q=\exp \biggl(-\frac{\omega^2 }{4}\biggr).
\end{equation*}
\notag
$$
Then we have
$$
\begin{equation*}
\frac{|c_{k}(\omega)|}{|c_{k+1}(\omega)|}=\frac{q^{-k^2} X_{k}}{q^{-(k+1)^2} X_{k+1}} =q^{2k+1} \frac{q^{(k+1/2)^2} - X_{k+1} }{X_{k+1}}=q^{2k+1} \biggl(\frac{q^{(k+1/2)^2} }{X_{k+1}}-1\biggr).
\end{equation*}
\notag
$$
Note that $ X_{k+1} \leqslant q^{(k+3/2)^2}$. Hence
$$
\begin{equation*}
q^{2k+1} \biggl(\frac{q^{(k+1/2)^2} }{X_{k+1}}-1\biggr) \geqslant q^{2k+1} \biggl(\frac{q^{(k+1/2)^2} }{q^{(k+3/2)^2}}-1\biggr)= q^{-1}-q^{2k+1},
\end{equation*}
\notag
$$
proving Theorem 2. 4.2. Proof of Corollaries 1–3To prove Corollary 1 we find $q$ and $\omega$ such that $q^{-1}-q^{2k+1} > 1$. Then we have
$$
\begin{equation*}
q^{2k+1} < q^{-1}-1, \qquad 2k+1 > \log_q (q^{-1}-1).
\end{equation*}
\notag
$$
Now $ 2k > \log_q (1-q) -2$, from which (3.3) is immediate. Corollary 2 is clear in view of estimate (3.2). Let us proceed with Corollary 3. From (3.3) we find that the sequence $|c_k (\omega)|$ decreases for all $k \geqslant 0$ if Since $q > 0$, we have Changing to $\omega$ and solving the resulting inequality we find that
$$
\begin{equation*}
\exp \biggl (\frac{\omega^2} {4} \biggr ) > \frac{\sqrt{5}+1}{2},
\end{equation*}
\notag
$$
which proves Corollary 3. 4.3. Proof of Theorem 3The formula for the uncertainty constant $u(\widetilde {g})$ is based on the following four lemmas. Proof. By (2.7) the function $\widetilde {g} (x,\omega_1, \omega_2)$ is even with respect to $x$. This implies (4.2). Equality (4.3) follows from the fact that by (2.1) and (2.7) the Fourier transform of our function agrees with the function itself with swapped parameters:
This proves Lemma 2. We set
$$
\begin{equation}
a_{\ell}(\omega)=\sum_{k'=-\infty}^{\infty} c_{\ell+k'}(\omega) c_{k'}(\omega),
\end{equation}
\tag{4.4}
$$
$$
\begin{equation}
A(\omega)=\sum_{\ell=-\infty}^{\infty} \exp\biggl(-\frac{{\ell}^2 \omega^2}{4}\biggr) a_{\ell}(\omega),
\end{equation}
\tag{4.5}
$$
$$
\begin{equation}
D (\omega)=\sum_{\ell=-\infty}^{\infty} \ell^2 \exp\biggl(-\frac{{\ell}^2 \omega^2}{4}\biggr) a_{\ell}(\omega) \nonumber
\end{equation}
\notag
$$
and
$$
\begin{equation}
b_{\ell} (\omega)=\sum_{k'=-\infty}^{\infty} k' c_{l+k'}(w) c_{k'}(w).
\end{equation}
\tag{4.6}
$$
Proof. We use (2.7). Changing to $n=\ell+k'$ in (4.4) we have
$$
\begin{equation*}
a_{\ell}(\omega)=\sum_{k'} c_{\ell+k'}(\omega) c_{k'}(\omega)=\sum_{n} c_{n}(\omega) c_{n-\ell}(\omega).
\end{equation*}
\notag
$$
Replacing $n$ by $k'$ we obtain
$$
\begin{equation}
a_{\ell}(\omega)=\sum_{k'} c_{k'-\ell}(\omega) c_{k'}(\omega)=a_{-\ell}(\omega).
\end{equation}
\tag{4.7}
$$
Now we turn to the second equality:
$$
\begin{equation*}
b_{\ell}(\omega)=\sum_{k'} k' c_{\ell+k'}(\omega) c_{k'}(\omega)=\sum_{k'} k' c_{\ell+k'}(\omega) c_{-k'}(\omega).
\end{equation*}
\notag
$$
Changing to $k=-k'$ we find that
$$
\begin{equation*}
b_{\ell}(\omega)=- \sum_{k} k c_{\ell-k}(\omega) c_{k}(\omega)=- \sum_{k} k c_{-\ell+k}(\omega) c_{k}(\omega)=- b_{-\ell}(\omega).
