Q stringlengths 4 3.96k | A stringlengths 1 3k | Result stringclasses 4
values |
|---|---|---|
Lemma 1.18. The periods \( {\omega }_{k} \) and the constants \( {\eta }_{k} \) are related by the following equation:\n\n\[ \n{\eta }_{1}{\omega }_{2} - {\eta }_{2}{\omega }_{1} = {2\pi }\mathrm{i}. \n\] | Proof. If we pull the center \( c = \left( {{\omega }_{1} + {\omega }_{2}}\right) /2 \) of the parallelogram spanned by \( {\omega }_{1} \) and \( {\omega }_{2} \) to the origin, the result is another parallelogram \( R = {R}_{\Lambda } \) with the following vertices:\n\n\[ \n- \frac{1}{2}\left( {{\omega }_{1} + {\omeg... | Yes |
Proposition 1.19. Each \( \sigma \) is an entire function whose zero set is exactly the lattice \( \Lambda = \left\{ {\omega }_{mn}\right\} \) . Furthermore, \( \sigma \) is odd and quasiperiodic in the following sense:\n\n\[ \sigma \left( {z + {\omega }_{k}}\right) = - {\mathrm{e}}^{{\eta }_{k}\left( {z + \left( {{\om... | Proof. It follows from a standard argument involving the Weierstrass product (see Sect. 1.1) that the infinite product in (1.14) converges to an entire function \( \sigma \) and the convergence is uniform and absolute on any compact subset of the complex plane. It is also clear that the zero set of \( \sigma \) is exac... | Yes |
Corollary 1.21. For any \( \alpha > 0 \), the Weierstrass function \( \sigma \) associated to \( {\Lambda }_{\alpha } \) has the following properties:\n\n(a) The function \( \left| {\sigma \left( z\right) }\right| {\mathrm{e}}^{-\frac{\alpha }{2}{\left| z\right| }^{2}} \) is doubly periodic with periods \( \sqrt{\pi /\... | Proof. Property (a) follows from the quasiperiodicity of \( \sigma \) ; see (1.15) and (1.16). Property (b) then follows from (a) and the fact that each point in \( {\Lambda }_{\alpha } \) is a simple zero of \( \sigma \) . | No |
Theorem 1.23. If \( \sigma \left( {\zeta, x}\right) \) and \( f\left( x\right) \) are regular enough, then we have\n\n\[ \sigma \left( {D, X}\right) f\left( x\right) = \frac{\alpha }{\pi }{\int }_{\mathbb{R}}{\int }_{\mathbb{R}}\sigma \left( {\zeta ,\frac{x + y}{2}}\right) {\mathrm{e}}^{{2\alpha }\mathrm{i}\left( {x - ... | Proof. The Fourier inversion formula\n\n\[ {\int }_{\mathbb{R}}{\int }_{\mathbb{R}}{\mathrm{e}}^{{2\pi }\mathrm{i}\left( {u - v}\right) \zeta }f\left( v\right) \mathrm{d}v\mathrm{\;d}\zeta = f\left( u\right) \]\n\ncan be expressed in the language of distributions as\n\n\[ {\int }_{\mathbb{R}}{\mathrm{e}}^{{2\pi }\mathr... | Yes |
Lemma 1.27. We have\n\n\[ \rho \left( {{p}_{1},{q}_{1}}\right) \rho \left( {{p}_{2},{q}_{2}}\right) = {\mathrm{e}}^{\alpha \mathrm{i}\left( {{p}_{1}{q}_{2} - {p}_{2}{q}_{1}}\right) }\rho \left( {{p}_{1} + {p}_{2},{q}_{1} + {q}_{2}}\right) \]\n\nfor all real numbers \( {p}_{1},{q}_{1},{p}_{2} \), and \( {q}_{2} \) . | Proof. This follows directly from the definition of \( \rho \left( {p, q}\right) \) in (1.21). Details are left to the reader. | No |
Lemma 1.28. We have\n\n\[ \rho \left( {{p}_{1},{q}_{1}}\right) \rho \left( {{p}_{2},{q}_{2}}\right) = {\mathrm{e}}^{{2\alpha }\mathrm{i}\left( {{p}_{1}{q}_{2} - {p}_{2}{q}_{1}}\right) }\rho \left( {{p}_{2},{q}_{2}}\right) \rho \left( {{p}_{1},{q}_{1}}\right) \]\n\nfor all real numbers \( {p}_{1},{q}_{1},{p}_{2} \), and... | Proof. This is a direct consequence of Lemma 1.27. | No |
Theorem 1.29. Suppose \( \alpha \) is any positive parameter and pseudodifferential operators are defined as in the previous section. For any real \( p \) and \( q \), the pseudodifferential operator \( {\mathrm{e}}^{{2\alpha }\mathrm{i}\left( {{pD} + {qX}}\right) } \) is a unitary operator on \( {L}^{2}\left( {\mathbb... | Proof. By (1.21), the action of each \( u\left( {z, t}\right) \) on \( {L}^{2}\left( {\mathbb{R},\mathrm{d}x}\right) \), where \( z \in \mathbb{C} \) and \( t \in \mathbb{R} \), is a unimodular constant times a certain translation of \( \mathbb{R} \) . Since any translation of \( \mathbb{R} \) is a unitary operator on ... | Yes |
Proposition 2.1. For any nonnegative integer \( n \), let\n\n\[ \n{e}_{n}\left( z\right) = \sqrt{\frac{{\alpha }^{n}}{n!}}{z}^{n}.\n\]\n\nThen the set \( \left\{ {e}_{n}\right\} \) is an orthonormal basis for \( {F}_{\alpha }^{2} \) . | Proof. A calculation with polar coordinates shows that \( \left\{ {e}_{n}\right\} \) is an orthonormal set. Given \( f \in {F}_{\alpha }^{2} \) and \( n \geq 0 \), we have\n\n\[ \n{\left\langle f,{e}_{n}\right\rangle }_{\alpha } = \mathop{\lim }\limits_{{R \rightarrow \infty }}{\int }_{\left| z\right| < R}f\left( z\rig... | Yes |
Proposition 2.2. The reproducing kernel of \( {F}_{\alpha }^{2} \) is given by\n\n\[ \n{K}_{\alpha }\left( {z, w}\right) = {\mathrm{e}}^{{\alpha z}\bar{w}},\;z, w \in \mathbb{C}. \n\] | Proof. For any \( f \in {F}_{\alpha }^{2} \), we have\n\n\[ \nf\left( 0\right) = {\left\langle f,{e}_{0}\right\rangle }_{\alpha } = {\int }_{\mathbb{C}}f\left( z\right) \mathrm{d}{\lambda }_{\alpha }\left( z\right) . \n\]\n\nFix any \( w \in \mathbb{C} \) and replace \( f\left( z\right) \) by \( f\left( {w - z}\right) ... | Yes |
Corollary 2.3. The orthogonal projection\n\n\[ \n{P}_{\alpha } : {L}^{2}\left( {\mathbb{C},\mathrm{d}{\lambda }_{\alpha }}\right) \rightarrow {F}_{\alpha }^{2} \]\n\n is an integral operator. More specifically,\n\n\[ \n{P}_{\alpha }f\left( z\right) = {\int }_{\mathbb{C}}{K}_{\alpha }\left( {z, w}\right) f\left( w\right... | Proof. Fix \( f \in {L}^{2}\left( {\mathbb{C},\mathrm{d}{\lambda }_{\alpha }}\right) \) and \( z \in \mathbb{C} \) . We have\n\n\[ \n{P}_{\alpha }f\left( z\right) = {\left\langle {P}_{\alpha }f,{K}_{z}\right\rangle }_{\alpha } = {\left\langle f,{P}_{\alpha }{K}_{z}\right\rangle }_{\alpha } = {\left\langle f,{K}_{z}\rig... | Yes |
Corollary 2.4. Suppose \( f \geq 0 \) or \( f \in {L}^{1}\left( {\mathbb{C},\mathrm{d}{\lambda }_{\alpha }}\right) \) . Then for any \( z \in \mathbb{C} \), we have\n\n\[ \n{\int }_{\mathbb{C}}f\left( {z \pm w}\right) \mathrm{d}{\lambda }_{\alpha }\left( w\right) = {\int }_{\mathbb{C}}f\left( w\right) {\left| {k}_{z}\l... | Proof. It is clear that\n\n\[ \n{\int }_{\mathbb{C}}f\left( {z \pm w}\right) \mathrm{d}{\lambda }_{\alpha }\left( w\right) = \frac{\alpha }{\pi }{\int }_{\mathbb{C}}f\left( {z \pm w}\right) {\mathrm{e}}^{-\alpha {\left| w\right| }^{2}}\mathrm{\;d}A\left( w\right)\n\]\n\n\[ \n= \frac{\alpha }{\pi }{\int }_{\mathbb{C}}f\... | Yes |
Corollary 2.5. Suppose \( \\alpha > 0 \) and \( \\beta \) is real. Then\n\n\[ \n{\\int }_{\\mathbb{C}}\\left| {\\mathrm{e}}^{{\\beta z}\\bar{a}}\\right| \\mathrm{d}{\\lambda }_{\\alpha }\\left( z\\right) = {\\mathrm{e}}^{{\\beta }^{2}{\\left| a\\right| }^{2}/{4\\alpha }}\n\]\n\nfor all \( a \\in \\mathbb{C} \) . | Proof. It follows from the definition of the reproducing kernel that\n\n\[ \n{K}_{\\alpha }\\left( {a, a}\\right) = {\\int }_{\\mathbb{C}}{\\left| {K}_{\\alpha }\\left( a, z\\right) \\right| }^{2}\\mathrm{\\;d}{\\lambda }_{\\alpha }\\left( z\\right) ,\\;a \\in \\mathbb{C}.\n\]\n\nReplacing \( a \) by \( {\\beta a}/\\le... | No |
