Fourier Series

a₁ a²u analogous approximated b₁ boundary conditions bounded variation C₁ C₂ calculation Cauchy sequence Chapter complete and orthogonal complete orthogonal systems complex numbers consequently considered constant converges uniformly defined Denote derivative easily verify eigenfunctions Example exists expansion f₁ find the Fourier follows formula Fourier coefficients Fourier series Fourier transform function f(x ƒ and g H₁ Hence Hilbert space Hint inequality integrable function kernel L₁ L₁(a L₂ Let f(x lim f(x limit non-negative norm nx dx obtain orthogonal system orthonormal system Parseval's equality partial sums periodic function periodic with period Problem proof of Theorem properties prove reader real number Remark right hand side scalar product Section series converges sin nx space H space L₂(a term by term trigonometric polynomial trigonometric series vanish vectors weight function zero π π ди