\end{equation*}
\notag
$$
Next, replacing $\ell+k'$ by $n$ in (4.6), this establishes the relation
$$
\begin{equation*}
\begin{aligned} \, b_{\ell}(\omega) &=\sum_{n} (n-\ell) c_{n}(\omega) c_{n-\ell}(\omega)=\sum_{n} n c_{n}(\omega) c_{n-\ell}(\omega) - \ell \sum_{n} c_{n}(\omega) c_{n-\ell}(\omega) \\ &=b_{-\ell}(\omega) - \ell a_{-\ell}(\omega)=b_{-\ell}(\omega) - \ell a_{\ell}(\omega). \end{aligned}
\end{equation*}
\notag
$$
Now (4.7) follows from the equalities
This proves the lemma. Let us write the norm of $\widetilde {g} (x,\omega_1, \omega_2) $ using the above series. Proof. Squaring the absolute value of the function in question we obtain
$$
\begin{equation*}
\begin{aligned} \, & | \widetilde {g} (x,\omega_1, \omega_2)|^2=\frac{1}{4\pi n^{2}} \\ &\qquad\qquad \times \sum_{k, m, k', m'} c_{k}(\omega_1) c_{k'}(\omega_1) {c}_{m}(\omega_2) {c}_{m'}(\omega_2) g_{k, m} (x,\omega_1, \omega_2) g^*_{k', m'} (x,\omega_1, \omega_2). \end{aligned}
\end{equation*}
\notag
$$
Next, by (2.2)
$$
\begin{equation*}
\begin{aligned} \, \| \widetilde {g} (x,\omega_1, \omega_2)\|_{L_2}^2 &=\frac{\sqrt{\pi}}{4\pi n^{2}} \sum_{k, k'} c_{k}(\omega_1) c_{k'}(\omega_1) \exp\biggl(-\frac{(k -k')^2 \omega_1^2}{4} \biggr) \\ &\qquad \times \sum_{m, m'} c_{m}(\omega_2) c_{m'}(\omega_2) \exp\biggl(-\frac{(m-m')^2 \omega_2^2}{4}\biggr). \end{aligned}
\end{equation*}
\notag
$$
Changing to $\ell=k -k'$, $k=\ell +k'$, and using (4.4), this gives
$$
\begin{equation*}
\sum_{k, k'} c_{k}(\omega_1) c_{k'}(\omega_1) \exp\biggl(-\frac{(k -k')^2 \omega_1^2}{4}\biggr)= \sum_{\ell} \exp\biggl(-\frac{{\ell}^2 \omega_1^2}{4}\biggr) a_{\ell}(\omega_{1}).
\end{equation*}
\notag
$$
Hence The other series is transformed similarly: Now the lemma follows from the above formulae because the Fourier transform is unitary. Lemma 5. The following formula holds: Proof. Proceeding as in the proof of Lemma 4 and taking the relation $\omega_{1} \omega_2=4 \pi n$, $n \in \mathbb{Z}$, into account we evaluate the auxiliary integrals first. We have
$$
\begin{equation*}
\begin{aligned} \, &(x^2 g_{k, m} (x,\omega_1, \omega_2), g_{k', m'} (x,\omega_1, \omega_2) ) \\ &\qquad = \frac{\sqrt{\pi}}{4} \exp\biggl(-\frac{(k-k')^2 \omega_1^2}{4} \biggr) \exp\biggl(-\frac{(m-m')^2 \omega_2^2}{4} \biggr) \\ &\qquad\qquad \times \exp \biggl(\frac{i(k+k')(m-m') \omega_1\omega_2}{2}\biggr) \bigl(2-((m-m') \omega_2 - i (k+k') \omega_1)^2\bigr) \\ &\qquad =\frac{\sqrt{\pi}}{4} \exp\biggl(-\frac{(k-k')^2 \omega_1^2}{4} \biggr) \exp\biggl(-\frac{ (m-m')^2 \omega_2^2}{4} \biggr) \\ &\qquad\qquad \times \bigl(2-((m-m') \omega_2 - i (k+k') \omega_1)^2\bigr). \end{aligned}
\end{equation*}
\notag
$$
The last expression is simplified as follows:
$$
\begin{equation*}
\begin{aligned} \, &((m-m') \omega_2 - i (k+k') \omega_1 )^2 \\ &\qquad =(m-m')^2 \omega_2^2 - 8 i (m-m') (k+k') \pi n - (k-k')^2 \omega_1^2 - 4 k k' \omega_1^{2}. \end{aligned}
