Then, l (x0 ) does not depend on the choice of the family (Oi )i I in .NET

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Then, l (x0 ) does not depend on the choice of the family (Oi )i I
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This result makes it possible to de ne the local exponent by using any intervals family containing x0
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DEFINITION 13 Let f be a function de ned in a neighborhood of x0 Let {In }n N be a decreasing sequence of open intervals converging to x0 By de nition, the local H lder exponent of f at x0 , noted l (x0 ), is: l (x0 ) = sup l (In ) = lim l (In )
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Let us brie y note that the local exponent is related to a notion of critical exponent of fractional derivation [KOL 01] We may understand the difference between p and l as follows: let us suppose that there exists a single couple (y, z) such that (f, B(x, )) = f (y) f (z) Then p results from the comparison between (f, B(x, )) and , whereas for l , we compare (f, B(x, )) to |y z| This is particularly clear in the case of the chirp, where the distance between the points (y, z) realizing the oscillation tends to zero much faster than the size of the ball around 0 Accordingly, it is easy to demonstrate that l (0) = /(1 + ) for the chirp The exponent l thus sees oscillations around 0: for xed , the chirp is more irregular (in the sense of l ) when is larger The local exponent possesses an advantage over p : it is stable under the action of pseudo-differential operators However, as well as p , l cannot by itself completely characterize the irregularity around a point Moreover, l is, in a certain sense, less precise than p The results presented below support this assertion PROPOSITION 12 For any continuous f and for all x:
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The following two theorems describe the structure of the H lder functions, ie the functions which associate with any x the exponents of f at x THEOREM 14 Let g : R R+ be a function The two assertions below are equivalent: g is the lower limit of a sequence of continuous functions; there exists a continuous function f whose pointwise H lder function p (x) satis es p (x) = g(x) for all x THEOREM 15 Let g : R R+ be a function The following two assertions are equivalent: g is a lower semi-continuous (LSC) function; there exists a continuous function f whose local H lder function l (x) satis es l (x) = g(x) for any x
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NOTE 12 Let us recall that a function f : D R R is LSC if, for any x D and for any sequence (xn ) in D tending to x: lim inf f (xn )
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Figure 12 shows a generalization of the Weierstrass function de ned on [0, 1] for which p (x) = l (x) = x for any x This function is de ned as Wg (x) = i=0 nx cos( n x), with > 1
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Figure 12 Generalized Weierstrass function for which p (x) = l (x) = x
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Since the class of lower limits of continuous function is much larger than that of lower semi-continuous functions, we observe that p generally supplies more information than l For example, p can vary much faster than l In particular, it is possible to construct a continuous function whose pointwise H lder function coincides with the indicator function of the set of rational numbers It is everywhere discontinuous, and thus its local H lder function is constantly equal to 0 The following results describe more precisely the relations between l and p PROPOSITION 13 Let f : I R be a continuous function, and assume that there exists > 0 such that f C (I) Then, there exists a subset D of I such that: D is dense, uncountable and has Hausdorff dimension zero; for any x D, p (x) = l (x)
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