Lemma 2.6. Suppose \( \alpha > 0,\zeta \in \mathbb{C} - \{ 0\} \), and \( 0 < p \leq \infty \) . Then the dilation operator \( f\left( z\right) \mapsto f\left( {\zeta z}\right) \) is an isometry from \( {L}_{\alpha }^{p} \) onto \( {L}_{{\left| \zeta \right| }^{2}\alpha }^{p} \), and it is an isometry from \( {F}_{\alp... | Proof. This follows from a simple change of variables. | No |
Proposition 2.9. Suppose \( 0 < p < \infty, f \in {F}_{\alpha }^{p} \), and \( {f}_{r}\left( z\right) = f\left( {rz}\right) \). Then:\n\n(a) \( {\begin{Vmatrix}{f}_{r} - f\end{Vmatrix}}_{p,\alpha } \rightarrow 0 \) as \( r \rightarrow {1}^{ - } \).\n\n(b) There is a sequence \( \left\{ {p}_{n}\right\} \) of polynomials... | Proof. Suppose \( \left\{ {g}_{n}\right\} \) and \( g \) are functions in \( {L}^{p}\left( {X,\mathrm{\;d}\mu }\right) \) such that\n\n\[ \n{g}_{n}\left( x\right) \rightarrow g\left( x\right) ,\;n \rightarrow \infty ,\n\]\n\nalmost everywhere. Then it is well known that\n\n\[ \n\mathop{\lim }\limits_{{n \rightarrow \in... | Yes |
Theorem 2.10. If \( 0 < p < q < \infty \), then \( {F}_{\alpha }^{p} \subset {F}_{\alpha }^{q} \), and the inclusion is proper and continuous. Moreover, \( {F}_{\alpha }^{p} \subset {f}_{\alpha }^{\infty } \), and the inclusion is proper and continuous. | Proof. For any entire function \( f \), we consider the integral\n\n\[ \parallel f{\parallel }_{q,\alpha }^{q} = \frac{q\alpha }{2\pi }{\int }_{\mathbb{C}}{\left| f\left( z\right) {\mathrm{e}}^{-\alpha {\left| z\right| }^{2}/2}\right| }^{q}\mathrm{\;d}A\left( z\right) .\n\]\nIt follows from the pointwise estimate in Co... | Yes |
For any positive parameters \( \alpha \) and \( \gamma \), the set of functions of the form\n\n\[ f\left( z\right) = \mathop{\sum }\limits_{{k = 1}}^{n}{c}_{k}{K}_{\gamma }\left( {z,{w}_{k}}\right) = \mathop{\sum }\limits_{{k = 1}}^{n}{c}_{k}{\mathrm{e}}^{{\gamma z}{\bar{w}}_{k}}, \]\n\nis dense in \( {F}_{\alpha }^{p}... | Proof. Since the points \( {w}_{k} \) are arbitrary, we may assume that \( \gamma = \alpha \) .\n\nThe result is obvious when \( p = 2 \) . In fact, if a function \( h \) in \( {F}_{\alpha }^{2} \) is orthogonal to each function \( f\left( z\right) = {K}_{\alpha }\left( {z, w}\right) \), then \( h\left( w\right) = 0 \)... | Yes |
Theorem 2.12. Let \( f \in {F}_{\alpha }^{p} \) with \( 0 < p \leq \infty \) . Then \( f \) is of order less than or equal to 2. When \( f \) is of order 2, it must be of type less than or equal to \( \alpha /2 \) . | Proof. By Corollary 2.8, there exists a positive constant \( C \) such that\n\n\[ \left| {f\left( z\right) }\right| \leq C{\mathrm{e}}^{\alpha {\left| z\right| }^{2}/2} \]\n\nfor all \( z \in \mathbb{C} \) . In particular, \( M\left( r\right) \leq C{\mathrm{e}}^{\alpha {r}^{2}/2} \) for all \( r > 0 \) . It follows tha... | Yes |
Lemma 2.13. Suppose \( 1 \leq p < \infty \) and \( 1/p + 1/q = 1 \) . If an integral operator\n\n\[ \n{Tf}\left( x\right) = {\int }_{X}H\left( {x, y}\right) f\left( y\right) \mathrm{d}\mu \left( y\right)\n\]\n\nis bounded on \( {L}^{p}\left( {X,\mathrm{\;d}\mu }\right) \), then its adjoint\n\n\[ \n{T}^{ * } : {L}^{q}\l... | Proof. This is a standard result in real analysis. See [113] for example. | No |
Lemma 2.14. Suppose \( H\left( {x, y}\right) \) is a positive kernel and\n\n\[ \n{Tf}\left( x\right) = {\int }_{X}H\left( {x, y}\right) f\left( y\right) \mathrm{d}\mu \left( y\right) \n\] \n\nis the associated integral operator. Let \( 1 < p < \infty \) with \( 1/p + 1/q = 1 \) . If there exist a positive function \( h... | Proof. See [250] for example. | No |
Lemma 2.16. Suppose \( 1 < p < \infty \) and \( {P}_{\alpha } \) is bounded on \( {L}^{p}\left( {\mathbb{C},\mathrm{d}{\lambda }_{\beta }}\right) \). Then \( {p\alpha } > \beta \) . | Proof. If \( p > 1 \) and \( {P}_{\alpha } \) is bounded on \( {L}^{p}\left( {\mathbb{C},\mathrm{d}{\lambda }_{\beta }}\right) \), then \( {P}_{\alpha }^{ * } \) is bounded on \( {L}^{q}\left( {\mathbb{C},\mathrm{d}{\lambda }_{\beta }}\right) \) , where \( 1/p + 1/q = 1 \) . Applying the formula for \( {P}_{\alpha }^{ ... | Yes |
Lemma 2.17. If \( {P}_{\alpha } \) is bounded on \( {L}^{1}\left( {\mathbb{C},\mathrm{d}{\lambda }_{\beta }}\right) \), then \( \alpha = {2\beta } \) . | Proof. Fix any \( a \in \mathbb{C} \) and consider the function\n\n\[ \n{f}_{a}\left( z\right) = \frac{{\mathrm{e}}^{{\alpha z}\bar{a}}}{\left| {\mathrm{e}}^{{\alpha z}\bar{a}}\right| },\;z \in \mathbb{C}.\n\]\n\nObviously, \( {\begin{Vmatrix}{f}_{a}\end{Vmatrix}}_{\infty } = 1 \) for every \( a \in \mathbb{C} \) . On ... | Yes |
Lemma 2.19. Suppose \( 2 < p < \infty \) and \( {P}_{\alpha } \) is bounded on \( {L}^{p}\left( {\mathbb{C},\mathrm{d}{\lambda }_{\beta }}\right) \). Then \( {p\alpha } = {2\beta } \). | Proof. If \( {P}_{\alpha } \) is a bounded operator on \( {L}^{p}\left( {\mathbb{C},\mathrm{d}{\lambda }_{\beta }}\right) \), then \( {P}_{\alpha }^{ * } \) is also bounded on \( {L}^{q}\left( {\mathbb{C},\mathrm{d}{\lambda }_{\beta }}\right) \), where \( 1 < q < 2 \) and \( 1/p + 1/q = 1 \). It follows from (2.5) that... | Yes |
Theorem 2.21. If \( 1 \leq p < \infty \) and \( {p\alpha } = {2\beta } \), then\n\n\[{\int }_{\mathbb{C}}{\left| {P}_{\alpha }f\right| }^{p}\mathrm{\;d}{\lambda }_{\beta } \leq {\int }_{\mathbb{C}}{\left| {Q}_{\alpha }f\right| }^{p}\mathrm{\;d}{\lambda }_{\beta } \leq {2}^{p}{\int }_{\mathbb{C}}{\left| f\right| }^{p}\m... | Proof. With the choice of \( \delta \) in (2.13), the constants in (2.9) and (2.12) both reduce to 2. Therefore, Schur’s test tells us that, in the case when \( 1 < p < \infty \), the norm of \( {Q}_{\alpha } \) on \( {L}^{p}\left( {\mathbb{C},\mathrm{d}{\lambda }_{\beta }}\right) \) does not exceed 2 .\n\nWhen \( p = ... | Yes |
For any \( \alpha > 0 \) and \( 1 \leq p \leq \infty \), the operator \( {P}_{\alpha } \) is a bounded projection from \( {L}_{\alpha }^{p} \) onto \( {F}_{\alpha }^{p} \) . Furthermore, \( {\begin{Vmatrix}{P}_{\alpha }f\end{Vmatrix}}_{p,\alpha } \leq 2\parallel f{\parallel }_{p,\alpha } \) for all \( f \in {L}_{\alpha... | The case \( 1 \leq p < \infty \) follows from Theorem 2.21. The case \( p = \infty \) follows from Corollary 2.5. | No |
Theorem 2.24. Suppose \( 0 < p \leq 1 \) and \( \beta > 0 \) . Then the dual space of \( {F}_{\alpha }^{p} \) can be identified with \( {F}_{\beta }^{\infty } \) under the integral pairing\n\n\[ \langle f, g{\rangle }_{\gamma } = \mathop{\lim }\limits_{{R \rightarrow \infty }}\frac{\gamma }{\pi }{\int }_{\left| z\right... | Proof. First, assume that \( g \in {F}_{\beta }^{\infty } \) and \( F \) is defined by \( F\left( f\right) = \langle f, g{\rangle }_{\gamma } \) . To show that \( F \) extends to a bounded linear functional on \( {F}_{\alpha }^{p} \), we use (2.16) to rewrite\n\n\[ F\left( f\right) = \frac{\alpha }{\pi }{\int }_{\mathb... | Yes |
Theorem 2.27. Suppose \( w,{w}_{0} \), and \( {w}_{1} \) are positive weight functions on the complex plane. If \( 1 \leq {p}_{0} \leq {p}_{1} \leq \infty \) and \( 0 \leq \theta \leq 1 \), then\n\n\[ \n{\left\lbrack {L}^{{p}_{0}}\left( \mathbb{C},{w}_{0}\mathrm{\;d}A\right) ,{L}^{{p}_{1}}\left( \mathbb{C},{w}_{1}\math... | This result is very useful and widely known. See [216] for a proof. | No |
