When does a continuous function's “Fourier series” converge pointwise almost everywhere to the function?
$begingroup$
Let $G$ be a compact topological group. By the Peter-Weyl theorem, the complex Hilbert space $L^2(G)$ is the Hilbert space direct sum of the spaces of matrix coefficients of all the irreducible representations of $G$. When $G$ is commutative, these irreducible representations are one-dimensional, and the Peter-Weyl theorem just says that the unitary characters of $G$ form an orthonormal basis of $L^2(G)$.
For example, when $G = mathbb R/mathbb Z$, the functions $f_n(x) = e^{2pi i nx}$ are an orthonormal basis of $L^2(G)$. Thus if $f in L^2(G)$, then there are unique complex numbers $c_n$, with $sumlimits_n |c_n|^2 < infty$, which gives $f$ its Fourier expansion:
$$f = sumlimits_{nin mathbb Z} c_n f_n$$
where the sum on the right hand side converges in the $L^2$-norm to $f$. It is a much deeper theorem that when $f$ is continuous, the right hand side also converges pointwise to $f$ almost everywhere.
Is the analogue of this deep theorem known for other compact groups? That is, suppose we take an orthonormal basis $f_i : i in I$ of $L^2(G)$ via matrix coefficients of irreducible representations as in the Peter-Weyl theorem, so that any $f in L^2(G)$ has a "Fourier expansion" for uniquely determined $c_i in mathbb C$,
$$f = sumlimits_{i in I} c_i f_i$$
so the right hand side converges to $f$ in the $L^2$-norm. Suppose that $f$ is continuous. Then, do we know that the right hand side converges pointwise to $f$ almost everywhere?
If this is not known in general, is it known for, say, $G = mathbb A_k/k$ for $k$ a global field?
number-theory reference-request representation-theory fourier-series locally-compact-groups
asked Jan 2 at 4:35
D_SD_S
14.3k61756
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add a comment |
$begingroup$
Let $G$ be a compact topological group. By the Peter-Weyl theorem, the complex Hilbert space $L^2(G)$ is the Hilbert space direct sum of the spaces of matrix coefficients of all the irreducible representations of $G$. When $G$ is commutative, these irreducible representations are one-dimensional, and the Peter-Weyl theorem just says that the unitary characters of $G$ form an orthonormal basis of $L^2(G)$.
For example, when $G = mathbb R/mathbb Z$, the functions $f_n(x) = e^{2pi i nx}$ are an orthonormal basis of $L^2(G)$. Thus if $f in L^2(G)$, then there are unique complex numbers $c_n$, with $sumlimits_n |c_n|^2 < infty$, which gives $f$ its Fourier expansion:
$$f = sumlimits_{nin mathbb Z} c_n f_n$$
where the sum on the right hand side converges in the $L^2$-norm to $f$. It is a much deeper theorem that when $f$ is continuous, the right hand side also converges pointwise to $f$ almost everywhere.
Is the analogue of this deep theorem known for other compact groups? That is, suppose we take an orthonormal basis $f_i : i in I$ of $L^2(G)$ via matrix coefficients of irreducible representations as in the Peter-Weyl theorem, so that any $f in L^2(G)$ has a "Fourier expansion" for uniquely determined $c_i in mathbb C$,
$$f = sumlimits_{i in I} c_i f_i$$
so the right hand side converges to $f$ in the $L^2$-norm. Suppose that $f$ is continuous. Then, do we know that the right hand side converges pointwise to $f$ almost everywhere?
If this is not known in general, is it known for, say, $G = mathbb A_k/k$ for $k$ a global field?
number-theory reference-request representation-theory fourier-series locally-compact-groups
asked Jan 2 at 4:35
D_SD_S
14.3k61756
$endgroup$
add a comment |
$begingroup$
Let $G$ be a compact topological group. By the Peter-Weyl theorem, the complex Hilbert space $L^2(G)$ is the Hilbert space direct sum of the spaces of matrix coefficients of all the irreducible representations of $G$. When $G$ is commutative, these irreducible representations are one-dimensional, and the Peter-Weyl theorem just says that the unitary characters of $G$ form an orthonormal basis of $L^2(G)$.
