Representing rectangular function using divided differences.












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I was reading the chapter 9 of A practical guide to splines (de Boor, C.) were the autor defines a B-spline of order 1, basically a rectangular function i.e. a function taking value $1$ if the variable $x$ is contained in the interval $(t_j, t_{j+1})$ and $0$ elsewhere using divided differences as follows:



$B_1(x)=(t_{j+1}-t_j)[t_j,t_{j+1}](cdot -x)^0_+$



where $[t_j,...,t_{j+k}]f$ indicates the k-th divided difference of the function $f$ and $(cdot)_+$ indicates the positive part of the argument (if the argument is negative then the function returns as result $0$).



This definition is okey to me, basically we have that $B_1(x)=1$ if $t_j<x<t_{j+1}$.



Then the autor says that another way to write it following the definition of divided differences is:



$B_1(x)=(cdot-t_{j+1})_+^0-(cdot-t_{j})_+^0$



Following this definition I obtain that, assuming (as the autor makes) $t_j<t_{j+1}$, the first term will be $1$ if $x>t_{j+1}$ and given our assumption also the second term will be $1$ leading to a result of $0$.
If instead $t_j<x<t_{j+1}$ the first term will be $0$ and the second term will be $1$, but this is a rectangular function with a minus in front.



Am I missing something?










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    I was reading the chapter 9 of A practical guide to splines (de Boor, C.) were the autor defines a B-spline of order 1, basically a rectangular function i.e. a function taking value $1$ if the variable $x$ is contained in the interval $(t_j, t_{j+1})$ and $0$ elsewhere using divided differences as follows:



    $B_1(x)=(t_{j+1}-t_j)[t_j,t_{j+1}](cdot -x)^0_+$



    where $[t_j,...,t_{j+k}]f$ indicates the k-th divided difference of the function $f$ and $(cdot)_+$ indicates the positive part of the argument (if the argument is negative then the function returns as result $0$).



    This definition is okey to me, basically we have that $B_1(x)=1$ if $t_j<x<t_{j+1}$.



    Then the autor says that another way to write it following the definition of divided differences is:



    $B_1(x)=(cdot-t_{j+1})_+^0-(cdot-t_{j})_+^0$



    Following this definition I obtain that, assuming (as the autor makes) $t_j<t_{j+1}$, the first term will be $1$ if $x>t_{j+1}$ and given our assumption also the second term will be $1$ leading to a result of $0$.
    If instead $t_j<x<t_{j+1}$ the first term will be $0$ and the second term will be $1$, but this is a rectangular function with a minus in front.



    Am I missing something?










    share|cite|improve this question











    $endgroup$















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      0








      0





      $begingroup$


      I was reading the chapter 9 of A practical guide to splines (de Boor, C.) were the autor defines a B-spline of order 1, basically a rectangular function i.e. a function taking value $1$ if the variable $x$ is contained in the interval $(t_j, t_{j+1})$ and $0$ elsewhere using divided differences as follows:



      $B_1(x)=(t_{j+1}-t_j)[t_j,t_{j+1}](cdot -x)^0_+$



      where $[t_j,...,t_{j+k}]f$ indicates the k-th divided difference of the function $f$ and $(cdot)_+$ indicates the positive part of the argument (if the argument is negative then the function returns as result $0$).



      This definition is okey to me, basically we have that $B_1(x)=1$ if $t_j<x<t_{j+1}$.



      Then the autor says that another way to write it following the definition of divided differences is:



      $B_1(x)=(cdot-t_{j+1})_+^0-(cdot-t_{j})_+^0$



      Following this definition I obtain that, assuming (as the autor makes) $t_j<t_{j+1}$, the first term will be $1$ if $x>t_{j+1}$ and given our assumption also the second term will be $1$ leading to a result of $0$.
      If instead $t_j<x<t_{j+1}$ the first term will be $0$ and the second term will be $1$, but this is a rectangular function with a minus in front.



      Am I missing something?










      share|cite|improve this question











      $endgroup$




      I was reading the chapter 9 of A practical guide to splines (de Boor, C.) were the autor defines a B-spline of order 1, basically a rectangular function i.e. a function taking value $1$ if the variable $x$ is contained in the interval $(t_j, t_{j+1})$ and $0$ elsewhere using divided differences as follows:



      $B_1(x)=(t_{j+1}-t_j)[t_j,t_{j+1}](cdot -x)^0_+$



      where $[t_j,...,t_{j+k}]f$ indicates the k-th divided difference of the function $f$ and $(cdot)_+$ indicates the positive part of the argument (if the argument is negative then the function returns as result $0$).



      This definition is okey to me, basically we have that $B_1(x)=1$ if $t_j<x<t_{j+1}$.



      Then the autor says that another way to write it following the definition of divided differences is:



      $B_1(x)=(cdot-t_{j+1})_+^0-(cdot-t_{j})_+^0$



      Following this definition I obtain that, assuming (as the autor makes) $t_j<t_{j+1}$, the first term will be $1$ if $x>t_{j+1}$ and given our assumption also the second term will be $1$ leading to a result of $0$.
      If instead $t_j<x<t_{j+1}$ the first term will be $0$ and the second term will be $1$, but this is a rectangular function with a minus in front.



      Am I missing something?







      recurrence-relations spline






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      share|cite|improve this question













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      edited Dec 6 '18 at 16:17







      Ramiro Scorolli

















      asked Dec 6 '18 at 16:11









      Ramiro ScorolliRamiro Scorolli

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      637114






















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