Corollary 2.28. Suppose \( 1 \leq {p}_{0} \leq {p}_{1} \leq \infty \) and \( 0 \leq \theta \leq 1 \) . Then for any positive weight parameters \( {\alpha }_{0} \) and \( {\alpha }_{1} \), we have\n\n\[{\left\lbrack {L}_{{\alpha }_{0}}^{{p}_{0}},{L}_{{\alpha }_{1}}^{{p}_{1}}\right\rbrack }_{\theta } = {L}_{\alpha }^{p}\... | Proof. Since \( {L}_{\alpha }^{p} = {L}^{p}\left( {\mathbb{C},\mathrm{d}{\lambda }_{{p\alpha }/2}}\right) \), it follows from the Stein-Weiss interpolation theorem that\n\n\[{\left\lbrack {L}_{{\alpha }_{1}}^{{p}_{0}},{L}_{{\alpha }_{2}}^{{p}_{1}}\right\rbrack }_{\theta } = {\left\lbrack {L}^{{p}_{0}}\left( \mathbb{C},... | Yes |
Theorem 2.29. Suppose \( 1 \leq {p}_{0} \leq {p}_{1} \leq \infty \) and \( 0 \leq \theta \leq 1 \) . Then\n\n\[{\left\lbrack {F}_{\alpha }^{{p}_{0}},{F}_{\alpha }^{{p}_{1}}\right\rbrack }_{\theta } = {F}_{\alpha }^{p}\]\n\nwhere\n\n\[ \frac{1}{p} = \frac{1 - \theta }{{p}_{0}} + \frac{\theta }{{p}_{1}} \] | Proof. The inclusion\n\n\[{\left\lbrack {F}_{\alpha }^{{p}_{0}},{F}_{\alpha }^{{p}_{1}}\right\rbrack }_{\theta } \subset {F}_{\alpha }^{p}\]\n\nfollows from the definition of complex interpolation, the fact that each \( {F}_{\alpha }^{{p}_{k}} \) is a closed subspace of \( {L}_{\alpha }^{{p}_{k}} \), and the fact that ... | Yes |
Theorem 2.30. Suppose \( 1 \leq {p}_{0} \leq {p}_{1} \leq \infty \) and \( 0 \leq \theta \leq 1 \) . Then for any positive weight parameters \( {\alpha }_{0} \) and \( {\alpha }_{1} \), we have\n\n\[{\left\lbrack {F}_{{\alpha }_{0}}^{{p}_{0}},{F}_{{\alpha }_{1}}^{{p}_{1}}\right\rbrack }_{\theta } = {F}_{\alpha }^{p}\]\... | Proof. For any \( \zeta \in \mathbb{C} \), consider the dilation operator \( {S}_{\zeta } \) defined by\n\n\[{S}_{\zeta }f\left( z\right) = f\left( {{\left( \frac{{\alpha }_{0}}{{\alpha }_{1}}\right) }^{\left( {\zeta - \theta }\right) /2}z}\right) .\]\n\nAccording to Lemma 2.6, \( {S}_{\zeta } \) is an isometry from \(... | Yes |
Theorem 2.31. Suppose \( 1 \leq p \leq \infty \) . Then for any positive weight parameters \( \alpha ,\beta \) , and \( \gamma \), we have:\n\n(a) \( {P}_{\alpha }{L}_{\beta }^{p} \subset {F}_{\gamma }^{p} \) if and only if \( {\alpha }^{2}/\gamma \leq {2\alpha } - \beta \) .\n\n(b) \( {P}_{\alpha }{L}_{\beta }^{p} = {... | Proof. It is easy to see that a necessary condition for \( {P}_{\alpha }{L}_{\beta }^{p} \subset {F}_{\gamma }^{p},1 \leq p \leq \infty \), is that \( {2\alpha } > \beta \) . So for the rest of the proof, we always assume that \( {2\alpha } > \beta \) .\n\nIf \( {\alpha }^{2}/\gamma \leq {2\alpha } - \beta \), it follo... | Yes |
Lemma 2.32. For any positive parameters \( \alpha, p \), and \( R \), there exists a positive constant \( C = C\left( {p,\alpha, R}\right) \) such that\n\n\[{\left| f\left( a\right) {\mathrm{e}}^{-\alpha {\left| a\right| }^{2}/2}\right| }^{p} \leq \frac{C}{{r}^{2}}{\int }_{B\left( {a, r}\right) }{\left| f\left( z\right... | Proof. Let \( I \) denote the integral above. Then\n\n\[I = {\int }_{B\left( {a, r}\right) }{\left| f\left( z\right) \right| }^{p}{\mathrm{e}}^{-{p\alpha }{\left| z\right| }^{2}/2}\mathrm{\;d}A\left( z\right)\]\n\n\[= {\int }_{\left| w\right| < r}{\left| f\left( w + a\right) \right| }^{p}{\mathrm{e}}^{-{p\alpha }{\left... | Yes |
Lemma 2.33. Each \( {k}_{a} \) is also a unit vector in \( {F}_{\alpha }^{p} \), where \( 0 < p \leq \infty \) . | Proof. It follows from the definition of the norm in \( {F}_{\alpha }^{p} \) and the reproducing formula in \( {F}_{{p\alpha }/2}^{2} \) that\n\n\[ \n{\begin{Vmatrix}{k}_{a}\end{Vmatrix}}_{p,\alpha }^{p} = \frac{p\alpha }{2\pi }{\int }_{\mathbb{C}}{\left| {k}_{a}\left( z\right) {\mathrm{e}}^{-\frac{1}{2}\alpha {\left| ... | Yes |
Theorem 2.34. Let \( 0 < p \leq \infty \) . There exists a positive constant \( {r}_{0} \) such that for any \( 0 < r < {r}_{0} \), the space \( {F}_{\alpha }^{p} \) consists exactly of the following functions:\n\n\[ f\left( z\right) = \mathop{\sum }\limits_{{w \in r{\mathbb{Z}}^{2}}}{c}_{w}{k}_{w}\left( z\right) \]\n\... | Proof. If \( 0 < p \leq 1 \) and \( f \) is given by (2.17) with \( \left\{ {c}_{w}\right\} \in {l}^{p} \), then by Hölder’s inequality,\n\n\[ {\left| f\left( z\right) {\mathrm{e}}^{-\alpha {\left| z\right| }^{2}/2}\right| }^{p} \leq \mathop{\sum }\limits_{{w \in r{\mathbb{Z}}^{2}}}{\left| {c}_{w}\right| }^{p}{\left| {... | Yes |
Lemma 2.35. Suppose \( 0 < r < 1,0 < p \leq 1 \), and \( m \) is a nonnegative integer: For any entire function \( f \), we define a sequence\n\n\[ \left\{ {{\left( Sf\right) }_{w, k} : w \in r{\mathbb{Z}}^{2},0 \leq k \leq m}\right\} \]\n\nby\n\n\[ {\left( Sf\right) }_{w, k} = \frac{\alpha }{\pi }{\int }_{{S}_{r} + w}... | Proof. For any \( w \in r{\mathbb{Z}}^{2}, z \in {S}_{r} + w \), and \( 1 \leq k \leq m \), we have\n\n\[ {\left| {\left( Sf\right) }_{w, k}\right| }^{p} = \frac{{\alpha }^{p}}{{\pi }^{p}}\left| {\;{\int }_{{S}_{r} + w}{\mathrm{e}}^{\alpha \mathrm{i}\operatorname{Im}z\left( {\bar{z} - \bar{w}}\right) - \frac{\alpha }{2... | Yes |
Lemma 2.36. Suppose \( 0 < r < 1,0 < p \leq 1 \), and \( m \) is a nonnegative integer: For every sequence\n\n\[ c = \left\{ {{c}_{w, k} : w \in r{\mathbb{Z}}^{2},0 \leq k \leq m}\right\} \]\n\ndefine a function \( {Tc} \) by\n\n\[ {Tc}\left( z\right) = \mathop{\sum }\limits_{{w \in r{\mathbb{Z}}^{2}}}\mathop{\sum }\li... | Proof. It is obvious that the series converges to an entire function \( f\left( z\right) \) uniformly on compact subsets of \( \mathbb{C} \) . Since \( 0 < p < 1 \), it follows from Hölder’s inequality that\n\n\[ {\left| f\left( z\right) \right| }^{p} \leq \mathop{\sum }\limits_{{w \in r{\mathbb{Z}}^{2}}}\mathop{\sum }... | Yes |
Lemma 2.37. Let \( {r}_{0} \) be the number from Theorem 2.34 in the case \( p = \infty \) . Suppose \( 0 < r < {r}_{0} \) and \( 0 < p \leq 1 \) . Then every monomial \( {z}^{k} \) can be represented as\n\n\[ \n{z}^{k} = \mathop{\sum }\limits_{{w \in r{\mathbb{Z}}^{2}}}{c}_{w}{k}_{w}\left( z\right) \n\] \n\nwhere \( \... | Proof. Fix \( \rho \in \left( {r,{r}_{0}}\right) \) . By the already-proved case \( p = \infty \) of Theorem 2.34, every monomial \( {z}^{k} \) can be represented as\n\n\[ \n{z}^{k} = \mathop{\sum }\limits_{{w \in \rho {\mathbb{Z}}^{2}}}{c}_{w}{k}_{w}\left( z\right) \n\] \n\nwhere \( \left\{ {c}_{w}\right\} \in {l}^{\i... | Yes |
Proposition 2.38. Let \( 0 < p \leq \infty \) . We have\n\n\[{\begin{Vmatrix}{W}_{a}f\end{Vmatrix}}_{p,\alpha } = {\begin{Vmatrix}{U}_{a}f\end{Vmatrix}}_{p,\alpha } = \parallel f{\parallel }_{p,\alpha }\n\]\n\nfor all \( a \in \mathbb{C} \) and \( f \in {F}_{\alpha }^{p} \) . Furthermore, both \( {W}_{a} \) and \( {U}_... | Proof. It is easy to check that\n\n\[{\mathrm{e}}^{-\frac{\alpha }{2}{\left| z\right| }^{2}}\left| {{W}_{a}f\left( z\right) }\right| = {\mathrm{e}}^{-\frac{\alpha }{2}{\left| z - a\right| }^{2}}\left| {f\left( {z - a}\right) }\right|\]\n\nand\n\n\[{\mathrm{e}}^{-\frac{\alpha }{2}{\left| z\right| }^{2}}\left| {{U}_{a}f\... | Yes |
