1.6. Differentiation theorems 113 We are taking f to be complex valued, but it is clear from taking real and imaginary parts that it suﬃces to prove the claim when f is real-valued, and we shall thus assume this for the rest of the argument. The conclusion (1.20) we want to prove is a convergence theorem—an assertion that for all functions f in a given class (in this case, the class of absolutely integrable functions f : R → R), a certain sequence of linear expressions Thf (in this case, the right averages Thf(x) = 1 h [x,x+h] f(t) dt) converge in some sense (in this case, pointwise almost everywhere) to a specified limit (in this case, f). There is a general and very useful argument to prove such convergence theorems, known as the density argument. This argument requires two ingredients, which we state informally as follows: (i) A verification of the convergence result for some “dense subclass” of “nice” functions f, such as continuous functions, smooth functions, simple functions, etc. By “dense”, we mean that a general function f in the original class can be approximated to arbitrary accuracy in a suitable sense by a function in the nice subclass. (ii) A quantitative estimate that upper bounds the maximal fluctuation of the linear expressions Thf in terms of the “size” of the function f (where the precise definition of “size” depends on the nature of the approximation in the first ingredient). Once one has these two ingredients, it is usually not too hard to put them together to obtain the desired convergence theorem for general functions f (not just those in the dense subclass). We illustrate this with a simple example: Proposition 1.6.13 (Translation is continuous in L1). Let f : Rd → C be an absolutely integrable function, and for each h ∈ Rd, let fh : Rd → C be the shifted function fh(x) := f(x − h). Then fh converges in L1 norm to f as h → 0, thus lim h→0 Rd |fh(x) − f(x)| dx = 0. Proof. We first verify this claim for a dense subclass of f, namely the func- tions f which are continuous and compactly supported (i.e. they vanish outside of a compact set). Such functions are continuous, and thus fh con- verges uniformly to f as h → 0. Furthermore, as f is compactly supported, the support of fh −f stays uniformly bounded for h in a bounded set. From this we see that fh also converges to f in L1 norm as required.

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