\end{equation*}
\notag
$$
Using (4.5) we find that
$$
\begin{equation}
\nonumber (x^2 \widetilde {g} (x,\omega_1, \omega_2), \widetilde {g} (x,\omega_1,\omega_2)) =\frac{1}{4\pi n^{2}}\biggl[\frac{\sqrt{\pi}}{2} A(\omega_1)A(\omega_2)
\end{equation}
\notag
$$
$$
\begin{equation}
\nonumber \qquad\qquad -\frac{\sqrt{\pi}\, \omega_2^2}{4} A(\omega_1) \sum_{m, m'} c_{m}(\omega_2) c_{m'}(\omega_2) (m-m')^2 \exp\biggl(-\frac{(m-m')^2\omega_2^2}{4} \biggr)
\end{equation}
\notag
$$
$$
\begin{equation}
\qquad\qquad + \frac{\sqrt{\pi}\, \omega_1^2}{4} A(\omega_2) \sum_{k, k'} c_{k}(\omega_1) c_{k'}(\omega_1) (k-k')^2 \exp\biggl(-\frac{(k-k')^2\omega_1^2}{4} \biggr)
\end{equation}
\tag{4.9}
$$
$$
\begin{equation}
\nonumber \qquad\qquad + i 2 \sqrt{\pi}\, \pi n \sum_{k, k'} c_{k}(\omega_1) c_{k'}(\omega_1) (k+k') \exp\biggl(-\frac{(k-k')^2 \omega_1^2}{4} \biggr)
\end{equation}
\notag
$$
$$
\begin{equation}
\qquad\qquad\qquad \times \sum_{m, m'} c_{m}(\omega_2) c_{m'}(\omega_2) (m-m') \exp\biggl(-\frac{(m-m')^2 \omega_2^2}{4} \biggr)
\end{equation}
\tag{4.10}
$$
$$
\begin{equation}
\qquad\qquad + \sqrt{\pi}\, \omega_1^2 A(\omega_2) \sum_{k, k'} c_{k}(\omega_1) c_{k'}(\omega_1) k k' \exp\biggl(-\frac{(k-k')^2 \omega_1^2}{4}\biggr)\biggr].
\end{equation}
\tag{4.11}
$$
Changing to $\ell=k -k'$, $k=\ell +k'$, series (4.9) is transformed as Proceeding similarly, we have
In view of (4.8) the product of the series in (4.10) is zero. Changing to $\ell=k -k'$, $k= \ell +k'$, the series (4.11) is transformed as follows:
$$
\begin{equation*}
\begin{aligned} \, &\sum_{k, k'} c_{k}(\omega_1) c_{k'}(\omega_1) k k' \exp\biggl(-\frac{(k-k')^2 \omega_1^2}{4}\biggr) \\ &\qquad =\sum_{\ell} \exp\biggl(-\frac{\ell^2 \omega_1^2}{4}\biggr) \sum_{k'} (\ell+k') k' c_{\ell+k'}(\omega_1) c_{k'}(\omega_1) \\ &\qquad =\sum_{\ell} \ell \exp\biggl(-\frac{\ell^2 \omega_1^2}{4}\biggr) b_{\ell}(\omega_1) + \sum_{\ell} \exp\biggl(-\frac{\ell^2 \omega_1^2}{4}\biggr) \sum_{k'} {k'}^2 c_{\ell+k'}(\omega_1) c_{k'}(\omega_1). \end{aligned}
\end{equation*}
\notag
$$
Changing the order of summation and using the properties of the coefficients $c_k (\omega)$ in (2.6) we can calculate the last series. We have
$$
\begin{equation*}
\begin{aligned} \, &\sum_{\ell} \exp\biggl(-\frac{\ell^2 \omega_1^2}{4}\biggr) \sum_{k'} {k'}^2 c_{\ell+k'}(\omega_1) c_{k'}(\omega_1) \\ &\qquad =\sum_{k'} {k'}^2 c_{k'}(\omega_1) \sum_{\ell} c_{\ell+k'}(\omega_1) \exp\biggl(-\frac{ \ell^2 \omega_1^2}{4}\biggr)=0. \end{aligned}
\end{equation*}
\notag
$$
The remanding terms in (4.11) are simplified using (4.7) as follow:
$$
\begin{equation*}
\sum_{\ell} \ell \exp\biggl(-\frac{\ell^2 \omega_1^2}{4}\biggr) b_{\ell}(\omega_1)= -\frac{1}{2} D(\omega_1).
\end{equation*}
\notag
$$
So we have
$$
\begin{equation*}
\sum_{k, k'} c_{k}(\omega_1) c_{k'}(\omega_1) k k' \exp\biggl(-\frac{(k -k')^2 \omega_1^2}{4} \biggr)=-\frac{1}{2} D(\omega_1).