Theorem 2.39. The mapping \( \left( {a,\theta }\right) \mapsto {\mathrm{e}}^{\mathrm{i}\theta }{W}_{a} \) is a unitary representation of the Heisenberg group \( \mathbb{H} \) on the Fock space \( {F}_{\alpha }^{2} \) . | Proof. For any two points \( a \) and \( b \) in \( \mathbb{C} \), we easily check that\n\n\[ \n{W}_{a}{W}_{b} = {\mathrm{e}}^{-\alpha \operatorname{iIm}\left( {a\bar{b}}\right) }{W}_{a + b} = {\mathrm{e}}^{\alpha \operatorname{iIm}\left( {\bar{a}b}\right) }{W}_{a + b}.\n\]\n\n(2.22)\n\nThis shows that \( \left( {a,\th... | Yes |
Proposition 2.40. The Fock space \( {F}_{\alpha }^{\infty } \) is maximal in the sense that if \( X \) is any Banach space of entire functions with the following properties:\n\n(a) \( {\begin{Vmatrix}{W}_{a}f\end{Vmatrix}}_{X} = \parallel f{\parallel }_{X} \) for all \( a \in \mathbb{C} \) and \( f \in X \) ,\n\n(b) th... | Proof. Condition (a) implies that \( {W}_{a}f \in X \) for every \( f \in X \) and every \( a \in \mathbb{C} \) . Combining this with condition (b), we see that for every \( a \in \mathbb{C} \), the point evaluation \( f \mapsto f\left( a\right) \) is also a bounded linear functional on \( X \), and\n\n\[ \n{\mathrm{e}... | Yes |
Proposition 2.41. The Fock space \( {F}_{\alpha }^{1} \) is minimal in the sense that if \( X \) is a Banach space of entire functions with the following properties:\n\n(a) \( {\begin{Vmatrix}{W}_{a}f\end{Vmatrix}}_{X} = \parallel f{\parallel }_{X} \) for all \( a \in \mathbb{C} \) and \( f \in X \) ,\n\n(b) \( X \) co... | Proof. Since \( X \) contains all constant functions, applying \( {W}_{a} \) to the constant function 1 shows that for each \( a \in \mathbb{C} \), the function\n\n\[ \n{k}_{a}\left( z\right) = {\mathrm{e}}^{\alpha \bar{a}z - \frac{\alpha }{2}{\left| a\right| }^{2}} \n\]\n\nbelongs to \( X \) . Furthermore, \( {\begin{... | Yes |
Proposition 2.42. Suppose \( H \) is a nontrivial separable Hilbert space of entire functions with the following properties:\n\n(a) \( {\begin{Vmatrix}{W}_{a}f\end{Vmatrix}}_{H} = \parallel f{\parallel }_{H} \) for all \( a \in \mathbb{C} \) and \( f \in H \) .\n\n(b) \( f \mapsto f\left( 0\right) \) is a bounded linea... | Proof. Since \( H \) contains at least one function that is not identically zero, it follows from conditions (a) and (b) that for any \( z \in \mathbb{C} \), the mapping \( f \mapsto f\left( z\right) \) is a nonzero bounded linear functional on \( H \) . Furthermore, for any compact subset \( S \) of \( \mathbb{C} \), ... | Yes |
Proposition 3.1. Let \( L\left( {F}_{\alpha }^{2}\right) \) be the Banach space of all bounded linear operators on \( {F}_{\alpha }^{2} \) . Then \( T \mapsto \widetilde{T} \) is a bounded linear mapping from \( L\left( {F}_{\alpha }^{2}\right) \) into \( {L}^{\infty }\left( \mathbb{C}\right) \) . Furthermore, the mapp... | Proof. Everything is obvious except the one-to-one part. To see this, assume that \( T \) is a bounded linear operator on \( {F}_{\alpha }^{2} \) and that \( \left\langle {T{k}_{z},{k}_{z}}\right\rangle = 0 \) for all \( z \in \mathbb{C} \) . Then \( \left\langle {T{K}_{z},{K}_{z}}\right\rangle = 0 \) for all \( z \in ... | Yes |
Proposition 3.2. If \( T \) is compact on \( {F}_{\alpha }^{2} \), then \( \widetilde{T}\left( z\right) \rightarrow 0 \) as \( z \rightarrow \infty \) . | Proof. It is easy to see that \( {k}_{z} \rightarrow 0 \) weakly in \( {F}_{\alpha }^{2} \) as \( z \rightarrow \infty \) . This gives the desired result. | No |
Proposition 3.3. If \( S \) is a trace-class operator or a positive operator, then\n\n\[ \operatorname{tr}\left( S\right) = \frac{\alpha }{\pi }{\int }_{\mathbb{C}}\widetilde{S}\left( z\right) \mathrm{d}A\left( z\right) \]\n\n(3.2)\n\nFurthermore, a positive operator \( S \) belongs to the trace class if and only if th... | Proof. First, assume that \( S \) is positive, say \( S = {T}^{2} \) for some \( T \geq 0 \) . Then for any orthonormal basis \( \left\{ {e}_{n}\right\} \), it follows from Fubini’s theorem that\n\n\[ \operatorname{tr}\left( S\right) = \mathop{\sum }\limits_{{n = 1}}^{\infty }{\left\langle S{e}_{n},{e}_{n}\right\rangle... | Yes |
Lemma 3.4. Suppose \( T \) is a positive operator on a Hilbert space \( H \) and \( x \) is a unit vector in \( H \) . Then \( \left\langle {{T}^{p}x, x}\right\rangle \geq \langle {Tx}, x{\rangle }^{p} \) for \( p \geq 1 \) and \( \left\langle {{T}^{p}x, x}\right\rangle \leq \langle {Tx}, x{\rangle }^{p} \) for all \( ... | Proof. See Proposition 1.31 of [250]. | No |
Proposition 3.5. If \( p \geq 1 \) and \( T \) is in the Schatten class \( {S}_{p} \), then \( \widetilde{T} \) belongs to \( {L}^{p}\left( {\mathbb{C},\mathrm{d}A}\right) \) . | Proof. If \( T \) is in the trace class, then we can write\n\n\[ T = {T}_{1} - {T}_{2} + \mathrm{i}\left( {{T}_{3} - {T}_{4}}\right) \]\n\nwhere each \( {T}_{k} \) is a positive trace-class operator. By Proposition 3.3 above, the function\n\n\[ \widetilde{T} = {\widetilde{T}}_{1} - {\widetilde{T}}_{2} + \mathrm{i}{\wid... | Yes |
Proposition 3.6. Suppose \( 0 < p \leq 1 \) and \( T \) is a positive operator on \( {F}_{\alpha }^{2} \) . If \( \widetilde{T} \in \) \( {L}^{p}\left( {\mathbb{C},\mathrm{d}A}\right) \), then \( T \) belongs to the Schatten class \( {S}_{p} \) . | Proof. Since \( T \) is positive, it belongs to the Schatten class \( {S}_{p} \) if and only if \( {S}^{p} \) is in the trace class. The desired result then follows from Proposition 3.3 and Lemma 3.4. | No |
Corollary 3.8 Let \( T \) be any bounded linear operator on \( {F}_{\alpha }^{2} \) . Then\n\n\[ \left| {\widetilde{T}\left( z\right) - \widetilde{T}\left( w\right) }\right| \leq 2\sqrt{\alpha }\parallel T\parallel \left| {z - w}\right| \] \n\nfor all \( z \) and \( w \) in \( \mathbb{C} \) . | Proof. It is easy to see that\n\n\[ 1 - {\left| \left\langle {k}_{z},{k}_{w}\right\rangle \right| }^{2} = 1 - {\mathrm{e}}^{-\alpha {\left| z - w\right| }^{2}} \leq \alpha {\left| z - w\right| }^{2} \] \n\nfor all \( z \) and \( w \) . The desired Lispchitz estimate is then obvious. | No |
Proposition 3.9. The mapping \( S \mapsto {K}_{S} \) has the following properties:\n\n(1) \( {K}_{S + T} = {K}_{S} + {K}_{T},{K}_{cS} = c{K}_{S} \) .\n\n(2) \( {K}_{S}\left( {\cdot, z}\right) \in {F}_{\alpha }^{2} \) .\n\n(3) \( {K}_{{S}^{ * }}\left( {w, z}\right) = \overline{{K}_{S}\left( {z, w}\right) } \) .\n\n(4) \... | Proof. Properties (1)-(5) and (8) are direct consequences of the definition of \( {K}_{S} \) in (3.3) and the definition of the Berezin transform. Property (6) follows from (3.3) and the Cauchy-Schwarz inequality, and it implies property (7). Since the Berezin transform \( S \mapsto \widetilde{S} \) is one-to-one, we s... | Yes |
