Topics in Harmonic Analysis Lecture 1: The Fourier transform Po-Lam Yung The Chinese University of Hong Kong
Outline Fourier series on T: L 2 theory Convolutions The Dirichlet and Fejer kernels Pointwise convergence of Fourier series Other modes of convergence Multiplier operators: a prelude The role played by the group of translations Analogue on R n The groups of dilations and modulations
Fourier series on T: L 2 theory Let T = R/Z be the unit circle. {Functions on T} = {periodic functions on R}. The n-th Fourier coefficient of an L 1 function on T is given by f (n) := f (x)e 2πinx dx, n Z. T A remarkable insight of J. Fourier was that perhaps every function f on T can be represented by a Fourier series: a n e 2πinx. n Z One can make this rigorous, for instance, when one restricts attention to L 2 functions on T (which form a Hilbert space).
The claim is that for every f L 2 (T), the Fourier series of f converges to f in L 2 (T) norm. In other words, for every f L 2 (T), we have N n= N f (n)e 2πinx f (x) in L 2 (T) as N. Underlying this claim are two important concepts, namely orthogonality and completeness.
Clearly {e 2πinx } n Z is an orthonormal system on L 2 (T), and f (n) = f (x), e 2πinx L 2 (T). General principles about orthogonality then shows that for every N N, S N f (x) := f (n)e 2πinx n N is the orthogonal projection of f onto the linear subspace V N of L 2 (T) spanned by {e 2πinx : n N}. In other words, f = (f S N f ) + S N f with S N f V N and (f S N f ) V N, and hence f (n) 2 f 2 L 2 (T). n Z (The last inequality is sometimes called Bessel s inequality.)
The aforementioned property about orthogonal projection can be rephrased as f S N f L 2 (T) = min { f p N L 2 (T) : p N span{e 2πinx } n N }. On the other hand, one can show that the set of all finite linear combinations of {e 2πinx } n Z (i.e. the set of all trigonometric polynomials) is dense in L 2 (T) (more to follow below). Thus {e 2πinx } n Z form a complete orthonormal system on L 2 (T). It follows that for every f L 2 (T), we have S N f f in L 2 (T) as N. To see this desired density, we study convolutions and multiplier operators.
Convolutions The convolutions of two L 1 functions on T is another L 1 function, given by f g(x) = f (x y)g(y)dy. Convolutions are associative and commutative: T f (g h) = (f g) h, f g = g f. If f L p (T) for some p [1, ] and g L 1 (T), then f g L p (T). More generally, if p, q, r [1, ] and 1 + 1 r = 1 p + 1 q then f g L r (T) f L p (T) g L q (T). (Young s convolution inequality)
For f L 1 (T), let f (n) = T f (x)e 2πinx dx as before. Then f g(n) = f (n)ĝ(n) whenever f, g L 1 (T). So if K L 1 (T), then the convolution operator f f K can be understood via the Fourier transform of K: indeed f K(x) = n Z 2πinx f (n) K(n)e whenever f is in (say) L 2 (T).
The Dirichlet and Fejer kernels For each N N, let D N and F N be the Dirichlet and Fejer kernels respectively, defined by { { 1 if n N D N (n) = 0 if n > N, 1 n FN (n) = N if n N 0 if n > N. Then D N (x) = F N (x) = n N n N e 2πinx ( 1 n ) e 2πinx N
If f L 1 (T), then S N f = f D N, and S 0 f + S 1 f + + S N 1 f = f F N. N One has a closed formula for F N : indeed F N (x) = 1 N ( ) sin Nπx 2. sin πx From this we see that F N 0 for all N, and δ y 1/2 F N L 1 = F N (0) = 1, F N (y)dy 0 as N. Thus {F N } N N form a family of good kernels.
