Mixing
Convergence of sum $f_n(x)=sum_{l,k} w_{l,n} w_{l,k} x^k$ , with $w$ expansion coefs of an orthonormal system
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Good day,
Let ${P_k}$ be a complete orthonormal system (Fourier series, Legendre-Fourier series, etc..) on interval $(a,b)$ which can be expanded into powers :
$$ P_n = sum_{k=0}^infty w_{n,k}x^k$$
It seems the following sum is always divergent:
$$f_n(x)=sum_{l,k} w_{l,n} w_{l,k} x^k$$
Is there some easy way to prove it? (assuming it is true)
Motivation comes from studying (raw-)moment expansion of a function. If $C_m$ is functional which associates $m$-th raw moment to a function $g$, then the above-mentioned functions $f_n$ would have a delta property
$$C_m(f_n)=delta_{m,n}$$
and thus would allow for an easy and elegant expansion of $g$ in terms of $f_n$. Unfortunately $f_n$ does not seem to exist.
convergence summation orthonormal orthogonal-polynomials
asked Jan 7 at 11:23
F. JatpilF. Jatpil
1347
$endgroup$
add a comment |
$begingroup$
Good day,
Let ${P_k}$ be a complete orthonormal system (Fourier series, Legendre-Fourier series, etc..) on interval $(a,b)$ which can be expanded into powers :
$$ P_n = sum_{k=0}^infty w_{n,k}x^k$$
It seems the following sum is always divergent:
$$f_n(x)=sum_{l,k} w_{l,n} w_{l,k} x^k$$
Is there some easy way to prove it? (assuming it is true)
Motivation comes from studying (raw-)moment expansion of a function. If $C_m$ is functional which associates $m$-th raw moment to a function $g$, then the above-mentioned functions $f_n$ would have a delta property
$$C_m(f_n)=delta_{m,n}$$
and thus would allow for an easy and elegant expansion of $g$ in terms of $f_n$. Unfortunately $f_n$ does not seem to exist.
convergence summation orthonormal orthogonal-polynomials
asked Jan 7 at 11:23
F. JatpilF. Jatpil
1347
$endgroup$
add a comment |
$begingroup$
Good day,
Let ${P_k}$ be a complete orthonormal system (Fourier series, Legendre-Fourier series, etc..) on interval $(a,b)$ which can be expanded into powers :
$$ P_n = sum_{k=0}^infty w_{n,k}x^k$$
It seems the following sum is always divergent:
$$f_n(x)=sum_{l,k} w_{l,n} w_{l,k} x^k$$
Is there some easy way to prove it? (assuming it is true)
Motivation comes from studying (raw-)moment expansion of a function. If $C_m$ is functional which associates $m$-th raw moment to a function $g$, then the above-mentioned functions $f_n$ would have a delta property
$$C_m(f_n)=delta_{m,n}$$
and thus would allow for an easy and elegant expansion of $g$ in terms of $f_n$. Unfortunately $f_n$ does not seem to exist.
convergence summation orthonormal orthogonal-polynomials
asked Jan 7 at 11:23
F. JatpilF. Jatpil
1347
$endgroup$
Good day,
Let ${P_k}$ be a complete orthonormal system (Fourier series, Legendre-Fourier series, etc..) on interval $(a,b)$ which can be expanded into powers :
$$ P_n = sum_{k=0}^infty w_{n,k}x^k$$
It seems the following sum is always divergent:
$$f_n(x)=sum_{l,k} w_{l,n} w_{l,k} x^k$$
Is there some easy way to prove it? (assuming it is true)
Motivation comes from studying (raw-)moment expansion of a function. If $C_m$ is functional which associates $m$-th raw moment to a function $g$, then the above-mentioned functions $f_n$ would have a delta property
$$C_m(f_n)=delta_{m,n}$$
and thus would allow for an easy and elegant expansion of $g$ in terms of $f_n$. Unfortunately $f_n$ does not seem to exist.
convergence summation orthonormal orthogonal-polynomials
convergence summation orthonormal orthogonal-polynomials
asked Jan 7 at 11:23
F. JatpilF. Jatpil
1347
asked Jan 7 at 11:23
F. JatpilF. Jatpil
1347
edited Jan 9 at 8:08
F. Jatpil
asked Jan 7 at 11:23
F. JatpilF. Jatpil
1347
asked Jan 7 at 11:23
F. JatpilF. Jatpil
1347
asked Jan 7 at 11:23
F. JatpilF. Jatpil
1347
1347
add a comment |
add a comment |
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