Proposition 3.10. Let \( S \) and \( T \) be bounded operators on \( {F}_{\alpha }^{2} \) . Then\n\n\[ \n{K}_{ST}\left( {w, z}\right) = {\int }_{\mathbb{C}}{K}_{S}\left( {u, z}\right) {K}_{T}\left( {w, u}\right) \mathrm{d}{\lambda }_{\alpha }\left( u\right) \n\]\n\nfor all \( w \) and \( z \) in \( \mathbb{C} \) . | Proof. It follows from (3.3) that\n\n\[ \n{K}_{ST}\left( {w, z}\right) = {\left\langle {T}^{ * }{S}^{ * }{K}_{z},{K}_{w}\right\rangle }_{\alpha } = {\left\langle {S}^{ * }{K}_{z}, T{K}_{w}\right\rangle }_{\alpha } \n\]\n\n\[ \n= {\int }_{\mathbb{C}}{S}^{ * }{K}_{z}\left( u\right) \overline{T{K}_{w}\left( u\right) }\mat... | Yes |
Proposition 3.11. If \( S \) is a positive or trace-class operator, then\n\n\[ \operatorname{tr}\left( S\right) = {\int }_{\mathbb{C}}\overline{{K}_{S}\left( {z, z}\right) }\mathrm{d}{\lambda }_{\alpha }\left( z\right) \] | Proof. This follows from Proposition 3.3 and property (8) in Proposition 3.9. | No |
Corollary 3.12. Let \( S \) and \( T \) be bounded linear operators on \( {F}_{\alpha }^{2} \) such that \( {ST} \) is trace class. Then\n\n\[ \n\operatorname{tr}\left( {ST}\right) = {\int }_{\mathbb{C}}\mathrm{d}{\lambda }_{\alpha }\left( w\right) {\int }_{\mathbb{C}}\overline{{K}_{S}\left( {z, w}\right) }\overline{{K... | Proof. This is a direct consequence of Propositions 3.10 and 3.11. | No |
Theorem 3.14. The function \( u\left( {x, y, t}\right) = {H}_{t}f\left( z\right) \), where \( z = x + \mathrm{i}y \), satisfies the heat equation\n\n\[ \frac{{\partial }^{2}u}{\partial {x}^{2}} + \frac{{\partial }^{2}u}{\partial {y}^{2}} = 4\frac{\partial u}{\partial t} \] | Proof. With \( z = x + \mathrm{i}y \) and \( w = u + \mathrm{i}v \), we have\n\n\[ u\left( {x, y, t}\right) = \frac{1}{\pi t}{\int }_{{\mathbb{R}}^{2}}f\left( {u, v}\right) {\mathrm{e}}^{-\frac{1}{t}\left\lbrack {{\left( x - u\right) }^{2} + {\left( y - v\right) }^{2}}\right\rbrack }\mathrm{d}u\mathrm{\;d}v. \]\n\nDiff... | Yes |
Lemma 3.16 Suppose that \( n \) is a positive integer and \( f \) is a function on \( {\mathbb{R}}^{n} \) such that the function\n\n\[ x \mapsto f\left( x\right) {\mathrm{e}}^{\left| tx\right| }{\mathrm{e}}^{-{x}^{2}} \]\n\nis integrable on \( {\mathbb{R}}^{n} \) with respect to Lebesgue measure \( \mathrm{d}x \) for a... | Proof. Since\n\n\[ {\mathrm{e}}^{\mathrm{i}{tx}} = \mathop{\sum }\limits_{{k = 0}}^{\infty }\frac{{\left( \mathrm{i}tx\right) }^{k}}{k!} \]\n\nand\n\n\[ \left| {\mathop{\sum }\limits_{{k = 0}}^{N}\frac{{\left( \mathrm{i}tx\right) }^{k}}{k!}}\right| \leq \mathop{\sum }\limits_{{k = 0}}^{\infty }\frac{{\left| tx\right| }... | Yes |
Proposition 3.17. The Berezin transform \( {B}_{\alpha } \) is linear and order preserving. Furthermore, if \( {B}_{\alpha }f = 0 \) and \( f \) satisfies the condition that\n\n\[ \n{\int }_{\mathbb{C}}\left| {f\left( z\right) }\right| {\mathrm{e}}^{\left| tz\right| }{\mathrm{e}}^{-\alpha {\left| z\right| }^{2}}\mathrm... | Proof. It is clear that each \( {B}_{\alpha } \) is linear and order preserving.\n\nIf \( {B}_{\alpha }f = 0 \) and \( f \) satisfies the integral condition (3.12), then differentiating under the integral sign gives\n\n\[ \n\frac{{\partial }^{n + m}}{\partial {z}^{n}\partial {\bar{z}}^{m}}{B}_{\alpha }f\left( 0\right) ... | Yes |
Proposition 3.20. Let \( \alpha > 0 \) and \( 1 \leq p < \infty \) . Then\n\n(a) \( {B}_{\alpha } : {L}^{\infty }\left( \mathbb{C}\right) \rightarrow {L}^{\infty }\left( \mathbb{C}\right) \) is a contraction.\n\n(b) \( {B}_{\alpha } : {C}_{0}\left( \mathbb{C}\right) \rightarrow {C}_{0}\left( \mathbb{C}\right) \) is a c... | Proof. Part (a) is obvious. If \( f \in {C}_{c}\left( \mathbb{C}\right) \), namely, if \( f \) is a continuous function on \( \mathbb{C} \) with compact support, then it is easy to see that \( {B}_{\alpha }f \in {C}_{0}\left( \mathbb{C}\right) \) . Thus, part (b) follows from (a) and the fact that \( {C}_{c}\left( \mat... | No |
Proposition 3.21. Let \( 0 < \beta < \alpha \) and \( 1 \leq p < \infty \) . Then\n\n(a) \( {B}_{\alpha }f \in {L}^{\infty }\left( \mathbb{C}\right) \) implies \( {B}_{\beta }f \in {L}^{\infty }\left( \mathbb{C}\right) \) with\n\n\[ \n{\begin{Vmatrix}{B}_{\beta }f\end{Vmatrix}}_{\infty } \leq {\begin{Vmatrix}{B}_{\alph... | Proof. Choose a positive \( \gamma \) such that \( 1/\gamma + 1/\alpha = 1/\beta \) . By Corollary 3.15, we have \( {B}_{\beta } = {B}_{\gamma }{B}_{\alpha } \) . The desired result then follows from Proposition 3.20. | Yes |
Proposition 3.22. If \( 0 < \beta < \alpha ,0 < p < \infty \), and \( f \geq 0 \) . Then\n\n\[ \n{B}_{\alpha }f\left( z\right) \leq \frac{\alpha }{\beta }{B}_{\beta }f\left( z\right) ,\;z \in \mathbb{C}.\n\]\n\nConsequently:\n\n(a) \( {B}_{\beta }f \in {L}^{\infty }\left( \mathbb{C}\right) \) implies that \( {B}_{\alph... | Proof. Since \( f \geq 0 \) and \( 0 < \beta < \alpha \), we have\n\n\[ \n{B}_{\alpha }f\left( z\right) = \frac{\alpha }{\pi }{\int }_{\mathbb{C}}f\left( w\right) {\mathrm{e}}^{-\alpha {\left| z - w\right| }^{2}}\mathrm{\;d}A\left( w\right)\n\]\n\n\[ \n\leq \frac{\alpha }{\pi }{\int }_{\mathbb{C}}f\left( w\right) {\mat... | Yes |
Theorem 3.23. Suppose \( \alpha \) and \( \beta \) are positive weight parameters and \( f \geq 0 \) on \( \mathbb{C} \) . For \( 0 < p \leq \infty \), we have\n\n(a) \( {B}_{\alpha }f \in {L}^{p}\left( {\mathbb{C},\mathrm{d}A}\right) \) if and only if \( {B}_{\beta }f \in {L}^{p}\left( {\mathbb{C},\mathrm{d}A}\right) ... | Proof. Part (a) in the case \( 1 \leq p \leq \infty \) and part (b) follow from Propositions 3.21 and 3.22. Part (a) in the case \( 0 < p < 1 \) will be proved in Chap. 6. | No |
Proposition 3.24. If \( f \) is a function such that the Berezin transform \( {B}_{\alpha }f \) is well defined, then for any \( a \in \mathbb{C} \), we have\n\n(i) \( {B}_{\alpha }\left( {f \circ {t}_{a}}\right) = \left( {{B}_{\alpha }f}\right) \circ {t}_{a} \).\n\n(ii) \( {B}_{\alpha }\left( {f \circ {\tau }_{a}}\rig... | Proof. By (3.7), we have\n\n\[ \widetilde{f \circ {t}_{a}}\left( z\right) = {\int }_{\mathbb{C}}f \circ {t}_{a}\left( {z + w}\right) \mathrm{d}{\lambda }_{\alpha }\left( w\right) \]\n\n\[ = {\int }_{\mathbb{C}}f\left( {a + z + w}\right) \mathrm{d}{\lambda }_{\alpha }\left( w\right) \]\n\n\[ = \widetilde{f}\left( {a + z... | Yes |
Proposition 3.26. Suppose \( f \) is a harmonic function on \( \mathbb{C} \) satisfying condition \( \left( {I}_{1}\right) \) . Then \( \widetilde{f} = f \) . | Proof. If \( f \) is harmonic, then \( f \circ {t}_{z} \) is harmonic for every \( z \) . It follows from the mean value theorem for harmonic functions that\n\n\[ f \circ {t}_{z}\left( 0\right) = {\int }_{\mathbb{C}}f \circ {t}_{z}\left( w\right) \mathrm{d}{\lambda }_{\alpha }\left( w\right) . \]\n\nThis shows that \( ... | Yes |
Proposition 3.27. If \( f \in {L}^{\infty }\left( \mathbb{C}\right) \), then the following conditions are equivalent:\n\n(a) \( \widetilde{f} = f \) .\n\n(b) \( f \) is harmonic.\n\n(c) \( f \) is constant. | Proof. Since \( f \) is bounded, the equivalence of (b) and (c) follows from the wellknown maximum modulus principle for harmonic functions. If \( f \) is constant, then clearly \( \widetilde{f} = f \) . If \( \widetilde{f} = f \), then \( {\widetilde{f}}^{\left( n\right) } = f \) for all positive integers \( n \) . By... | Yes |