\end{equation*}
\notag
$$
As a result,
$$
\begin{equation*}
\begin{aligned} \, & (x^2 \widetilde {g} (x,\omega_1, \omega_2), \widetilde {g} (x,\omega_1, \omega_2) ) \\ &\qquad = \frac{1}{4\pi n^{2}}\biggl(\frac{\sqrt{\pi}}{2} A(\omega_1) A(\omega_2) - \frac{\sqrt{\pi}\, \omega_2^2}{4} A(\omega_1) D(\omega_2) - \frac{\sqrt{\pi}\, \omega_1^2}{4} A(\omega_2) D(\omega_1)\biggr). \end{aligned}
\end{equation*}
\notag
$$
Lemma 5 is proved. From Lemmas 2, 4 and 5 we obtain the following formula for the uncertainty constant:
$$
\begin{equation*}
u({\widetilde {g}} (x,\omega_1, \omega_2))=\frac{1}{2} - \frac{\omega_2^2}{4} \frac{ D(\omega_2)}{A(\omega_2)} - \frac{\omega_1^2}{4} \frac{D(\omega_1)}{A(\omega_1)}.
\end{equation*}
\notag
$$
In addition, the radii for the function ${\widetilde {g}} (x,\omega_1, \omega_2)$ and for its Fourier transform are equal. 4.4. Proof of Theorem 4For the coefficients $c_k (\omega)$, we have the explicit formula (2.5); however, there is no such formula for $c_k^{\bot} (\omega)$ as far as we know. So let us use a general theorem on the properties of the Fourier coefficients of analytic functions (see [36], Ch. 1): the function
$$
\begin{equation*}
\frac{1}{\sqrt{\vartheta_{3}(x/2,\exp{(-\omega^{2}/4)})}}
\end{equation*}
\notag
$$
is analytic on the interval $[-\pi,\pi]$, and therefore its Fourier coefficients $c_{k}^{ \bot}(\omega)$ satisfy for some constants $M>0$ and $0<r<1$. This estimate implies that each series involved in the proof is absolutely convergent. The proof of the formulae in Theorem 4 is more involved than the proof in Theorem 3, because no formula (2.6) is available for the coefficients $c_{k}^{\bot}(\omega)$ (unlike the $c_{k} (\omega)$). The remaining transformations in the proofs of these two theorems are completely analogous, and we omit them.
§ 5. DiscussionA distinctive feature of the present paper is that the uncertainty constants of the window functions of the dual and tight frames are found not by evaluating the functions themselves, but rather in terms of the coefficients $c_k (\omega)$ and $c_k^{\bot} (\omega)$. This is why we have placed emphasis on the properties of these coefficients. Table 1 presents the uncertainty constants (calculated with the help of the formulae from Theorem 2) for the window functions of dual frames. We see that the constants increase with increasing disproportion in the parameters $\alpha_1$ and $\alpha_2$ of the frame, while an increase of the parameter $n$, which characterizes the degree of overcompleteness of the frame, results in the substantial deceleration of the growth. In addition, numerical calculations show that the uncertainty constant of the dual frame tends to $1/2$ as $n\to \infty$. This can be explained by the fact that the dual frame tends to the original frame in the $L_2(\mathbb{R})$-norm for finer grid spacings (see [37], Corollary 3.6.12). Table 1.The uncertainty constants for the window function of the dual frame
Let us consider in more detail the coefficients involved in the construction of a tight frame. For a fairly wide range of the parameters $\omega$ (from $0.4$ to $10$) and $k$ ($|k| \leqslant 200$) numerical calculations produce the following results:
Computations were performed to within $10^{-17}$. It seemed impractical to extend the range of $\omega$ because, for example, for $\omega=0.4$, the coefficients are too large:
$$
\begin{equation*}
c_0(\omega)=4.469 \cdot 10^{23}\quad\text{and} \quad c_0^{\bot} (\omega)=1.276 \cdot 10^{11},
\end{equation*}
\notag
$$
while for $\omega=10$ the coefficients decay very rapidly:
$$
\begin{equation*}
c_0(\omega)=c_0^{\bot} (\omega)=1, \qquad c_1(\omega)=-1.389 \cdot 10^{-11}\quad\text{and} \quad c_1^{\bot} (\omega)=-6.944 \cdot 10^{-12}.
\end{equation*}
\notag
$$
The above values are correct to within roundoff. Our conjecture is that the $c_k^{\bot} (\omega)$ should have the same properties for all relations between the parameters. Table 2 shows the uncertainty constants for the window functions of tight frames for the parameters as in Table 1. A comparison of the uncertainty constants shows that the localization of window functions is much better in the case of a tight frame. For both frames the case of a square time-frequency window (that is, $\omega_1=\omega_{2}$) is optimal. Table 2.The uncertainty constants for the window function of a tight frame
If, for a fixed degree $n$ of overcompleteness of the frame, the ratio of the parameters of the window $\omega_2 / \omega_{1}$ tends to $\infty$, then by [38] the window function of the tight frame $g^{\bot} (x,\omega_{1},\omega_{2})$ tends to the sample function. 
Citation in format AMSBIB
\Bibitem{KisMinNov24} |
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