Mixing
Decomposing functions to Taylor-Fourier series
$begingroup$
A great many functions can be expressed as a series of the form
$$ U_0(x) + U_1(x) x + U_2(x) frac{1}{2!}x(x-1) + ... $$
Where $U_r(x)$ are integrable periodic functions with period $1$. Call such functions "1 periodic normal" functions. Note that the $U_r(x)$ being periodic can be decomposed into their fourier series as:
$$ U_r(x) = sum_{k=-infty}^{infty} a_{r,k} e^{2pi i k x} $$
And so 1-periodic normal functions have a general form as:
$$ sum_{k=-infty}^{infty} a_{0,k} e^{2pi i k x} + left( sum_{k=-infty}^{infty} a_{1,k} e^{2pi i k x} right) x + ... $$
In the event that $U_1, U_2 ... $ are equal to $0$ it follows that we can use fourier analysis to determine the coefficients of $U_0$.
In particular when $U_1, U_2 ... $ are equal to 0, then the operator
$$ f rightarrow 2 int_{0}^{1}f(x) e^{ipi Jx} dx $$
Gives the coefficient $a_{j,0}$ of our series.
Suppose we have no guarantees about non-zero $U_r$ how could we systematically determine the $a_{j,r}$ coefficients of our series?
complex-analysis fourier-analysis recurrence-relations
edited Jan 24 at 23:37
Bernard
123k741116
asked Jan 24 at 23:33
frogeyedpeasfrogeyedpeas
7,59972054
$endgroup$
add a comment |
$begingroup$
A great many functions can be expressed as a series of the form
$$ U_0(x) + U_1(x) x + U_2(x) frac{1}{2!}x(x-1) + ... $$
Where $U_r(x)$ are integrable periodic functions with period $1$. Call such functions "1 periodic normal" functions. Note that the $U_r(x)$ being periodic can be decomposed into their fourier series as:
$$ U_r(x) = sum_{k=-infty}^{infty} a_{r,k} e^{2pi i k x} $$
And so 1-periodic normal functions have a general form as:
$$ sum_{k=-infty}^{infty} a_{0,k} e^{2pi i k x} + left( sum_{k=-infty}^{infty} a_{1,k} e^{2pi i k x} right) x + ... $$
In the event that $U_1, U_2 ... $ are equal to $0$ it follows that we can use fourier analysis to determine the coefficients of $U_0$.
In particular when $U_1, U_2 ... $ are equal to 0, then the operator
$$ f rightarrow 2 int_{0}^{1}f(x) e^{ipi Jx} dx $$
Gives the coefficient $a_{j,0}$ of our series.
Suppose we have no guarantees about non-zero $U_r$ how could we systematically determine the $a_{j,r}$ coefficients of our series?
complex-analysis fourier-analysis recurrence-relations
edited Jan 24 at 23:37
Bernard
123k741116
asked Jan 24 at 23:33
frogeyedpeasfrogeyedpeas
7,59972054
$endgroup$
add a comment |
$begingroup$
A great many functions can be expressed as a series of the form
$$ U_0(x) + U_1(x) x + U_2(x) frac{1}{2!}x(x-1) + ... $$
Where $U_r(x)$ are integrable periodic functions with period $1$. Call such functions "1 periodic normal" functions. Note that the $U_r(x)$ being periodic can be decomposed into their fourier series as:
$$ U_r(x) = sum_{k=-infty}^{infty} a_{r,k} e^{2pi i k x} $$
And so 1-periodic normal functions have a general form as:
$$ sum_{k=-infty}^{infty} a_{0,k} e^{2pi i k x} + left( sum_{k=-infty}^{infty} a_{1,k} e^{2pi i k x} right) x + ... $$
In the event that $U_1, U_2 ... $ are equal to $0$ it follows that we can use fourier analysis to determine the coefficients of $U_0$.
In particular when $U_1, U_2 ... $ are equal to 0, then the operator
$$ f rightarrow 2 int_{0}^{1}f(x) e^{ipi Jx} dx $$
Gives the coefficient $a_{j,0}$ of our series.
Suppose we have no guarantees about non-zero $U_r$ how could we systematically determine the $a_{j,r}$ coefficients of our series?
complex-analysis fourier-analysis recurrence-relations
edited Jan 24 at 23:37
Bernard
123k741116
asked Jan 24 at 23:33
frogeyedpeasfrogeyedpeas
7,59972054
$endgroup$
A great many functions can be expressed as a series of the form
$$ U_0(x) + U_1(x) x + U_2(x) frac{1}{2!}x(x-1) + ... $$
Where $U_r(x)$ are integrable periodic functions with period $1$. Call such functions "1 periodic normal" functions. Note that the $U_r(x)$ being periodic can be decomposed into their fourier series as:
$$ U_r(x) = sum_{k=-infty}^{infty} a_{r,k} e^{2pi i k x} $$
And so 1-periodic normal functions have a general form as:
$$ sum_{k=-infty}^{infty} a_{0,k} e^{2pi i k x} + left( sum_{k=-infty}^{infty} a_{1,k} e^{2pi i k x} right) x + ... $$
In the event that $U_1, U_2 ... $ are equal to $0$ it follows that we can use fourier analysis to determine the coefficients of $U_0$.
In particular when $U_1, U_2 ... $ are equal to 0, then the operator
$$ f rightarrow 2 int_{0}^{1}f(x) e^{ipi Jx} dx $$
Gives the coefficient $a_{j,0}$ of our series.
Suppose we have no guarantees about non-zero $U_r$ how could we systematically determine the $a_{j,r}$ coefficients of our series?
complex-analysis fourier-analysis recurrence-relations
complex-analysis fourier-analysis recurrence-relations
edited Jan 24 at 23:37
Bernard
123k741116
asked Jan 24 at 23:33
frogeyedpeasfrogeyedpeas
7,59972054
edited Jan 24 at 23:37
Bernard
123k741116
asked Jan 24 at 23:33
frogeyedpeasfrogeyedpeas
7,59972054
edited Jan 24 at 23:37
Bernard
123k741116
edited Jan 24 at 23:37
Bernard
123k741116
edited Jan 24 at 23:37
Bernard
123k741116
123k741116
asked Jan 24 at 23:33
frogeyedpeasfrogeyedpeas
7,59972054
asked Jan 24 at 23:33
frogeyedpeasfrogeyedpeas
7,59972054
asked Jan 24 at 23:33
frogeyedpeasfrogeyedpeas
7,59972054
7,59972054
add a comment |
add a comment |
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