The Geometry of Planar Fourier Expansions

Let A be an appropriate planar domain and let f be a piecewise smooth function on R2. We discuss the rate of convergence of S f(x) = Z A b f( ) exp(2 i x)d in terms of the interaction between the geometry of A and the geometry of the singularities of f . The most subtle case is when x belongs to the singular set of f and here Hilbert transform techniques play an important role. The pointwise convergence of one-dimensional Fourier series of piecewise smooth functions is one of the best known topics in analysis. In higher dimensions the problem is harder and also of di erent nature. This is basically due to the failure of the Riemann localization principle, since the convergence at a given point does not depend exclusively on the regularity of the function in a neighborhood of the point. See [15] and [14, VII.4]. The purpose of this paper is to characterize the convergence properties of two-dimensional Fourier integrals of piecewise smooth functions in terms of certain natural geometric features. On this topic the classical reference is [1], but for more recent works see [2], [3], [5], [6], [8],[9], [10], [11], [12]. De ne the Fourier transform and the spherical sums of Fourier integral of integrable functions as b f( ) = Z Rn f(y) exp( 2 i y)dy; S f(x) = Z fj j< g b f( ) exp(2 ix )d : In analogy with the one dimensional case one expects that the spherical sums of piecewise smooth functions converge everywhere to the 1 2 L. BRANDOLINI, L. COLZANI, A. IOSEVICH, AND G. TRAVAGLINI function expanded, but indeed the situation is slightly more complicated. The simplest example to illustrate this fact is the characteristic function of the n-dimensional unit ball centered at the origin B = B(0; 1). Its Fourier transform can be expressed in terms of a Bessel function, b B( ) = j j n=2 Jn=2 (2 j j), and it is possible to show that the spherical sums S B(x) converge at every point x 6= 0 when ! +1. This follows for example by an equiconvergence result between Fourier-Bessel expansions and the classical one dimensional trigonometric expansions. See [4]. On the other hand, the spherical sums at the origin x = 0, S B(0) = jfj j = 1gj Z 0 Jn=2 (2 r) rn=2 1dr converge for n = 2, oscillate for n = 3, and are unbounded for n > 3. More precisely, in the planar case n = 2, one can prove that when ! +1, S B(x) = 8>><>>>>: 1 +O 1=2 when x = 0, B(x) +O ( 1) when x 6= 0 and jxj 6 = 1, 1=2 +O ( 1) when jxj = 1. Observe that the worst rate of convergence 1=2 takes place at the origin, while at all other points the rate is 1. In [11] there is a more sophisticated example: The spherical sums of the characteristic function of an ellipse converge with speed 1, except that at points of the evolute, an astroid. At the four vertexes of the astroid the speed of convergence is 3=4 and at the other points is 5=6. What is suggested by these examples is that the rate of convergence of S f(x) may not be optimal if the geometry of the disc used to de ne the summation operators interacts badly with the geometry of the singularities of the function expanded, and this leads one to ask whether the situation can be ameliorated by replacing the disc with a suitable domain, or if the same phenomena occur. In particular, in this paper we de ne summation methods more general than spherical sums and we apply these summation methods to piecewise smooth functions. Given a bounded planar domain A containing the origin, de ne the two dimensional partial sums of Fourier integral of integrable functions as THE GEOMETRY OF PLANAR FOURIER EXPANSIONS 3