As a result, if f is continuous on T, then f F N f uniformly on T. Hence by approximating by continuous functions, we see that trigonometric polynomials are dense in L 2 (T). This establishes in full the elementary L 2 theory of the Fourier transform on T. In particular, now we have Plancherel s theorem: f (n) 2 = f 2 L 2 (T) whenever f L 2 (T). n Z
Pointwise convergence of Fourier series We can now prove the Riemann-Lebesgue lemma: for f L 1 (T), we have sup f (n) f L 1 (T). n Z Indeed f (n) 0 as n ±. This allows one to show that lim f (n)e 2πinx = f (x) N n N whenever f is Hölder continuous of some positive order at x T (actually we only need a Dini condition at x: indeed the condition f (x + t) f (x) dt < t will suffice.) t 1/2
On the other hand, there exists a continuous function on T whose Fourier series diverges at say 0 T. So mere continuity of f does not guarantee everywhere pointwise convergence of Fourier series! On the pathological side, there even exists an L 1 function on T, whose Fourier series diverges everywhere (Kolmogorov). However, a remarkable theorem of Carleson says that the Fourier series of an L 2 function on T converges pointwise almost everywhere (a.e.). (Later Hunt showed the same for L p (T), 1 < p <.) We will come back to this briefly towards the end of this lecture.
Other modes of convergence To sum up, we have considered L 2 norm convergence, pointwise convergence and a.e. pointwise convergence of Fourier series. Other modes of convergence of Fourier series also give rise to interesting questions. Examples include L p norm convergence, uniform convergence, absolute convergence, and various summability methods. If f C 2 (T), then clearly the Fourier series of f converges absolutely and uniformly (since f (n) n 2 ). However, indeed the Fourier series of f converges absolutely and uniformly already, as long as f is Hölder continuous of some order > 1/2 on T. The question of L p norm convergence on T, for 1 < p <, is also well understood, and will be considered in Lecture 4. It is interesting to note that an analogous question in higher dimensions (for T n, n > 1) is much deeper, and is related to some excellent open problems in the area.
Multiplier operators: a prelude Given a bounded function m : Z C, the map f (x) T m f (x) := m(n) f (n)e 2πinx n Z defines a bounded linear operator on L 2 (T). Examples: convolution with the Dirichlet or the Fejer kernels. The analysis of such operators often benefit by taking the inverse Fourier transform of m, as we have seen in the case of convolution with the Fejer kernel. We will be interested, for instance, in the boundedness of many multiplier operators on L p (T) for various p [1, ]. e.g. The uniform L p boundedness of convolutions with the Dirichlet kernels is crucial in the discussion of L p norm convergence of the Fourier series of a function in L p (T). We note in passing that all multiplier operators commute with translations (and so do all convolution operators).
The role played by the group of translations Let s take a step back and look at Fourier series on L 2 (T). There we expand functions in terms of complex exponentials {e 2πinx } n Z. But why complex exponentials? An explanation can be given in terms of the underlying group structure on T = R/Z. Recall T is an abelian group under addition. Thus if y T, then the operator τ y : L 2 (T) L 2 (T), defined by τ y f (x) = f (x + y), is a unitary operator on L 2 (T), and the group homomorphism y T τ y B(L 2 (T)) is continuous if we endow the strong topology on B(L 2 (T)).
Thus the map y τ y defines a unitary representation of the compact group T on the (complex) Hilbert space L 2 (T). The Peter-Weyl theorem then provides a splitting of L 2 (T) into an orthogonal direct sum of irreducible finite-dimensional representations. Since T is abelian, all irreducible finite-dimensional representations are 1-dimensional, by Schur s lemma. In other words, there exists an orthonormal family of functions {f n } n Z of L 2 (T), such that L 2 (T) = n Z Cf n, and such that f n is an eigenfunction of τ y for all y T.
For n Z, let χ n (y) be the eigenvalue of τ y on f n, i.e. let χ n : T C be a continuous function such that τ y f n = χ n (y)f n for all y T. Then χ n (y + y ) = χ n (y)χ n (y ) for all y, y T, i.e. χ n is a character on T. But if χ: R C is a continuous function with χ(x + x ) = χ(x)χ(x ) for all x, x R, then there exists a C such that χ(x) = e ax for all x R (just show that χ(x) = e ax holds for x = mx 0 /2 n whenever m Z, n N and x 0 is sufficiently close to 0, and use continuity); if in addition χ is periodic with period 1, then the a above must be in 2πiZ. Thus without loss of generality, we may label the f n s, so that χ n (y) = e 2πiny for all n Z and all y T.
