Let $0 < p\le \infty$ and $\nu \in \mathbb{Z}$. Suppose that $g$ is a (smooth) function satisfying $\mbox{supp }\hat g \subset \lbrace\xi: |\xi| \le 2^{\nu + 1}\rbrace$. Then $$\sum_{k\in\mathbb{Z}} \sup_{x\in [k2^{-\nu},(k+1)2^{-\nu}]} |g(x)|^p \lesssim 2^{\nu} \|g\|_p^p.$$
Question: Does an analogous inequality hold if the support condition on $\hat g$ is relaxed? Say, if we assume that $g_\nu = \varphi\left(2^{\nu}\cdot\right)$ (EDIT: This definition of $g$ is modulo a scaling factor in order for the $p$ norms to scale properly with $\nu$ e.g. so that $\|g_\nu\|_p = 1$ for all $\nu$), 2^{-\nu}\varphi\left(2^{\nu}\cdot\right)$, where$\varphi$is some smooth function? I am mainly interested in the case$p<1$. EDIT: I will try a concrete example which is in the same spirit: Assume that$\varphi$is some nice function (but not with compact frequency support). When does an inequality of the form $$\sum_{k\in \mathbb{Z}} |c_k|^p \lesssim \|\sum_{k\in \mathbb{Z}}c_k\varphi(\cdot - k)\|_p^p ?$$ hold? Certainly if$\hat \varphi$has compact frequency support, this follows from the PP inequality. But also if e.g.$\varphi$is a B-spline, so compact frequency support of$\varphi$is not necessary. 3 added 503 characters in body Hi! The Plancerel-Polya inequality can be stated as follows: Let$0 < p\le \infty$and$ \nu \in \mathbb{Z}$. Suppose that$g$is a (smooth) function satisfying$\mbox{supp }\hat g \subset \lbrace\xi: |\xi| \le 2^{\nu + 1}\rbrace$. Then $$\sum_{k\in\mathbb{Z}} \sup_{x\in [k2^{-\nu},(k+1)2^{-\nu}]} |g(x)|^p \lesssim 2^{\nu} \|g\|_p^p.$$ Question: Does an analogous inequality hold if the support condition on$\hat g$is relaxed? Say, if we assume that$g_\nu = \varphi\left(2^{\nu}\cdot\right)$(EDIT: This definition of$g$is modulo a scaling factor in order for the$p$norms to scale properly with$\nu$e.g. so that$\|g_\nu\|_p = 1$for all$\nu$), where$\varphi$is some smooth function? I am mainly interested in the case$p<1$. EDIT: I will try a concrete example which is in the same spirit: Assume that$\varphi$is some nice function (but not with compact frequency support). When does an inequality of the form $$\sum_{k\in \mathbb{Z}} |c_k|^p \lesssim \|\sum_{k\in \mathbb{Z}}c_k\varphi(\cdot - k)\|_p^p ?$$ hold? Certainly if$\hat \varphi$has compact frequency support, this follows from the PP inequality. But also if e.g.$\varphi$is a B-spline, so compact frequency support of$\varphi$is not necessary. 2 added 181 characters in body; deleted 12 characters in body Hi! The Plancerel-Polya inequality can be stated as follows: Let$0 < p\le \infty$and$ \nu \in \mathbb{Z}$. Suppose that$g$is a (smooth) function satisfying$\mbox{supp }\hat g \subset \lbrace\xi: |\xi| \le 2^{\nu + 1}\rbrace$. Then $$\sum_{k\in\mathbb{Z}} \sup_{x\in [k2^{-\nu},(k+1)2^{-\nu}]} |g(x)|^p \lesssim 2^{\nu} \|g\|_p^p.$$ Question: Does an analogous inequality hold if the support condition on$\hat g$is relaxed? Say, if we assume that$g g_\nu = \varphi\left(2^{\nu}\cdot\right)$, varphi\left(2^{\nu}\cdot\right)$ (EDIT: This definition of $g$ is modulo a scaling factor in order for the $p$ norms to scale properly with $\nu$ e.g. so that $\|g_\nu\|_p = 1$ for all $\nu$), where $\varphi$ is some smooth function? I am mainly interested in the case $p<1$.