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Smooth $q$-adic decomposition of unity for $q > 1$ large
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Smooth dyadic decompositions of unity are classical and very applicable tools in analysis. This is usually an identity of the form
$$
sum_j in mathbbZ varphi(t/2^j) = 1
$$
for all $t in mathbbR$ (except possibly when $t=0$) and some $varphi$ smooth compactly supported on (say) $[2^-1, 2]$ and with bounded derivatives. Thus $varphi(t/2^j)$ is supported on $[2^j-1, 2^j+1]$.
I wonder how one could go about generalizing this to the case when $2$ is replaced with some large $q$. Thus something of the form
$$
sum_j in mathbbZvarphi(t/q^j) = 1.
$$
However I want that each $varphi(t/q^j)$ has derivatives decaying proportionately with the length of its support. If one were to follow the dyadic example above, one should expect $varphi(t/q^j)$ to be supported on an interval of length $asymp q^j+1$ but its first derivative is $ll 1/q^j$ (if I'm not mistaken). Can we construct $varphi$ in such a way that we also get $|(varphi(cdot/q^j))^(k)|_infty ll_k |textsupp(varphi(cdot/q^j))|^-k$ for each $k geq 0$. Here $|textsupp(varphi(cdot/q^j))| $ denotes the length of the support of $t mapsto varphi(t/q^j)$. Thanks.
calculus analysis fourier-analysis harmonic-analysis
asked Jul 17 at 20:36
user152169
906714
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up vote
1
down vote
favorite
Smooth dyadic decompositions of unity are classical and very applicable tools in analysis. This is usually an identity of the form
$$
sum_j in mathbbZ varphi(t/2^j) = 1
$$
for all $t in mathbbR$ (except possibly when $t=0$) and some $varphi$ smooth compactly supported on (say) $[2^-1, 2]$ and with bounded derivatives. Thus $varphi(t/2^j)$ is supported on $[2^j-1, 2^j+1]$.
I wonder how one could go about generalizing this to the case when $2$ is replaced with some large $q$. Thus something of the form
$$
sum_j in mathbbZvarphi(t/q^j) = 1.
$$
However I want that each $varphi(t/q^j)$ has derivatives decaying proportionately with the length of its support. If one were to follow the dyadic example above, one should expect $varphi(t/q^j)$ to be supported on an interval of length $asymp q^j+1$ but its first derivative is $ll 1/q^j$ (if I'm not mistaken). Can we construct $varphi$ in such a way that we also get $|(varphi(cdot/q^j))^(k)|_infty ll_k |textsupp(varphi(cdot/q^j))|^-k$ for each $k geq 0$. Here $|textsupp(varphi(cdot/q^j))| $ denotes the length of the support of $t mapsto varphi(t/q^j)$. Thanks.
calculus analysis fourier-analysis harmonic-analysis
asked Jul 17 at 20:36
user152169
906714
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Smooth dyadic decompositions of unity are classical and very applicable tools in analysis. This is usually an identity of the form
$$
sum_j in mathbbZ varphi(t/2^j) = 1
$$
for all $t in mathbbR$ (except possibly when $t=0$) and some $varphi$ smooth compactly supported on (say) $[2^-1, 2]$ and with bounded derivatives. Thus $varphi(t/2^j)$ is supported on $[2^j-1, 2^j+1]$.
I wonder how one could go about generalizing this to the case when $2$ is replaced with some large $q$. Thus something of the form
$$
sum_j in mathbbZvarphi(t/q^j) = 1.
$$
However I want that each $varphi(t/q^j)$ has derivatives decaying proportionately with the length of its support. If one were to follow the dyadic example above, one should expect $varphi(t/q^j)$ to be supported on an interval of length $asymp q^j+1$ but its first derivative is $ll 1/q^j$ (if I'm not mistaken). Can we construct $varphi$ in such a way that we also get $|(varphi(cdot/q^j))^(k)|_infty ll_k |textsupp(varphi(cdot/q^j))|^-k$ for each $k geq 0$. Here $|textsupp(varphi(cdot/q^j))| $ denotes the length of the support of $t mapsto varphi(t/q^j)$. Thanks.
calculus analysis fourier-analysis harmonic-analysis
asked Jul 17 at 20:36
user152169
906714
Smooth dyadic decompositions of unity are classical and very applicable tools in analysis. This is usually an identity of the form
$$
sum_j in mathbbZ varphi(t/2^j) = 1
$$
for all $t in mathbbR$ (except possibly when $t=0$) and some $varphi$ smooth compactly supported on (say) $[2^-1, 2]$ and with bounded derivatives. Thus $varphi(t/2^j)$ is supported on $[2^j-1, 2^j+1]$.
I wonder how one could go about generalizing this to the case when $2$ is replaced with some large $q$. Thus something of the form
$$
sum_j in mathbbZvarphi(t/q^j) = 1.
$$
However I want that each $varphi(t/q^j)$ has derivatives decaying proportionately with the length of its support. If one were to follow the dyadic example above, one should expect $varphi(t/q^j)$ to be supported on an interval of length $asymp q^j+1$ but its first derivative is $ll 1/q^j$ (if I'm not mistaken). Can we construct $varphi$ in such a way that we also get $|(varphi(cdot/q^j))^(k)|_infty ll_k |textsupp(varphi(cdot/q^j))|^-k$ for each $k geq 0$. Here $|textsupp(varphi(cdot/q^j))| $ denotes the length of the support of $t mapsto varphi(t/q^j)$. Thanks.
calculus analysis fourier-analysis harmonic-analysis
asked Jul 17 at 20:36
user152169
906714
edited Jul 17 at 20:48
asked Jul 17 at 20:36
user152169
906714
asked Jul 17 at 20:36
user152169
906714
asked Jul 17 at 20:36
user152169
906714
906714
add a comment |Â
add a comment |Â
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