By Bisac J.
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A 23, p. 166f). 3. THEOREM 3 (Fatou's lemma). If a sequence of measurable and non-negative functions fx(x), f2(x), . . , / „ ( x ) , . . 2) j F(x) dx < infi jfn(x) dx\ E \ \E (see Natanson, réf. A23, p. 155). THEOREM 4. Iffi(x),f2(x), . . , / „ ( x ) , . . 1) holds in the sense that both terms of the equality become equal to + oo. Indeed, iff(x) is summable, then this assertion immediately follows from Theorem 2. Even if f(x) is not summable, then, supposing (f)N = f{x) at f{x) < N and (f)N = Natf(x) > N, we see that j (f)N dx -» oo as N -> oo.
2) and substituting the values for an and bn from the Fourier formulae or in a similar manner to that by which the Fourier formulae themselves were produced. 3) by e~inx and integrating term by term, we find that jf(x)e-inxdx —n But rn = Σ ck j k— — oo —n ( 0, eHk~n)xdx. if k φ n, , x t The free term of the series must be written in the form a0/2 for a0 to be obtained from an when /i=0. t t Strictly speaking, these formulae were already known to Euler, but Fourier began to use them systematically; therefore they are traditionally called Fourier formulae and the corresponding series Fourier series.
3) (see Notation for definition of norm). 3). ~j. 4) 19 HOLDERS INEQUALITY Note 1. Supposing p = q = 2, we obtain the well-known Bunyakovskii's inequality ß ]\f\2dx\ I f (x)