For example, when $G = mathbb R/mathbb Z$, the functions $f_n(x) = e^{2pi i nx}$ are an orthonormal basis of $L^2(G)$. Thus if $f in L^2(G)$, then there are unique complex numbers $c_n$, with $sumlimits_n |c_n|^2 < infty$, which gives $f$ its Fourier expansion:
$$f = sumlimits_{nin mathbb Z} c_n f_n$$
where the sum on the right hand side converges in the $L^2$-norm to $f$. It is a much deeper theorem that when $f$ is continuous, the right hand side also converges pointwise to $f$ almost everywhere.
Is the analogue of this deep theorem known for other compact groups? That is, suppose we take an orthonormal basis $f_i : i in I$ of $L^2(G)$ via matrix coefficients of irreducible representations as in the Peter-Weyl theorem, so that any $f in L^2(G)$ has a "Fourier expansion" for uniquely determined $c_i in mathbb C$,
$$f = sumlimits_{i in I} c_i f_i$$
so the right hand side converges to $f$ in the $L^2$-norm. Suppose that $f$ is continuous. Then, do we know that the right hand side converges pointwise to $f$ almost everywhere?
If this is not known in general, is it known for, say, $G = mathbb A_k/k$ for $k$ a global field?
number-theory reference-request representation-theory fourier-series locally-compact-groups
asked Jan 2 at 4:35
D_SD_S
14.3k61756
$endgroup$
Let $G$ be a compact topological group. By the Peter-Weyl theorem, the complex Hilbert space $L^2(G)$ is the Hilbert space direct sum of the spaces of matrix coefficients of all the irreducible representations of $G$. When $G$ is commutative, these irreducible representations are one-dimensional, and the Peter-Weyl theorem just says that the unitary characters of $G$ form an orthonormal basis of $L^2(G)$.
For example, when $G = mathbb R/mathbb Z$, the functions $f_n(x) = e^{2pi i nx}$ are an orthonormal basis of $L^2(G)$. Thus if $f in L^2(G)$, then there are unique complex numbers $c_n$, with $sumlimits_n |c_n|^2 < infty$, which gives $f$ its Fourier expansion:
$$f = sumlimits_{nin mathbb Z} c_n f_n$$
where the sum on the right hand side converges in the $L^2$-norm to $f$. It is a much deeper theorem that when $f$ is continuous, the right hand side also converges pointwise to $f$ almost everywhere.
Is the analogue of this deep theorem known for other compact groups? That is, suppose we take an orthonormal basis $f_i : i in I$ of $L^2(G)$ via matrix coefficients of irreducible representations as in the Peter-Weyl theorem, so that any $f in L^2(G)$ has a "Fourier expansion" for uniquely determined $c_i in mathbb C$,
$$f = sumlimits_{i in I} c_i f_i$$
so the right hand side converges to $f$ in the $L^2$-norm. Suppose that $f$ is continuous. Then, do we know that the right hand side converges pointwise to $f$ almost everywhere?
If this is not known in general, is it known for, say, $G = mathbb A_k/k$ for $k$ a global field?
number-theory reference-request representation-theory fourier-series locally-compact-groups
number-theory reference-request representation-theory fourier-series locally-compact-groups
asked Jan 2 at 4:35
D_SD_S
14.3k61756
asked Jan 2 at 4:35
D_SD_S
14.3k61756
asked Jan 2 at 4:35
D_SD_S
14.3k61756
asked Jan 2 at 4:35
D_SD_S
14.3k61756
asked Jan 2 at 4:35
D_SD_S
14.3k61756
14.3k61756
add a comment |
add a comment |
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