Does the 'continuous convolution relation' imply a discrete version?












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In particular, I want to know if $L^p(mathbb{R}^d)*L^q(mathbb{R}^d)subseteq L^r(mathbb{R}^d)$ implies $ell^{p}(mathbb{Z}^d)*ell^{q}(mathbb{Z}^d)subseteq ell^{r}(mathbb{Z}^d)$? My attempt is to try using the hypothesis on functions defined via sequences. That is, for any $a={a_n}_{nin mathbb{Z}^d} in ell^p(mathbb{Z}^d),$ $b={b_n }_{nin mathbb{Z}^d}in ell^{q}(mathbb{Z}^d)$, define $f_a =sum_{nin mathbb{Z}^d} a_nchi_{n}$, and $f_b=sum_{nin mathbb{Z}^d}b_n chi_{n}$ where for any $nin mathbb{Z}^d,$ $chi_{n}=chi_{n+[0,1)^{d}}$. I naively tried to show that $||a*b||_{ell^r}leq ||f_a*f_b||_{L^r},$ but failed miserably.










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    In particular, I want to know if $L^p(mathbb{R}^d)*L^q(mathbb{R}^d)subseteq L^r(mathbb{R}^d)$ implies $ell^{p}(mathbb{Z}^d)*ell^{q}(mathbb{Z}^d)subseteq ell^{r}(mathbb{Z}^d)$? My attempt is to try using the hypothesis on functions defined via sequences. That is, for any $a={a_n}_{nin mathbb{Z}^d} in ell^p(mathbb{Z}^d),$ $b={b_n }_{nin mathbb{Z}^d}in ell^{q}(mathbb{Z}^d)$, define $f_a =sum_{nin mathbb{Z}^d} a_nchi_{n}$, and $f_b=sum_{nin mathbb{Z}^d}b_n chi_{n}$ where for any $nin mathbb{Z}^d,$ $chi_{n}=chi_{n+[0,1)^{d}}$. I naively tried to show that $||a*b||_{ell^r}leq ||f_a*f_b||_{L^r},$ but failed miserably.










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      In particular, I want to know if $L^p(mathbb{R}^d)*L^q(mathbb{R}^d)subseteq L^r(mathbb{R}^d)$ implies $ell^{p}(mathbb{Z}^d)*ell^{q}(mathbb{Z}^d)subseteq ell^{r}(mathbb{Z}^d)$? My attempt is to try using the hypothesis on functions defined via sequences. That is, for any $a={a_n}_{nin mathbb{Z}^d} in ell^p(mathbb{Z}^d),$ $b={b_n }_{nin mathbb{Z}^d}in ell^{q}(mathbb{Z}^d)$, define $f_a =sum_{nin mathbb{Z}^d} a_nchi_{n}$, and $f_b=sum_{nin mathbb{Z}^d}b_n chi_{n}$ where for any $nin mathbb{Z}^d,$ $chi_{n}=chi_{n+[0,1)^{d}}$. I naively tried to show that $||a*b||_{ell^r}leq ||f_a*f_b||_{L^r},$ but failed miserably.










      share|cite|improve this question











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      In particular, I want to know if $L^p(mathbb{R}^d)*L^q(mathbb{R}^d)subseteq L^r(mathbb{R}^d)$ implies $ell^{p}(mathbb{Z}^d)*ell^{q}(mathbb{Z}^d)subseteq ell^{r}(mathbb{Z}^d)$? My attempt is to try using the hypothesis on functions defined via sequences. That is, for any $a={a_n}_{nin mathbb{Z}^d} in ell^p(mathbb{Z}^d),$ $b={b_n }_{nin mathbb{Z}^d}in ell^{q}(mathbb{Z}^d)$, define $f_a =sum_{nin mathbb{Z}^d} a_nchi_{n}$, and $f_b=sum_{nin mathbb{Z}^d}b_n chi_{n}$ where for any $nin mathbb{Z}^d,$ $chi_{n}=chi_{n+[0,1)^{d}}$. I naively tried to show that $||a*b||_{ell^r}leq ||f_a*f_b||_{L^r},$ but failed miserably.







      functional-analysis convolution






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      edited Dec 8 '18 at 12:43







      Kurome

















      asked Dec 8 '18 at 12:08









      KuromeKurome

      378114




      378114






















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