For any complex \( \zeta \), let\n\n\[ I\left( \zeta \right) = \frac{1}{\sqrt{\pi }}{\int }_{-\infty }^{\infty }{\mathrm{e}}^{{\zeta t} - {t}^{2}}\mathrm{\;d}t \]\n\nWe have \( I\left( \zeta \right) = {\mathrm{e}}^{{\zeta }^{2}/4} \) . | Proof. It is clear that \( I\left( \zeta \right) \) is an entire function of \( \zeta \). Differentiating under the integral sign, we obtain\n\n\[ {I}^{\prime }\left( \zeta \right) = \frac{1}{\sqrt{\pi }}{\int }_{-\infty }^{\infty }t{\mathrm{e}}^{{\zeta t} - {t}^{2}}\mathrm{\;d}t \]\n\n\[ = \frac{1}{\sqrt{\pi }}{\int }... | Yes |
Theorem 3.29. Suppose \( \mu \) is a positive Borel measure on \( \mathbb{C},0 < p < \infty \), and \( 0 < \) \( r < \infty \) . Then the following conditions are equivalent:\n\n(a) There exists a positive constant \( C \) such that\n\n\[ \n{\int }_{\mathbb{C}}{\left| f\left( w\right) {\mathrm{e}}^{-\frac{\alpha }{2}{\... | Proof. Fix a positive radius \( r \) and consider the lattice \( r{\mathbb{Z}}^{2} \) in \( \mathbb{C} \) . Let \( \left\{ {z}_{n}\right\} \) denote any fixed arrangement of this lattice into a sequence. For any entire function \( f \), we set\n\n\[ \nI\left( f\right) = {\int }_{\mathbb{C}}{\left| f\left( w\right) {\ma... | Yes |
Theorem 3.30. Suppose \( p > 0,\alpha > 0, r > 0 \), and \( \mu \) is a positive Borel measure on \( \mathbb{C} \) . Then the following conditions are equivalent:\n\n(i) \( \mu \) is a vanishing Fock-Carleson measure.\n\n(ii) \( {\int }_{\mathbb{C}}{\mathrm{e}}^{-\frac{p\alpha }{2}{\left| z - w\right| }^{2}}\mathrm{\;d... | Proof. By the proof of Theorem 3.29, there exists a positive constant \( C \) (independent of \( z \) ) such that\n\n\[ \mu \left( {B\left( {z, r}\right) }\right) \leq C{\int }_{\mathbb{C}}{\mathrm{e}}^{-\frac{p\alpha }{2}{\left| z - w\right| }^{2}}\mathrm{\;d}\mu \left( w\right) \]\n\nfor all \( z \in \mathbb{C} \) . ... | Yes |
Corollary 3.31. Suppose \( \mu \) is a positive Borel measure on \( \mathbb{C}, r > 0 \), and \( \left\{ {z}_{n}\right\} \) is any arrangement into a sequence of the lattice \( r{\mathbb{Z}}^{2} \) . Then\n\n(a) \( \mu \) is a Fock-Carleson measure if and only if \( \left\{ {\mu \left( {B\left( {{z}_{k}, r}\right) }\ri... | Taking \( p = 2 \) in Theorems 3.29 and 3.30, we see that a positive Borel measure \( \mu \) on \( \mathbb{C} \) is a Fock-Carleson measure if and only if \( \widetilde{\mu } \in {L}^{\infty }\left( \mathbb{C}\right) \), and \( \mu \) is a vanishing Fock-Carleson measure if and only if \( \widetilde{\mu } \in {C}_{0}\l... | Yes |
Lemma 3.32. Let \( 1 \leq p < \infty, r > 0 \), and \( f \) be a locally area-integrable function on \( \mathbb{C} \). Then \( f \in {\mathrm{{BMO}}}_{r}^{p} \) if and only if there exists some \( C > 0 \) such that for any \( z \in \mathbb{C} \), there is a complex constant \( {c}_{z} \) with\n\n\[ \n\frac{1}{\pi {r}^... | Proof. If \( f \in {\mathrm{{BMO}}}_{r}^{p} \), then (3.19) holds with \( C = \parallel f{\parallel }_{{\mathrm{{BMO}}}_{r}^{p}}^{p} \) and \( {c}_{z} = {\widehat{f}}_{r}\left( z\right) \).\n\nOn the other hand, if (3.19) holds, then by the triangle inequality for the \( {L}^{p} \) integral,\n\n\[ \nM{O}_{p, r}\left( f... | Yes |
Lemma 3.33. The space \( {\mathrm{{BO}}}_{r} \) is independent of \( r \) . Moreover, a continuous function \( f \) on the complex plane belongs to \( {\mathrm{{BO}}}_{r} \) if and only if there exists a constant \( C > 0 \) such that\n\n\[ \left| {f\left( z\right) - f\left( w\right) }\right| \leq C\left( {\left| {z - ... | Proof. If \( f \) satisfies the condition in (3.20), then clearly \( f \in {\mathrm{{BO}}}_{r} \).\n\nTo prove the other direction, assume that \( f \in {\mathrm{{BO}}}_{r} \). Thus, there exists a positive constant \( M \) such that\n\n\[ \left| {f\left( u\right) - f\left( v\right) }\right| \leq M \]\n\nwhenever \( \l... | Yes |
Theorem 3.36. Suppose \( f \in {\mathrm{{BMO}}}^{p} \) and \( 1 \leq p < \infty \) . Then \( \widetilde{f} \in \mathrm{{BO}} \) and \( f - \widetilde{f} \in {\mathrm{{BA}}}^{p} \) . | Proof. It is easy to see that there is a positive constant \( C \) such that\n\n\[ \left| {\widetilde{f}\left( z\right) - {\widehat{f}}_{r}\left( z\right) }\right| \leq \frac{1}{\pi {r}^{2}}{\int }_{B\left( {z, r}\right) }\left| {f\left( w\right) - \widetilde{f}\left( z\right) }\right| \mathrm{d}A\left( w\right) \]\n\n... | Yes |
Proposition 3.38. Suppose \( 1 \leq p < \infty \) and \( f \) is an entire function. Then \( f \in {\mathrm{{BMO}}}^{p} \) if and only if \( f \) is a linear polynomial. | Proof. When \( f \) is entire, we have \( {\widehat{f}}_{r} = f \) because of the mean value theorem. It follows from Theorem 3.34 (and its proof) that \( f = {\widehat{f}}_{r} \in \mathrm{{BO}} \) whenever \( f \in {\mathrm{{BMO}}}^{p} \) . Thus, there exists a positive constant \( C \) such that\n\n\[ \left| {f\left(... | Yes |
Theorem 3.39. Suppose \( 1 \leq p < \infty, r > 0 \), and \( f \) is locally area integrable. Then the following conditions are equivalent:\n\n(i) \( f \in {\mathrm{{VMO}}}^{p} = {\mathrm{{VMO}}}_{r}^{p} \) .\n\n(ii) \( {\operatorname{MO}}_{p}\left( f\right) \left( z\right) \rightarrow 0 \) as \( z \rightarrow \infty \... | We omit the proof. | No |
Theorem 4.2. Let \( I \) be any subset of \( \mathbb{C} \) of Lebesgue measure 1 whose boundary has Lebesgue measure 0 . Then we have\n\n\[ \n{D}^{ - }\left( Z\right) = \mathop{\liminf }\limits_{{r \rightarrow \infty }}\mathop{\inf }\limits_{{w \in \mathbb{C}}}\frac{n\left( {Z, w + {rI}}\right) }{{r}^{2}} \]\n\nand\n\n... | Proof. The proof is similar to that of Proposition 4.1. We will not need the full strength of the theorem and will omit its proof here. We refer the interested reader to [36] for details. | No |
Proposition 4.3. For any lattice\n\n\[ \n\\Lambda = \\left\\{ {\\omega + m{\\omega }_{1} + n{\\omega }_{2} : m \\in \\mathbb{Z}, n \\in \\mathbb{Z}}\\right\\} \n\]\nwe have\n\n\[ \n{D}^{ + }\\left( \\Lambda \\right) = {D}^{ - }\\left( \\Lambda \\right) = \\frac{1}{\\left| \\operatorname{Im}\\left( {\\omega }_{1}{\\bar{... | Proof. The fundamental region of the lattice \( \\Lambda \) is congruent to the parallelogram spanned by \( {\\omega }_{1} = {a}_{1} + \\mathrm{i}{a}_{2} \) and \( {\\omega }_{2} = {b}_{1} + \\mathrm{i}{b}_{2} \), whose area is\n\n\[ \n\\left| {\\det \\left( \\begin{array}{ll} {a}_{1} & {a}_{2} \\\\ {b}_{1} & {b}_{2} \... | Yes |
Proposition 4.4. Let \( Z = \left\{ {z}_{n}\right\} \) be a separated sequence and \( 0 < p < \infty \) . Then there exists a positive constant \( C \), independent of \( f \), such that\n\n\[ \mathop{\sum }\limits_{{n = 1}}^{\infty }{\left| f\left( {z}_{n}\right) {\mathrm{e}}^{-\alpha {\left| {z}_{n}\right| }^{2}/2}\r... | Proof. Let \( \delta = \delta \left( Z\right) \) be the separation constant of \( Z \) . By Lemma 2.32, there exists a positive constant \( C \), independent of \( n \) and \( f \), such that\n\n\[ {\left| f\left( {z}_{n}\right) {\mathrm{e}}^{-\alpha {\left| {z}_{n}\right| }^{2}/2}\right| }^{p} \leq C{\int }_{B\left( {... | Yes |
