2.6.2.42 Trigonometric Fourier series

1) Piecewise linear on the interval ( [0,10] ) function ( f (x) ), passing through the points ( (0 ; 10),(7 ; 7), (10 ; 7) ), expand on the interval ( [0,10] ) into a trigonometric Fourier series according to the system of functions: ( left{frac{1}{2} ; cos left(frac{2 pi k x}{10} ight) ; sin left(frac{2 pi k x}{10} ight) ight}, k=1,2, ldots ) 2) Continue ( f(x) ) through the origin in an even and odd manner and expand the extended function on the interval ( [-10 ; 10] ) into a Fourier series in the corresponding system of functions. 3) Construct graphs of three Fourier series. 4) For each series, find the values ​​of the Fourier coefficients ( a_{0}, a_{1}, a_{2}, a_{3}, b_{1}, b_{2}, b_{3}, b_{4} ). Calculate the squared norm of the difference in ( L_{2}[0,10] ) between ( f(x) ) and the 4th partial sum of the Fourier series and the squared norm of the difference in ( L_{2}[-10,10] ) between even and oddly extended functions and the 4th partial sums of the corresponding Fourier series. Compare and explain the results obtained.