Groups meet Analysis: the Fourier Algebra Matthew Daws Leeds York, June 2013 Matthew Daws (Leeds) The Fourier Algebra York, June 2013 1 / 24 Colloquium talk So I believe this is a talk to a general audience of Mathematicians. Some old advice for giving talks: the first 10 minutes should be aimed at the janitor; then at undergrads; then at graduates; then at researchers; then at specialists; and finish by talking to yourself. The janitor won’t understand me; and I’ll try not to talk to myself. I’m going to try just to give a survey talk about a particular area at the interface between algebra and analysis. Please ask questions! Matthew Daws (Leeds) The Fourier Algebra York, June 2013 2 / 24 Colloquium talk So I believe this is a talk to a general audience of Mathematicians. Some old advice for giving talks: the first 10 minutes should be aimed at the janitor; then at undergrads; then at graduates; then at researchers; then at specialists; and finish by talking to yourself. The janitor won’t understand me; and I’ll try not to talk to myself. I’m going to try just to give a survey talk about a particular area at the interface between algebra and analysis. Please ask questions! Matthew Daws (Leeds) The Fourier Algebra York, June 2013 2 / 24 Colloquium talk So I believe this is a talk to a general audience of Mathematicians. Some old advice for giving talks: the first 10 minutes should be aimed at the janitor; then at undergrads; then at graduates; then at researchers; then at specialists; and finish by talking to yourself. The janitor won’t understand me; and I’ll try not to talk to myself. I’m going to try just to give a survey talk about a particular area at the interface between algebra and analysis. Please ask questions! Matthew Daws (Leeds) The Fourier Algebra York, June 2013 2 / 24 Colloquium talk So I believe this is a talk to a general audience of Mathematicians. Some old advice for giving talks: the first 10 minutes should be aimed at the janitor; then at undergrads; then at graduates; then at researchers; then at specialists; and finish by talking to yourself. The janitor won’t understand me; and I’ll try not to talk to myself. I’m going to try just to give a survey talk about a particular area at the interface between algebra and analysis. Please ask questions! Matthew Daws (Leeds) The Fourier Algebra York, June 2013 2 / 24 Colloquium talk So I believe this is a talk to a general audience of Mathematicians. Some old advice for giving talks: the first 10 minutes should be aimed at the janitor; then at undergrads; then at graduates; then at researchers; then at specialists; and finish by talking to yourself. The janitor won’t understand me; and I’ll try not to talk to myself. I’m going to try just to give a survey talk about a particular area at the interface between algebra and analysis. Please ask questions! Matthew Daws (Leeds) The Fourier Algebra York, June 2013 2 / 24 Fourier transform Let f be a “well-behaved” function on the real line. Then the Fourier transform of f is Z 1 ^f (x) = f (t) e−2πixt dt: −∞ (You have to put a 2π somewhere!) Then we can reconstruct f from ^f by Z 1 f (t) = ^f (x) e2πixt dx: −∞ A basic tool in “applied” mathematics which we teach to undergraduates. Appears in probability theory as the Characteristic Function. Matthew Daws (Leeds) The Fourier Algebra York, June 2013 3 / 24 Fourier transform Let f be a “well-behaved” function on the real line. Then the Fourier transform of f is Z 1 ^f (x) = f (t) e−2πixt dt: −∞ (You have to put a 2π somewhere!) Then we can reconstruct f from ^f by Z 1 f (t) = ^f (x) e2πixt dx: −∞ A basic tool in “applied” mathematics which we teach to undergraduates. Appears in probability theory as the Characteristic Function. Matthew Daws (Leeds) The Fourier Algebra York, June 2013 3 / 24 Fourier transform Let f be a “well-behaved” function on the real line. Then the Fourier transform of f is Z 1 ^f (x) = f (t) e−2πixt dt: −∞ (You have to put a 2π somewhere!) Then we can reconstruct f from ^f by Z 1 f (t) = ^f (x) e2πixt dx: −∞ A basic tool in “applied” mathematics which we teach to undergraduates. Appears in probability theory as the Characteristic Function. Matthew Daws (Leeds) The Fourier Algebra York, June 2013 3 / 24 Gibbs “ringing” Matthew Daws (Leeds) The Fourier Algebra York, June 2013 4 / 24 Gibbs “ringing” Matthew Daws (Leeds) The