Lemma 4.5. Suppose \( 0 < p \leq \infty \) and \( Z = \left\{ {z}_{n}\right\} \) is an interpolating sequence for \( {F}_{\alpha }^{p} \). Then there exists a positive constant \( C \) with the following property: whenever \( \left\{ {v}_{n}\right\} \) is a sequence such that \( \left\{ {{v}_{n}{\mathrm{e}}^{-\alpha {\... | Proof. Let \( {X}_{p} \) denote the Banach space of sequences \( \left\{ {v}_{k}\right\} \) such that \( \left\{ {{v}_{k}{\mathrm{e}}^{-\frac{\alpha }{2}{\left| {z}_{k}\right| }^{2}}}\right\} \in \) \( {l}^{p} \). Let \( {J}_{Z} \) denote the space of all functions \( f \in {F}_{\alpha }^{p} \) such that \( f\left( z\r... | Yes |
Corollary 4.7. For \( 0 < p \leq \infty \), there is a positive constant \( C = C\left( {\alpha, p}\right) \) such that\n\n\[ \left| \right| S\left( {z}_{1}\right) \left| -\right| S\left( {z}_{2}\right) \left| \right| \leq C\left| {{z}_{1} - {z}_{2}}\right| \parallel f{\parallel }_{p,\alpha } \]\n\nfor all \( f \in {F}... | Proof. The case \( \left| {{z}_{1} - {z}_{2}}\right| \leq 1 \) follows from the lemma above (and its proof, which gives a version for \( p = \infty \) ), while the case \( \left| {{z}_{1} - {z}_{2}}\right| > 1 \) is obvious. | No |
Lemma 4.8. Suppose \( 0 < p \leq \infty \) and \( Z = \left\{ {z}_{n}\right\} \) is an interpolating sequence for \( {F}_{\alpha }^{p} \) . Then \( Z \) must be separated. | Proof. Fix any two different positive integers \( n \) and \( m \) . If \( \left| {{z}_{n} - {z}_{m}}\right| > 1 \), we do not do anything.\n\nIf \( \left| {{z}_{n} - {z}_{m}}\right| \leq 1 \), we consider the sequence \( \left\{ {a}_{k}\right\} \), where \( {a}_{n} = 1 \) and \( {a}_{k} = 0 \) for \( k \neq n \) . Sin... | Yes |
Lemma 4.9. Suppose \( 0 < p < \infty \) and \( Z = \left\{ {z}_{n}\right\} \) is any sequence of complex numbers. Then the following two conditions are equivalent:\n\n(a) There exists a positive constant \( C \) such that\n\n\[ \mathop{\sum }\limits_{{n = 1}}^{\infty }{\left| f\left( {z}_{n}\right) {\mathrm{e}}^{-\frac... | Proof. Condition (a) above simply says that the measure\n\n\[ \mu = \mathop{\sum }\limits_{{n = 1}}^{\infty }{\delta }_{{z}_{n}} \]\n\n is a Fock-Carleson measure for \( {F}_{\alpha }^{p} \), where \( {\delta }_{z} \) is the unit point mass at \( z \) . Therefore, according to (an obvious variant of) Theorem 3.29, cond... | Yes |
Proposition 4.12. For each \( n \geq 1 \), let \( {Z}_{n} \) be a separated sequence. If \( \delta = \) \( \mathop{\inf }\limits_{n}\delta \left( {Z}_{n}\right) > 0 \), then there exists a subsequence \( \left\{ {Z}_{{n}_{k}}\right\} \) and a separated sequence \( Z \) (possibly empty) such that \( \left\{ {Z}_{{n}_{k}... | Proof. We write \( {Z}_{n} = \left\{ {{z}_{n1},{z}_{n2},\cdots }\right\} \) with \( \left| {z}_{n1}\right| \leq \left| {z}_{n2}\right| \leq \cdots \) . If \( {z}_{n1} \rightarrow \infty \) as \( n \rightarrow \infty \) , then for every \( k \), we have \( {z}_{nk} \rightarrow \infty \) as \( n \rightarrow \infty \) . I... | Yes |
Proposition 4.13. Suppose each \( {Z}_{n} \) is a separated sequence with \( \delta = \mathop{\inf }\limits_{n}\delta \left( {Z}_{n}\right) > \) 0 . Write \( {Z}_{n} = \left\{ {{z}_{n1},{z}_{n2},\cdots }\right\} \) with \( \left| {z}_{n1}\right| \leq \left| {z}_{n2}\right| \leq \cdots \) . Then \( \left\{ {Z}_{n}\right... | Proof. It is clear from the definition that any one of the above conditions implies that \( \left\{ {Z}_{n}\right\} \) converges weakly to \( Z \) . The other implication follows from Proposition 4.12 and its proof, if we start out with an arbitrary subsequence of \( \left\{ {Z}_{n}\right\} \) . Here, we use the fact t... | Yes |
Corollary 4.17. Suppose \( 0 < p < \infty \) and \( Z \) is a separated sequence with separation constant \( \delta \) . If \( Z \) is sampling for \( {F}_{\alpha }^{p} \) and \( {Z}^{\prime } \) is another sequence such that \( \left\lbrack {Z,{Z}^{\prime }}\right\rbrack \) is sufficiently small, then \( {Z}^{\prime }... | Proof. This follows from Proposition 4.16. | No |
Proposition 4.18. Suppose \( \left\{ {Z}_{n}\right\} \) converges to \( Z \) weakly. Then\n\n\[ \n{N}_{p}\left( {Z,\alpha }\right) \leq \mathop{\liminf }\limits_{{n \rightarrow \infty }}{N}_{p}\left( {{Z}_{n},\alpha }\right) \n\] \n\nfor all \( 0 < p \leq \infty \) . | Proof. The case \( Z = \varnothing \) is obvious. Also, by working with a subsequence if necessary, we may assume that\n\n\[ \n\mathop{\liminf }\limits_{{n \rightarrow \infty }}{N}_{p}\left( {Z}_{n}\right) = \mathop{\lim }\limits_{{n \rightarrow \infty }}{N}_{p}\left( {Z}_{n}\right) < \infty . \n\] \n\nIn particular, w... | Yes |
Corollary 4.19. Suppose \( 0 < p \leq \infty \) and \( Z \) is a separated sequence. If \( Z \) is an interpolating sequence for \( {F}_{\alpha }^{p} \), then there exists a positive constant \( \sigma \) such that \( {Z}^{\prime } \) is interpolating for \( {F}_{\alpha }^{p} \) whenever \( \left\lbrack {{Z}^{\prime },... | Proof. This follows from Proposition 4.18. | No |
Proposition 4.20. The function \( {\sigma }_{\alpha } \) is quasiperiodic in the sense that\n\n\[ \n{W}_{{\omega }_{mn}}{\sigma }_{\alpha }\left( z\right) = {\left( -1\right) }^{m + n + {mn}}{\sigma }_{\alpha }\left( z\right) \n\] \n\nfor all \( z \) and \( {\omega }_{mn} \) . Consequently, if \n\n\[ \n{R}_{\alpha } = ... | Proof. See Proposition 1.20 and Corollary 1.21. | No |
Lemma 4.22. Let \( g \) be the function associated to \( Z = \left\{ {z}_{mn}\right\} \) . For any positive radius \( R \), there exists a positive constant \( C \) such that\n\n\[ \left| \frac{g\left( z\right) }{z - {z}_{mn}}\right| \leq C \]\n\nfor all \( \left( {m, n}\right) \) and all \( \left| z\right| \leq R \) . | Proof. It is clear that\n\n\[ \left| \frac{g\left( z\right) }{z - {z}_{mn}}\right| = \frac{\left| g\left( z\right) \right| }{d\left( {z, Z}\right) }\frac{d\left( {z, Z}\right) }{\left| z - {z}_{mn}\right| } \leq \frac{\left| g\left( z\right) \right| }{d\left( {z, Z}\right) }.\]\n\nThe desired result then follows from t... | No |
Lemma 4.23. Suppose \( Z \) is a sequence that is uniformly close to \( {\Lambda }_{\alpha } \) . Then, \( {D}^{ + }\left( Z\right) = {D}^{ - }\left( Z\right) = \alpha /\pi \) | Proof. Suppose \( Z = \left\{ {z}_{mn}\right\} ,{\Lambda }_{\alpha } = \left\{ {\omega }_{mn}\right\} \), and \( \left| {{z}_{mn} - {\omega }_{mn}}\right| \leq Q \) for all \( m \) and \( n \) , where \( Q \) is a positive constant. When \( r \) is much larger than \( Q \), the number of points in \( Z \cap B\left( {w,... | Yes |
Proposition 4.24. Let \( Z = \left\{ {z}_{mn}\right\} \) be a separated sequence in \( \mathbb{C} \) that is uniformly close to \( {\Lambda }_{\beta } \) and let \( g \) be the function associated to \( Z \) by (4.8). If \( \alpha < \beta \), then every function \( f \in {F}_{\alpha }^{\infty } \) can be written as\n\n... | Proof. Since \( \left| {f\left( {z}_{mn}\right) }\right| \leq C{\mathrm{e}}^{\alpha {\left| {z}_{mn}\right| }^{2}/2} \), it follows from (4.11) that\n\n\[ \left| \frac{f\left( {z}_{mn}\right) }{{g}^{\prime }\left( {z}_{mn}\right) }\right| \leq C\exp \left( {-\frac{1}{2}\left( {\beta - \alpha }\right) {\left| {z}_{mn}\r... | Yes |
