Complex form of Fourier's TheoremAny periodic function can be decomposed into a linear combination of complex exponentials.ProofConsider a complex exponential with period T 0 : e i 2 π T 0 t Hence, we have: ∫ 0 T 0 e i 2 π T 0 t d t = 0 … … ( 1 )Let us claim that for a periodic function x ( t ) we can write: x ( t ) ≈ ∑ k = − N N C k e i 2 π k T 0 t ,where 2 N + 1 is the number of frequency components used. As N → ∞ , we have: x ( t ) = ∑ k = − ∞ ∞ C k e i 2 π k T 0 t … … ( 2 )Consider: v k ( t ) := e i 2 π k T 0 t … … ( 3 )Then we have: v k ( t + T 0 ) = v k ( t )Hence, v k ( t ) is a periodic function. Now, ∫ 0 T 0 v k ( t ) v l ∗ ( t ) d t = ∫ 0 T 0 e i 2 π ( k − l ) T 0 t d t = 0 ( k ≠ l ) T 0 ( k = l ) … … ( 4 )Thus, v k ( t ) is orthogonal. Moreover, if we assume equation (2) to be valid , then we can multiply both its sides by v l ∗ ( t ) and integrate over T 0 we get: C k = 1 T 0 ∫ 0 T 0 x ( t ) v k ∗ ( t ) d t … … ( 5 )Putting this value in equation (2) we get back x ( t ). Hence, any periodic function can be decomposed into a linear combination of complex exponentials.Real form of Fourier's TheoremAn arbitrary periodic function F ( t ) with period T can be decomposed into a linear combination of the functions f n ( t ) and g n ( t ) where, f n ( t ) = sin ⁡ 2 π n t T g n ( t ) = cos ⁡ 2 π n t T Mathematically, F ( t ) = b 0 + b 1 g 1 ( t ) + b 2 g 2 ( t ) + … … + a 1 f 1 ( t ) + a 2 f 2 ( t ) + … … ,where n is a non-negative integer and all of a i , b i are real.ProblemIs there a similar proof for the real form of Fourier's Theorem as for the complex form? I couldn't get one on the internet or in any book. Does this mean that it can be derived from the complex form? If yes then how? Any help would be appreciated.

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Complex form of Fourier&#039;s TheoremAny periodic function can be decomposed into a linear combination of complex exponentials.ProofConsider a complex exponential with period       T    0  :      e          i                        2                      π                                    T          0                    t      Hence, we have:      ∫    0                  T        0                  e          i                        2                      π                                    T          0                    t        d  t  =  0  …  …  (  1  )Let us claim that for a periodic function   x  (  t  ) we can write:  x  (  t  )  ≈      ∑          k      =      −      N              N            C    k        e          i                        2                      π                    k                          T          0                    t        ,where   2  N  +  1 is the number of frequency components used. As   N  →      ∞  , we have:  x  (  t  )  =      ∑          k      =      −              ∞                            ∞                  C    k        e          i                        2                      π                    k                          T          0                    t        …  …  (  2  )Consider:      v    k    (  t  )  :=      e          i                        2                      π                    k                          T          0                    t        …  …  (  3  )Then we have:      v    k    (  t  +      T    0    )  =      v    k    (  t  )Hence,       v    k    (  t  ) is a periodic function. Now,      ∫    0                  T        0                  v    k    (  t  )      v    l    ∗    (  t  )  d  t  =      ∫    0                  T        0                  e          i                        2                      π                    (          k          −          l          )                          T          0                    t        d  t  =  0  (  k  ≠  l  )      T    0    (  k  =  l  )  …  …  (  4  )Thus,       v    k    (  t  ) is orthogonal. Moreover, if we assume equation (2) to be valid , then we can multiply both its sides by       v    l    ∗    (  t  ) and integrate over       T    0   we get:      C    k    =      1          T      0            ∫    0                  T        0              x  (  t  )      v    k    ∗    (  t  )  d  t  …  …  (  5  )Putting this value in equation (2) we get back   x  (  t  ). Hence, any periodic function can be decomposed into a linear combination of complex exponentials.Real form of Fourier&#039;s TheoremAn arbitrary periodic function   F  (  t  ) with period   T can be decomposed into a linear combination of the functions       f    n    (  t  ) and       g    n    (  t  ) where,      f    n    (  t  )  =  sin  ⁡            2              π            n      t        T        g    n    (  t  )  =  cos  ⁡            2              π            n      t        T  Mathematically,  F  (  t  )  =      b    0    +      b    1        g    1    (  t  )  +      b    2        g    2    (  t  )  +  …  …  +      a    1        f    1    (  t  )  +      a    2        f    2    (  t  )  +  …  …  ,where   n is a non-negative integer and all of       a    i  ,      b    i   are real.ProblemIs there a similar proof for the real form of Fourier&#039;s Theorem as for the complex form? I couldn&#039;t get one on the internet or in any book. Does this mean that it can be derived from the complex form? If yes then how? Any help would be appreciated.

barbesdestyle2k · Accepted Answer

The functions  1  ,  sin  ⁡  (  2  π  x      /    T  )  ,  cos  ⁡  (  2  π  x      /    T  )  ,  sin  ⁡  (  4  π  x      /    T  )  ,  cos  ⁡  (  4  π  x      /    T  )  ,  ⋯are mutually orthogonal. That is, if you choose any two different functions   f  ,  g from the above, then      ∫          −      T              T        f  (  x  )  g  (  x  )  d  x  =  0.And,      ∫          −      T              T            1    2    d  x  =  2  T        ∫          −      T              T            sin    2    ⁡  (  2  π  x      /    T  )  d  x  =  T        ∫          −      T              T            cos    2    ⁡  (  2  π  x      /    T  )  d  x  =  T  .That&#039;s enough to carry about the same argument that you used before. It should be noted that your argument assumes convergence of the series, and does not prove it.

Complex form of Fourier's Theorem Any periodic function can be decomposed into a linear combinatio

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