If f(x) and g(x) both satisfy the conditions of theorem 18, is theFourier series

Question

If f(x) and g(x) both satisfy the conditions of theorem 18, is theFourier series of f(x) + g(x) on (-L, L) the sum of the Fourierseries of f(x) and g(x) on the interval? Give reasons for youranswers.

Theorem 18:

If the function f and its derivative f' are piecewise continuousover the interval -L < x < L, then f equals its Fourierseries at all points of continuity. At a point c where a jumpdiscontinuity occurs in f, the Fourier series converges to theaverage
(f(c+ ) + f (c-) )/ 2

where f(c+ ) and f (c-)denote the right andleft limits of f at c respectively.

Explanation / Answer

I'm going to write F(f) to mean the Fourier series of f(x). I'm assuming that "the conditions of theorem 18" are that eachfunction and its derivative are piecewise continuous over thatinterval. Theorem 18 says that if those conditions are met, then f= F(f) at all points of continuity. Surely, if we begin byestablishing that f = F(f) and g = F(g), then f + g = F(f) + F(g).So, in continuous regions of both f(x) and g(x), the sum of thefunctions equals the sum of the Fourier series of thefunctions. I'm not sure how to deal with the jump discontinuity issue, though.It doesn't make sense to talk about the value of a function at adiscontinuous point, so I'm not sure how to talk about a sum of twovalues, at least one of which may be at a discontinuity. If f(x)and/or g(x) has discontinuities, the statement of your problemcan't be generally true.

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