By Kurt Mahler
Notre Dame Mathematical Lectures, No. 7.
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Additional resources for Lectures on diophantine approximations. g-adic numbers and Roth's theorem
Be a bounded sequence in Lp(T), 1 < p S 00, with bound M. Then there exist a subsequence nk and an Lp functionf, Ilfll, S M, such that for each g E Lp’(T), l/p + l/p’ = 1 (here p’ = 1 when p = 00). 5) holds, we say that fn, converges weakly to f in Lp(T). Prouf. We divide the proof into three steps: (i) There is a subsequence nk such that for all trigonometric polynomials g with rational coefficients exists. , all h E LP’(T). II. Cesciro Summability 36 (iii) (Riesz Representation theorem). Each bounded linear functional on LP'(T),as in (ii), can be represented as 'I L(h)=2T h ( t ) f ( t )dt, T M.
Proof. We may assume that L = 0 by replacing cj by cj - L if necessary. Observe that the cj’s have the following properties (i) lcjl s K , all j (ii) Given E > 0, there is a j o such that lcjl S 28 E provided j 3 jo. 2. Fejkr 's Kernel 29 It is now a simple matter to estimate the Cj's. Indeed Therefore, by first picking E, thus fixing j o , and then letting j + 00, we see that lC,l can be made arbitrarily small for j large. W Observe that the oscillating sequence cj = 1 + (-1)' has limit 1 in the (C, 1) sense.
13) holds for an arbitrary h E Lp'(T). But this is easy since, for a sequence of trigonometric polynomials h, converging to h in Lp',we have on one hand that L(h,) = L ( h ) and on the other hand, by Holder's inequality, that 'I lim n+a,2m T ( h ( t ) - h,,(t))f( t ) dt = 0. We are now in a position to characterize those trigonometric series which are the Fourier series of Lp(T) functions, 1 < p e 00. 3 (FejCr). , Ilu,,llp=sK. In this case ~ ~ K. f ~ ~ p ~ Proof. 4) it is enough to show the sufficiency of the statement.