Fourier Series

               Fourier Series 

Definition :

          If the numbers a₀,a₁....aₙ,....,b₁,...bₙ....

are derived from a function f by means of Euler_Fourier formulas :

                          π

           aₙ = 1/π ∫ f(x) cosnx dx ,n=0,1,2...

                        -π                                 .......(1)

                         π

          bₙ = 1/π ∫ f(x) sinnx dx , n=0,1,2...

                        -π

then the series 

                        ∞

          1/2 a₀ = Σ (aₙcosnx + bₙsinnx) ....(2)

                       n=1

  is called the Fourier Series of f or the Fourier Series Generated by f, and the co_efficient aₙ,bₙ defined by equation(1) as the Fourier Co_efficients of f .

                            ∞

  Where 1/2 a₀+ Σ (aₙcosnx + bₙsinnx)

                           n=1

is  called Trigonometric Series .

Explanation:

     It is to be noted that the Fourier Co_efficients have been obtained purely on the assumption that the function f is bounded and integrable on [-π,π] .

      Therefore the Fourier Series equation(2) is convergent .In fact the series may not converge at all , and even if it converges , the sum may not be f , though it often is and there is some justification for the hope that the series may converge uniformly , the definitions of Fourier constants suggest that its sum will be f , and that f is capable of a unique Fourier series expansion .

         Therefore, by Weierstrass's M_test , it follows that , if the series

                ∞

                Σ (|aₙ|+|bₙ|) converges , then

              n=1

the trigonometric series converges absolutely and uniformly on every closed interval [a,b] and that it is the Fourier Series of a continuous 2π_periodic function . We know that a periodic function of period λ is such that 

    f(x ± λ) = f(x) , for all x .