Fourier Algebra York, June 2013 5 / 24 Gibbs “ringing” Matthew Daws (Leeds) The Fourier Algebra York, June 2013 6 / 24 Fourier series ^ Given a periodic function f : R ! C the Fourier series of f is (f (n))n2Z where Z 1 ^f (n) = f (θ)e−2πinθ dθ: 0 We have the well-known “reconstruction”: 1 X f (θ) = ^f (n)e2πinθ: n=−∞ Of course, a great deal of classical analysis is concerned with the question of in what sense does this sum actually converge? Matthew Daws (Leeds) The Fourier Algebra York, June 2013 7 / 24 Convergence 1 X f (θ) = ^f (n)e2πinθ ?? n=−∞ If f is twice continuously differentiable, then the sum converges PN uniformly to f (that is, limN!1 n=−N ). (Fejer) If f is continuous, and we take Cesaro means, then we always get (uniform) convergence. (Kolmogorov) There is a (Lebesgue integrable) function f such that the sum diverges everywhere. (Carleson) If f is continuous then the sum converges almost everywhere. Matthew Daws (Leeds) The Fourier Algebra York, June 2013 8 / 24 Convergence 1 X f (θ) = ^f (n)e2πinθ ?? n=−∞ If f is twice continuously differentiable, then the sum converges PN uniformly to f (that is, limN!1 n=−N ). (Fejer) If f is continuous, and we take Cesaro means, then we always get (uniform) convergence. (Kolmogorov) There is a (Lebesgue integrable) function f such that the sum diverges everywhere. (Carleson) If f is continuous then the sum converges almost everywhere. Matthew Daws (Leeds) The Fourier Algebra York, June 2013 8 / 24 Convergence 1 X f (θ) = ^f (n)e2πinθ ?? n=−∞ If f is twice continuously differentiable, then the sum converges PN uniformly to f (that is, limN!1 n=−N ). (Fejer) If f is continuous, and we take Cesaro means, then we always get (uniform) convergence. (Kolmogorov) There is a (Lebesgue integrable) function f such that the sum diverges everywhere. (Carleson) If f is continuous then the sum converges almost everywhere. Matthew Daws (Leeds) The Fourier Algebra York, June 2013 8 / 24 Convergence 1 X f (θ) = ^f (n)e2πinθ ?? n=−∞ If f is twice continuously differentiable, then the sum converges PN uniformly to f (that is, limN!1 n=−N ). (Fejer) If f is continuous, and we take Cesaro means, then we always get (uniform) convergence. (Kolmogorov) There is a (Lebesgue integrable) function f such that the sum diverges everywhere. (Carleson) If f is continuous then the sum converges almost everywhere. Matthew Daws (Leeds) The Fourier Algebra York, June 2013 8 / 24 Convergence 1 X f (θ) = ^f (n)e2πinθ ?? n=−∞ If f is twice continuously differentiable, then the sum converges PN uniformly to f (that is, limN!1 n=−N ). (Fejer) If f is continuous, and we take Cesaro means, then we always get (uniform) convergence. (Kolmogorov) There is a (Lebesgue integrable) function f such that the sum diverges everywhere. (Carleson) If f is continuous then the sum converges almost everywhere. Matthew Daws (Leeds) The Fourier Algebra York, June 2013 8 / 24 A more “global” perspective Don’t want to look at single functions in isolation; but rather at spaces of functions. 2 R 1 2 Let’s consider L ([0; 1]); that is, functions f with 0 jf j < 1. This is a vector space. R 1 21=2 kf k = 0 jf j is a norm. So we get a metric d(f ; g) = kf − gk. With some help from Lebesgue, we get a complete space (so a Banach space; even a Hilbert space). (Parseval) In the Banach space L2([0; 1]), we always have that 1 X f = ^f (n)(e2πinθ): n=−∞ Matthew Daws (Leeds) The Fourier Algebra York, June 2013 9 / 24 A more “global” perspective Don’t want to look at single functions in isolation; but rather at spaces of functions. 2 R 1 2 Let’s consider L ([0; 1]); that is, functions f with 0 jf j < 1. This is a vector space. R 1 21=2 kf k = 0 jf j is a norm. So we get a metric d(f ; g) = kf − gk. With some help from Lebesgue, we get a complete space (so a Banach space; even a Hilbert space). (Parseval) In the Banach space L2([0; 1]), we always have that 1 X f = ^f (n)(e2πinθ): n=−∞ Matthew Daws (Leeds) The Fourier Algebra York, June 2013 9 / 24 A more “global” perspective Don’t want to look at single functions in isolation; but rather at spaces of functions. 2 R 1 2 Let’s consider L ([0; 1]); that is, functions f with 0 jf j < 1. This is a vector space. R 1 21=2 kf k = 0 jf j is a norm. So we get a metric d(f ; g) = kf − gk. With some help from Lebesgue, we get a complete space (so a Banach space; even a Hilbert space).