2.6.2.41 Trigonometric Fourier Series

1) the piece-linear linear on the interval ([0.10]) function (F (x)), passing through points ((0; 6), (3; 6), (10; 10)), lay out on the interval ([0.10]) into the trigonometric line of functions: (left {frac {1} {2}; cos left k x} {10} ight); sin left (frac {2 pi k x} {10} ight) ight}, k = 1.2, ldots) 2) continue (f (x)) through the beginning of the coordinates even and odd and decompose in the interval ([-10; 10]) the continued function in a series of Fourier according to the corresponding system of functions. 3) Build graphs of three rows of Fourier. 4) For each row, 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 square of the difference in the difference in (l_ {2} [0.10]) between (f (x)) and the 4th partial amount of the Fourier and the square of the difference norm in (l_ {2} [-10.10]) between the continued even and the 4 partial amounts of the corresponding rows of the Fourier. Compare and explain the resulting result.