For any fixed positive number \( r \), the sequence \( \left\{ {{\sigma }_{k}\left( r\right) }\right\} \) defined by\n\n\[ \n{\sigma }_{k}\left( r\right) = \frac{1}{k!}{\int }_{0}^{\alpha {r}^{2}}{t}^{k}{\mathrm{e}}^{-t}\mathrm{\;d}t \n\]\n\nis decreasing in \( k \) and tends to 0 as \( k \rightarrow \infty \) . | Proof. It is well known that the incomplete gamma function\n\n\[ \n\Gamma \left( {a, z}\right) = {\int }_{z}^{\infty }{t}^{a - 1}{\mathrm{e}}^{-t}\mathrm{\;d}t \n\]\n\nhas the property that\n\n\[ \n\Gamma \left( {k + 1, z}\right) = k!{\mathrm{e}}^{-z}\mathop{\sum }\limits_{{j = 0}}^{k}\frac{{z}^{j}}{j!}. \n\]\n\nIt fol... | Yes |
Lemma 4.28. If \( Z \) is a sampling sequence for \( {F}_{\alpha }^{\infty } \), then \( {D}^{ - }\left( Z\right) > \alpha /\pi \) . | Proof. By Lemma 4.10, \( Z \) contains a separated subsequence which is also sampling for \( {F}_{\alpha }^{\infty } \) . Therefore, by working with such a subsequence if necessary, we may assume that \( Z \) is already separated.\n\nIn view of Lemma 4.26, we just need to show that \( {D}^{ - }\left( Z\right) \geq \alp... | Yes |
Lemma 4.29. Suppose \( 0 < p \leq \infty \) and \( Z \) is a sampling sequence for \( {F}_{\alpha }^{p} \). Then \( Z \) is a set of uniqueness for \( {F}_{\alpha }^{\infty } \). | Proof. By Lemmas 4.10 and 4.11, we may assume that \( Z \) is separated.\n\nThe case \( p = \infty \) is obvious. Suppose \( 0 < p < \infty, Z \) is sampling for \( {F}_{\alpha }^{p} \), but \( Z \) is not a set of uniqueness for \( {F}_{\alpha }^{\infty } \). Then there exists a function \( f \in {F}_{\alpha }^{\infty... | Yes |
Lemma 4.30. Suppose \( 0 < p < \infty \) and \( Z \) is sampling for \( {F}_{\alpha }^{p} \) . Then \( {D}^{ - }\left( Z\right) > \alpha /\pi \) . | Proof. Again, by working with a subsequence of \( Z \) if necessary, we may assume that \( Z \) is already separated.\n\nRecall that \( W\left( Z\right) \) consists of all weak limits of translates of \( Z \) . Since every translation of \( Z \) is also a sampling sequence for \( {F}_{\alpha }^{p} \) with the same sepa... | Yes |
Lemma 4.31. Suppose \( 0 < \alpha < \beta \) and \( Z \) is a sequence with \( {D}^{ - }\left( Z\right) = \beta /\pi \) . There exists a subsequence \( {Z}^{\prime } \) of \( Z \) such that \( {Z}^{\prime } \) is uniformly close to \( {\Lambda }_{\gamma } \) for some \( \alpha < \gamma < \beta \) . | Proof. Fix \( \gamma \in \left( {\alpha ,\beta }\right) \) and choose \( \varepsilon > 0 \) such that \( \gamma + \varepsilon < \beta \) . The condition \( {D}^{ - }\left( Z\right) = \) \( \beta /\pi \) implies that there exists a positive number \( r \) such that any square of side length \( r \) contains at least \( ... | Yes |
Lemma 4.33. Suppose \( 0 < p \leq 1 \) and \( Z \) is a separated sequence with \( {D}^{ - }\left( Z\right) > \) \( \alpha /\pi \) . Then \( Z \) is sampling for \( {F}_{\alpha }^{p} \) . | Proof. With notation from the proof of the previous lemma, we use the assumption \( 0 < p \leq 1 \) to get\n\n\[ \n{\left| {W}_{{\omega }_{kl}}f\left( z\right) \right| }^{p} \leq \mathop{\sum }\limits_{{m, n}}{\left| \frac{{W}_{{\omega }_{kl}}f\left( {{z}_{mn} + {\omega }_{kl}}\right) }{{g}_{{\omega }_{kl}}^{\prime }\l... | Yes |
Lemma 4.34. Any separated sequence \( Z \) with \( {D}^{ - }\left( Z\right) > \alpha /\pi \) is a sampling sequence for \( {F}_{\alpha }^{\infty } \) . | Proof. With notation from the proof of the previous two lemmas, we have\n\n\[ \parallel f{\parallel }_{\infty ,\alpha } = \mathop{\sup }\limits_{{k, l}}{S}_{kl} \]\n\nwhere\n\n\[ {S}_{kl} = \sup \left\{ {{\mathrm{e}}^{-\frac{\alpha }{2}{\left| z\right| }^{2}}\left| {{W}_{{\omega }_{kl}}f\left( z\right) }\right| : z \in... | Yes |
Lemma 4.38. Let \( Z \) be a separated sequence in \( \mathbb{C} \) with \( {D}^{ + }\left( Z\right) = \beta /\pi \) and \( \beta < \alpha \) . We can expand \( Z \) to a separated sequence \( {Z}^{\prime } \) such that \( {Z}^{\prime } \) is uniformly close to a square lattice \( {\Lambda }_{\gamma } \) with \( \gamma... | Proof. Let \( \gamma \in \left( {\beta ,\alpha }\right) \) and choose \( \varepsilon > 0 \) such that \( \beta < \gamma - \varepsilon \) . The condition \( {D}^{ + }\left( Z\right) = \) \( \beta /\pi \) implies that there is some large \( r \) such that any square of side length \( r \) contains at most \( \left( {\gam... | Yes |
Lemma 4.40. Let \( 0 < p \leq \infty \) . There is no sequence in \( \mathbb{C} \) that is both sampling for \( {F}_{\alpha }^{p} \) and interpolating for \( {F}_{\alpha }^{p} \) . | Proof. Assume the contrary and let \( Z \) be a sequence that is both sampling and interpolating for \( {F}_{\alpha }^{p} \) . Then \( Z \) is separated and sampling for \( {F}_{\alpha + \varepsilon }^{p} \) for all sufficiently small \( \varepsilon \), because we have characterized sampling sequences for \( {F}_{\alph... | No |
Lemma 4.41. Suppose \( 0 < p \leq \infty \) and \( Z \) is interpolating for \( {F}_{\alpha }^{p} \) . If \( Z \) is a set of uniqueness for \( {F}_{\alpha }^{p} \), then it must be a sampling sequence for \( {F}_{\alpha }^{p} \) . | Proof. Since \( Z \) is interpolating for \( {F}_{\alpha }^{p} \), it must be separated by Lemma 4.8. Given any function \( f \in {F}_{\alpha }^{p} \), the sequence \( {w}_{n} = f\left( {z}_{n}\right) \) has the property that \( \left\{ {{w}_{n}{\mathrm{e}}^{-\alpha {\left| {z}_{n}\right| }^{2}/2}}\right\} \in \) \( {l... | Yes |
Corollary 4.43. Let \( 0 < p \leq \infty \) and let \( Z \) be a sampling sequence for \( {F}_{\alpha }^{p} \) . For any \( \zeta \in Z \), the sequence \( Z - \{ \zeta \} \) remains a sampling sequence for \( {F}_{\alpha }^{p} \) . | Proof. This is clear from the already-proved characterization of sampling sequences for \( {F}_{\alpha }^{p} \) in terms of the lower density because deleting a single point from a sequence does not alter the density of the sequence. | Yes |
Corollary 4.44. Let \( 0 < p \leq \infty \) . If \( Z = \left\{ {z}_{n}\right\} \) is an interpolating sequence for \( {F}_{\alpha }^{p} \) , then so is \( Z \cup \{ \zeta \} \) for any \( \zeta \notin Z \) . | Proof. By Corollary 4.42, there is a function \( g \in {F}_{\alpha }^{p} \) that is not identically zero but vanishes on \( Z \) . By dividing out an appropriate power of \( z - \zeta \) if necessary (which preserves membership in \( {F}_{\alpha }^{p} \) ), we may assume that \( g\left( \zeta \right) \neq 0 \) . Multip... | Yes |
Lemma 4.45. If \( Z \) is interpolating for \( {F}_{\alpha }^{p} \), where \( 0 < p \leq \infty \), then \( {\rho }_{p}\left( {z, Z}\right) > 0 \) when \( z \notin Z \) . | Proof. Actually, we only need to assume that \( Z \) is not a set of uniqueness (we already know that every interpolating sequence for \( {F}_{\alpha }^{p} \) is not a set of uniqueness for \( {F}_{\alpha }^{p} \) ). In fact, if \( f \) is any function in \( {F}_{\alpha }^{p} \) that is not identically zero and vanishe... | Yes |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.