Showing property for the derivative ∂ x T of a trigonometric polynomialLet be T = ∑ n = 0 N T ^ ( n ) e i n x a trigonometric polynomial of grade N without negative frequencies.I wanna show that ∂ x T = − i N ( F N ∗ T − T )Where F N ∗ T meas the convolution of the Fejer Kernel and T.might be easy..but I just can't work out the right conversion for this property..SO ∂ x T = ∑ n = 0 N T ^ ( n ) i n e i n x = ∑ n = 0 N ( 1 2 π ∫ − π π T ( y ) e − i n y d y ) e i n x i n = 1 2 π ∫ − π π T ( y ) ( ∑ n = 0 N e i n ( x − y ) ) i nfrom there on I get carried away in the wrong direction. is the derivative right?Also..the Fejer Kernel can be expressed as the mean arithmetic value of the dirichtlet kernel so: F N = 1 n + 1 ∑ k = 0 n D k ( x )Where D k = ∑ n = − k k e i n x is the Dirichtlet Kernel

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Showing property for the derivative       ∂    x    T of a trigonometric polynomialLet be  T  =      ∑          n      =      0        N              T      ^        (  n  )      e          i      n      x      a trigonometric polynomial of grade N without negative frequencies.I wanna show that      ∂    x    T  =  −  i  N  (      F    N    ∗  T  −  T  )Where       F    N    ∗  T meas the convolution of the Fejer Kernel and T.might be easy..but I just can&#039;t work out the right conversion for this property..SO      ∂    x    T  =      ∑          n      =      0        N              T      ^        (  n  )  i  n      e          i      n      x        =      ∑          n      =      0        N    (      1          2      π            ∫          −      π              π        T  (  y  )      e          −      i      n      y        d  y  )      e          i      n      x        i  n  =      1          2      π            ∫          −      π              π        T  (  y  )  (      ∑          n      =      0        N        e          i      n      (      x      −      y      )        )  i  nfrom there on I get carried away in the wrong direction. is the derivative right?Also..the Fejer Kernel can be expressed as the mean arithmetic value of the dirichtlet kernel so:      F    N    =      1          n      +      1            ∑          k      =      0        n        D    k    (  x  )Where       D    k    =      ∑          n      =      −      k        k        e          i      n      x       is the Dirichtlet Kernel

yen1291kp6 · Accepted Answer

Step 11). The last equality you wrote for the derivative is weird, since you have the in outside a term that involves summing in n, so that&#039;s a mistake.2). The definition of       F    N   should involved &#039;N&#039; rather than &#039;n&#039;.3). Either FN should be defined to be       1    N        ∑          k      =      0              N      −      1            D    k   or the result you wish to prove should be       ∂    x    T  =  −  i  (  N  +  1  )  (      F    N    ∗  T  −  T  ).I will keep your definition of       F    N   (with point (2) taken into account of course) and prove ∂xT=−i(N+1)(FN∗T−T). We show the fourier transform of both sides is the same -- this suffices.Step 2Note   [  −  i  (  N  +  1  )  (      F    N    ∗  T  −  T  )  ]                  ^        (  m  )  =  −  i  (  N  +  1  )  [                    F        N            ^        (  m  )            T      ^        (  m  )  −            T      ^        (  m  )  ] (the latter you can see from your first equality for the derivative). So, immediately, if   m  &amp;lt;  0 or if   m  ≥  N  +  1, both sides are 0 and we&#039;re good. So suppose   0  ≤  m  ≤  N. Then,                     F        N            ^        (  m  )  =      1          N      +      1            ∑          k      =      0              N      +      1                          D        k            ^        (  m  )  =      1          N      +      1            ∑          k      =      0              N      +      1            1          m      ≤      k        =      1          N      +      1        [  N  +  1  −  m  ], so we get   [  −  i  (  N  +  1  )  (      F    N    ∗  T  −  T  )  ]                  ^        (  m  )  =  −  i  (  N  +  1  )            T      ^        (  m  )            −      m              N      +      1        =  i  m            T      ^        (  m  ), as desired.

Showing property for the derivative partial_x T of a trigonometric polynomial

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