Fourier versus Wavelets: a Simple Approach to Lipschitz Regularity

We give a very simple proof of the caracterization of Lipschitz regularity of a function by the size of its Haar coefficients. It is well known that given a real function I periodic with period 27r satisfying a Lipschitz 0: condition for 0 < 0: ::::; 1, its kth Fourier coefficient is bounded by Ikl-a . More precisely, the following result holds (see for example Chapter 12 of [9]). (A) Let I be a 27r periodic real function satisfying a Lipschitz 0: condition for 0 < 0: ::::; 1, i. e., there exists a positive finite constant M such that, I I( x + h) I( x ) I ::::; Mlhl a , for every pair of real numbers hand x. Then, there exists a constant C such thai, for every k E LZ, ICd!ll::::; Clkla , where Cd!] = 217r J0271: I(x)e-ikxdx. The result is an easy consequence of the fact that J027r e-ikxdx = 0, for k f. O. Nevertheles, it does not constitute a characterization of Lipschitz 0:. This fact can easily be observed by t.aking the Fourier coefficients of the char~cteristic function of a subinterval of [0, 27r]. Moreover there is no way to characterize the regularity of a function in terms of the size of its Fourier coefficients, this is a very deep fact implied by the results in the article "Sur les coefficients de Fourier des fonctions continues" by J.P. Kahane, Y. Katznelson and K. de Leeuw, see [4]. On the other hand, we can easily obtain an analogous of (A) for the Haar coefficients. VIe define the Haar coefficients of a locally integrable function I as Ca,b = JIRJ(X)Ha,b(X)dx, where Ha,b(X) = a-1 / 2 H( x-;.b), a> 0, bE 1R and H is the Haar function i.e., H is defined by 1, for ° ::::; x < 1/2; by -1, for 1/2 ::::; x < 1 and 0 otherwise. More precisely we get the following result Supported by: CONICET and Programa.ci6n CAI+D, UNL.