Fourier Series

absolutely integrable function arithmetic means b₁ Bessel's inequality boundary conditions boundary value problem C₁ C₂ const constant continuous function converges to f(x converges uniformly du(x dx dy eigenfunctions equation 1.1 exists Expand extension of f(x finite number follows formula Fourier coefficients Fourier series Fourier-Bessel series function defined function of period graph hence implies inequality initial conditions integrable function f(x interval left-hand lemma Let f(x membrane mx dx nodal lines number of points nx dx obtain obviously odd function orthogonal system partial sums periodic extension piecewise smooth point of discontinuity Proof prove right-hand series converges series of f(x sin² smooth function square integrable function string sufficiently large term by term trigonometric series uniformly convergent vanishes vibrations whole x-axis zero π π ди Сп