Fourier Transform of B-splines linear combination, any application to spectral analysis of non stationary...
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In a framework of non-linear regression one could opt for using B-splines in order to approximate the values $y$ assumed by an unknown function $f(x)$. If we denote with $B_j(x,p)$ the value of the jth B-spline of degree p at point x we cab represent y using a linear combination of B-splines:
$$hat y=sum_j hat c_jB_j(x,p)$$
If I am interesting in estimating the Fourier Transform we would have:
$$mathscr{F}{hat y}=mathscr{F}{sum_j hat c_j B_j(x,p)}=sum_jhat c_jtimesmathscr{F}{B_j(x,p)}$$ and since $B_j(x,p)$ (with uniform knots) are defined as convultions the last expression could be written as:
$$sum_jhat c_jtimesmathscr{F}{B_j(x,0)}^p$$
were:
$B_j(x,0)= begin{cases}1 & t_i leqslant x < t_{i+1} \ 0 & otherwise end{cases}$
I am aware that the B-spline of order $0$ is equal to a Rectangular
function even though I do not see how to write the Fourier Transform
since we have different knots (equidistant).
Finally , I know that spectral analysis is not suitable for
non-stationary applications but could this method be useful when
dealing with spectral analysis of non-stationary series?
functional-analysis fourier-analysis fourier-transform convolution nonlinear-analysis
asked Nov 20 at 19:29
Ramiro Scorolli
63813
add a comment |
up vote
0
down vote
favorite
In a framework of non-linear regression one could opt for using B-splines in order to approximate the values $y$ assumed by an unknown function $f(x)$. If we denote with $B_j(x,p)$ the value of the jth B-spline of degree p at point x we cab represent y using a linear combination of B-splines:
$$hat y=sum_j hat c_jB_j(x,p)$$
If I am interesting in estimating the Fourier Transform we would have:
$$mathscr{F}{hat y}=mathscr{F}{sum_j hat c_j B_j(x,p)}=sum_jhat c_jtimesmathscr{F}{B_j(x,p)}$$ and since $B_j(x,p)$ (with uniform knots) are defined as convultions the last expression could be written as:
$$sum_jhat c_jtimesmathscr{F}{B_j(x,0)}^p$$
were:
$B_j(x,0)= begin{cases}1 & t_i leqslant x < t_{i+1} \ 0 & otherwise end{cases}$
I am aware that the B-spline of order $0$ is equal to a Rectangular
function even though I do not see how to write the Fourier Transform
since we have different knots (equidistant).
Finally , I know that spectral analysis is not suitable for
non-stationary applications but could this method be useful when
dealing with spectral analysis of non-stationary series?
functional-analysis fourier-analysis fourier-transform convolution nonlinear-analysis
asked Nov 20 at 19:29
Ramiro Scorolli
63813
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
In a framework of non-linear regression one could opt for using B-splines in order to approximate the values $y$ assumed by an unknown function $f(x)$. If we denote with $B_j(x,p)$ the value of the jth B-spline of degree p at point x we cab represent y using a linear combination of B-splines:
$$hat y=sum_j hat c_jB_j(x,p)$$
If I am interesting in estimating the Fourier Transform we would have:
$$mathscr{F}{hat y}=mathscr{F}{sum_j hat c_j B_j(x,p)}=sum_jhat c_jtimesmathscr{F}{B_j(x,p)}$$ and since $B_j(x,p)$ (with uniform knots) are defined as convultions the last expression could be written as:
$$sum_jhat c_jtimesmathscr{F}{B_j(x,0)}^p$$
were:
$B_j(x,0)= begin{cases}1 & t_i leqslant x < t_{i+1} \ 0 & otherwise end{cases}$
I am aware that the B-spline of order $0$ is equal to a Rectangular
function even though I do not see how to write the Fourier Transform
since we have different knots (equidistant).
Finally , I know that spectral analysis is not suitable for
non-stationary applications but could this method be useful when
dealing with spectral analysis of non-stationary series?
functional-analysis fourier-analysis fourier-transform convolution nonlinear-analysis
asked Nov 20 at 19:29
Ramiro Scorolli
63813
In a framework of non-linear regression one could opt for using B-splines in order to approximate the values $y$ assumed by an unknown function $f(x)$. If we denote with $B_j(x,p)$ the value of the jth B-spline of degree p at point x we cab represent y using a linear combination of B-splines:
$$hat y=sum_j hat c_jB_j(x,p)$$
If I am interesting in estimating the Fourier Transform we would have:
$$mathscr{F}{hat y}=mathscr{F}{sum_j hat c_j B_j(x,p)}=sum_jhat c_jtimesmathscr{F}{B_j(x,p)}$$ and since $B_j(x,p)$ (with uniform knots) are defined as convultions the last expression could be written as:
$$sum_jhat c_jtimesmathscr{F}{B_j(x,0)}^p$$
were:
$B_j(x,0)= begin{cases}1 & t_i leqslant x < t_{i+1} \ 0 & otherwise end{cases}$
I am aware that the B-spline of order $0$ is equal to a Rectangular
function even though I do not see how to write the Fourier Transform
since we have different knots (equidistant).
Finally , I know that spectral analysis is not suitable for
non-stationary applications but could this method be useful when
dealing with spectral analysis of non-stationary series?
functional-analysis fourier-analysis fourier-transform convolution nonlinear-analysis
functional-analysis fourier-analysis fourier-transform convolution nonlinear-analysis
asked Nov 20 at 19:29
Ramiro Scorolli
63813
asked Nov 20 at 19:29
Ramiro Scorolli
63813
asked Nov 20 at 19:29
Ramiro Scorolli
63813
asked Nov 20 at 19:29
Ramiro Scorolli
63813
asked Nov 20 at 19:29
Ramiro Scorolli
63813
63813
add a comment |
add a comment |
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