$\|\sum_ia_i g(x-i)\|_{L^p(\mathbb{R})}\le (\sup_x \{\sum_k|g(x-k)|\})\|a\|_{L^p(\mathbb{Z})}$

Question

Let $a=\{a_i\}$ be an arbitrary sequence of complex numbers with finitely many non-zero terms. Consider the function $f(x)=\sum_ia_i g(x-i)$, where $g$ is a good function. Prove that for any $p\in (1,\infty), \|f\|_{L^p(\mathbb{R})}\le C\|a\|_{L^p(\mathbb{Z})}$, where $C=\sup_x \{\sum_k|g(x-k)|\}$

Answer 1

I figure it out myself. First using change of variable to reduce to problem to the following inequality: $$ \sum_k(\sum_i|a_{i+k}||g(x-i)|)^p\le C^p\|a\|_{L^p(\mathbb{Z})}^p $$ for any $x\in[0,1]$. Then the above inequality can be proven by Minkowski's integral inequality.