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Latexed and removed non-relevant introductory sentence
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edit approved Jun 18, 2014 at 8:18
Loïc Teyssier
edit approved Jun 18, 2014 at 8:18
Loïc Teyssier
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Well I've heard a lot of good things about mathoverflow, and so decided to ask this question that emerged in our research.

Suppose we have a finite-norm vector $X$ in a HilberHilbert space, and we want to construct its expansion in a certain (infinite) basis $V_k$, $X=\sum a_k V_k$$X=\sum_k a_k V_k$. If the basis is orthonormal, then we know that $a_k=(X,V_k)$$a_k=<X,V_k>$ and also $|X|^2=\sum a_k^2$$||X||^2=\sum_k a_k^2$. In particular, we can say from that that $\sum a_k V_k$$\sum_k a_k V_k$ converges at least as fast as $a_k^2$ drops for large $k's$$k$'s.

Now the question is this, if the expansion is in a basis $V_k$ that is not orthogonal, are there known results about the speed of convergence of such a series, $X=\sum a_k V_k$$X=\sum_k a_k V_k$?

Well I've heard a lot of good things about mathoverflow, and so decided to ask this question that emerged in our research.

Suppose we have a finite-norm vector $X$ in a Hilber space, and we want to construct its expansion in a certain (infinite) basis $V_k$, $X=\sum a_k V_k$. If the basis is orthonormal, then we know that $a_k=(X,V_k)$ and also $|X|^2=\sum a_k^2$. In particular, we can say from that that $\sum a_k V_k$ converges at least as fast as $a_k^2$ drops for large $k's$.

Now the question is this, if the expansion is in a basis $V_k$ that is not orthogonal, are there known results about the speed of convergence of such a series, $X=\sum a_k V_k$?

Suppose we have a finite-norm vector $X$ in a Hilbert space, and we want to construct its expansion in a certain (infinite) basis $V_k$, $X=\sum_k a_k V_k$. If the basis is orthonormal, then we know that $a_k=<X,V_k>$ and also $||X||^2=\sum_k a_k^2$. In particular, we can say from that that $\sum_k a_k V_k$ converges at least as fast as $a_k^2$ drops for large $k$'s.

Now the question is this, if the expansion is in a basis $V_k$ that is not orthogonal, are there known results about the speed of convergence of such a series, $X=\sum_k a_k V_k$?

LaTeX formatting fixed
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edit approved Jun 18, 2014 at 8:13
leo monsaingeon
edit approved Jun 18, 2014 at 8:13
leo monsaingeon
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Well I've heard a lot of good things about mathoverflow, and so decided to ask this question that emerged in our research.

Suppose we have a finite-norm vector X$X$ in a Hilber space, and we want to construct its expansion in a certain (infinite) basis V_k$V_k$, X=sum a_k V_k$X=\sum a_k V_k$. If the basis is orthonormal, then we know that a_k=(X,V_k)$a_k=(X,V_k)$ and also |X|^2=sum a_k^2$|X|^2=\sum a_k^2$. In particular, we can say from that that sum a_k V_k$\sum a_k V_k$ converges at least as fast as a_k^2$a_k^2$ drops for large k's$k's$.

Now the question is this, if the expansion is in a basis V_k$V_k$ that is not orthogonal, are there known results about the speed of convergence of such a series, X=sum a_k V_k$X=\sum a_k V_k$?

Well I've heard a lot of good things about mathoverflow, and so decided to ask this question that emerged in our research.

Suppose we have a finite-norm vector X in a Hilber space, and we want to construct its expansion in a certain (infinite) basis V_k, X=sum a_k V_k. If the basis is orthonormal, then we know that a_k=(X,V_k) and also |X|^2=sum a_k^2. In particular, we can say from that that sum a_k V_k converges at least as fast as a_k^2 drops for large k's.

Now the question is this, if the expansion is in a basis V_k that is not orthogonal, are there known results about the speed of convergence of such a series, X=sum a_k V_k?

Well I've heard a lot of good things about mathoverflow, and so decided to ask this question that emerged in our research.

Suppose we have a finite-norm vector $X$ in a Hilber space, and we want to construct its expansion in a certain (infinite) basis $V_k$, $X=\sum a_k V_k$. If the basis is orthonormal, then we know that $a_k=(X,V_k)$ and also $|X|^2=\sum a_k^2$. In particular, we can say from that that $\sum a_k V_k$ converges at least as fast as $a_k^2$ drops for large $k's$.

Now the question is this, if the expansion is in a basis $V_k$ that is not orthogonal, are there known results about the speed of convergence of such a series, $X=\sum a_k V_k$?

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Yuriy M
Yuriy M
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Speed of convergence of vector expansions in non orthogonal basis

Well I've heard a lot of good things about mathoverflow, and so decided to ask this question that emerged in our research.

Suppose we have a finite-norm vector X in a Hilber space, and we want to construct its expansion in a certain (infinite) basis V_k, X=sum a_k V_k. If the basis is orthonormal, then we know that a_k=(X,V_k) and also |X|^2=sum a_k^2. In particular, we can say from that that sum a_k V_k converges at least as fast as a_k^2 drops for large k's.

Now the question is this, if the expansion is in a basis V_k that is not orthogonal, are there known results about the speed of convergence of such a series, X=sum a_k V_k?