Orthogonality of integer shifts and sum of fourier transforms
A function $psi in L_2(mathbb{R})$ is orthogonal to all integer shifts of a function $varphi in L_2(mathbb{R})$ if and if only
$$sum_{kin mathbb{z}} hat{varphi}(xi+k)overline{hat{psi}(xi+k)}equiv0$$
My approach:
$int_{mathbb{R}}psi(t)varphi(t+k)dt=0$ $forall k in mathbb{Z}$ given. Now
begin{align}
&sum_{kinmathbb{Z}}int_{mathbb{R}}psi(t)exp(-2pi i (xi+k)t)dt overline{int_{mathbb{R}}varphi(s)exp(-2pi i (xi+k)s )}ds\
implies &sum_{kinmathbb{Z}}int_{mathbb{R}}psi(t)exp(-2pi i (xi+k)t)dt int_{mathbb{R}}varphi(s)exp(2pi i (xi+k)s )ds\
implies &sum_{kinmathbb{Z}}int_{mathbb{R}}psi(t)varphi(s)dsdt int_{mathbb{R}}exp(2pi i (xi+k)s )exp(-2pi i (xi+k)t)dtds\
end{align}
The last step I am not so sure, I think this can be done if s and t are independent. I am open to further suggestions or corrections and ideas regarding how to proceed.
I got a hint from the prof that we need to use:
$$varphi(xi)=sum_{kin mathbb{Z}}|hat{psi}(xi+k)|^2$$
${psi(t-k)}_{kin mathbb{Z}}$ is a orthonormal system if and only if$varphiequiv1$
functional-analysis lp-spaces fourier-transform orthogonality
asked Dec 9 at 18:46
mm-crj
382212
add a comment |
A function $psi in L_2(mathbb{R})$ is orthogonal to all integer shifts of a function $varphi in L_2(mathbb{R})$ if and if only
$$sum_{kin mathbb{z}} hat{varphi}(xi+k)overline{hat{psi}(xi+k)}equiv0$$
My approach:
$int_{mathbb{R}}psi(t)varphi(t+k)dt=0$ $forall k in mathbb{Z}$ given. Now
begin{align}
&sum_{kinmathbb{Z}}int_{mathbb{R}}psi(t)exp(-2pi i (xi+k)t)dt overline{int_{mathbb{R}}varphi(s)exp(-2pi i (xi+k)s )}ds\
implies &sum_{kinmathbb{Z}}int_{mathbb{R}}psi(t)exp(-2pi i (xi+k)t)dt int_{mathbb{R}}varphi(s)exp(2pi i (xi+k)s )ds\
implies &sum_{kinmathbb{Z}}int_{mathbb{R}}psi(t)varphi(s)dsdt int_{mathbb{R}}exp(2pi i (xi+k)s )exp(-2pi i (xi+k)t)dtds\
end{align}
The last step I am not so sure, I think this can be done if s and t are independent. I am open to further suggestions or corrections and ideas regarding how to proceed.
I got a hint from the prof that we need to use:
$$varphi(xi)=sum_{kin mathbb{Z}}|hat{psi}(xi+k)|^2$$
${psi(t-k)}_{kin mathbb{Z}}$ is a orthonormal system if and only if$varphiequiv1$
functional-analysis lp-spaces fourier-transform orthogonality
asked Dec 9 at 18:46
mm-crj
382212
add a comment |
A function $psi in L_2(mathbb{R})$ is orthogonal to all integer shifts of a function $varphi in L_2(mathbb{R})$ if and if only
$$sum_{kin mathbb{z}} hat{varphi}(xi+k)overline{hat{psi}(xi+k)}equiv0$$
My approach:
$int_{mathbb{R}}psi(t)varphi(t+k)dt=0$ $forall k in mathbb{Z}$ given. Now
begin{align}
&sum_{kinmathbb{Z}}int_{mathbb{R}}psi(t)exp(-2pi i (xi+k)t)dt overline{int_{mathbb{R}}varphi(s)exp(-2pi i (xi+k)s )}ds\
implies &sum_{kinmathbb{Z}}int_{mathbb{R}}psi(t)exp(-2pi i (xi+k)t)dt int_{mathbb{R}}varphi(s)exp(2pi i (xi+k)s )ds\
implies &sum_{kinmathbb{Z}}int_{mathbb{R}}psi(t)varphi(s)dsdt int_{mathbb{R}}exp(2pi i (xi+k)s )exp(-2pi i (xi+k)t)dtds\
end{align}
The last step I am not so sure, I think this can be done if s and t are independent. I am open to further suggestions or corrections and ideas regarding how to proceed.
I got a hint from the prof that we need to use:
$$varphi(xi)=sum_{kin mathbb{Z}}|hat{psi}(xi+k)|^2$$
${psi(t-k)}_{kin mathbb{Z}}$ is a orthonormal system if and only if$varphiequiv1$
functional-analysis lp-spaces fourier-transform orthogonality
asked Dec 9 at 18:46
mm-crj
382212
A function $psi in L_2(mathbb{R})$ is orthogonal to all integer shifts of a function $varphi in L_2(mathbb{R})$ if and if only
$$sum_{kin mathbb{z}} hat{varphi}(xi+k)overline{hat{psi}(xi+k)}equiv0$$
My approach:
$int_{mathbb{R}}psi(t)varphi(t+k)dt=0$ $forall k in mathbb{Z}$ given. Now
begin{align}
&sum_{kinmathbb{Z}}int_{mathbb{R}}psi(t)exp(-2pi i (xi+k)t)dt overline{int_{mathbb{R}}varphi(s)exp(-2pi i (xi+k)s )}ds\
implies &sum_{kinmathbb{Z}}int_{mathbb{R}}psi(t)exp(-2pi i (xi+k)t)dt int_{mathbb{R}}varphi(s)exp(2pi i (xi+k)s )ds\
implies &sum_{kinmathbb{Z}}int_{mathbb{R}}psi(t)varphi(s)dsdt int_{mathbb{R}}exp(2pi i (xi+k)s )exp(-2pi i (xi+k)t)dtds\
end{align}
The last step I am not so sure, I think this can be done if s and t are independent. I am open to further suggestions or corrections and ideas regarding how to proceed.
I got a hint from the prof that we need to use:
$$varphi(xi)=sum_{kin mathbb{Z}}|hat{psi}(xi+k)|^2$$
${psi(t-k)}_{kin mathbb{Z}}$ is a orthonormal system if and only if$varphiequiv1$
functional-analysis lp-spaces fourier-transform orthogonality
functional-analysis lp-spaces fourier-transform orthogonality
asked Dec 9 at 18:46
mm-crj
382212
asked Dec 9 at 18:46
mm-crj
382212
edited Dec 11 at 18:35
asked Dec 9 at 18:46
mm-crj
382212
asked Dec 9 at 18:46
mm-crj
382212
asked Dec 9 at 18:46
mm-crj
382212
382212
add a comment |
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