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







share|cite|improve this question



























    0














    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







    share|cite|improve this question

























      0












      0








      0







      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







      share|cite|improve this question













      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






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Nov 20 '18 at 19:29









      Ramiro Scorolli

      655113




      655113






















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