This gives f n (x + y) = e 2πiny f n (x) for all n Z and all x, y T, so f n(x) = 2πinf n (x), from which we conclude that f n (x) = ce 2πinx for some c = 1. In other words, for each n Z, the function e 2πinx is an eigenvector of τ y for all y T, with eigenvalue e 2πiny. To recap: we have a family {τ y } y T of commuting unitary operators on L 2 (T), and the complex exponentials {e 2πinx } n Z provide a simultaneous diagonalization of these operators: for all n Z, and all y T. τ y e 2πinx = e 2πiny e 2πinx Note that the orthogonality of the complex exponentials follows from the Peter-Weyl theorem!
Also, the equation τ y e 2πinx = e 2πiny e 2πinx implies that e 2πinx are eigenfunctions of the derivative operator: d dx e2πinx = 2πine 2πinx. This is not too surprising since derivatives are infinitesimal translations! Another way of saying the same thing is that τ y f (n) = e 2πiny f (n) for all y T and n Z; hence differentiation and multiplication are interwined by the Fourier transform: f (n) = 2πin f (n) if f C 1 (T). The fact that the Fourier series constitutes a spectral decomposition of the derivative operator is what makes it so powerful in the study of differential equations.
Analogue on R n The Fourier transform of an L 1 function on R n is defined by f (ξ) = f (x)e 2πix ξ dx, ξ R n. R n We have f L (R n ) f L 1 (R n ); indeed the Fourier transform of an L 1 function is continuous on R n. If f, g L 1 (R n ), their convolution is defined by f g(x) = f (x y)g(y)dy. R n We have f g = g f and if f, g L 1 (R n ). f g(ξ) = f (ξ)ĝ(ξ)
The space of Schwartz functions on R n is defined as the space of all smooth functions, whose derivatives of all orders are rapidly decreasing at infinity. It is denoted S(R n ). One can restrict the Fourier transform on S(R n ); indeed the Fourier transform maps S(R n ) into itself. If f, g are Schwartz functions on R n, then Fubini s theorem gives f (y)ĝ(y)dy = R n f (ξ)g(ξ)dξ. R n Replacing g(ξ) by g(ξ)e 2πix ξ, it follows that f ĝ(x) = f (ξ)g(ξ)e 2πix ξ dξ R n for all x R n.
f ĝ(x) = f (ξ)g(ξ)e 2πix ξ dξ R n Applying this with g(ξ) = e πt ξ 2 (the heat kernel) and letting t 0, one can show that the Fourier inversion formula holds for Schwartz functions: f (x) = f (ξ)e 2πix ξ dξ R n for every x R n, whenever f S(R n ). Hence the Fourier transform defines a bijection on Schwartz functions on R n, and we have f g(x) = f (ξ)ĝ(ξ)e 2πix ξ dξ R n whenever f, g S(R n ).
We now also have f (x)g(x)dx = f (ξ)ĝ(ξ)dξ R n R n whenever f, g S(R n ). This allows one to show that the Fourier transform, initially defined on S(R n ), extends as a unitary operator on L 2 (R n ), and the Plancherel formula holds: whenever f L 2 (R n ). f L 2 (R n ) = f L 2 (R n ) A multiplier operator on R n is of the form f (m f )ˇ where m is a bounded measurable function on R n. We will come across examples of such in Lecture 2. These are automatically bounded on L 2 (R n ). We will study their mapping properties on L p (R n ) in Lecture 4.
The groups of modulations and dilations R n is an abelian group under addition. It acts on (say, L 2 ) functions on R n by translation (as in the case of the unit circle T): τ y f (x) := f (x + y), y R n But it also acts on functions on R n by modulation: Λ ξ f (x) := e 2πix ξ f (x), ξ R n. The actions are interwined by the Fourier transform F: Fτ y = Λ y F for all y R n. In particular, at least for Schwartz functions f on R n, we have j f (ξ) = 2πiξ j f (ξ), for 1 j n.
The multiplicative group R + = (0, ) also acts on functions on R n by dilations: D t f (x) := f (tx), t R +. It interacts with the Fourier transform as follows: FD t = t n D 1/t F for all t > 0. In harmonic analysis we often study operators that commutes with translations. Examples include derivative operators (such as f f ), and convolution operators (such as f f x (n 2) ). Such operators often come with some invariance under dilations: e.g. D t f = t 2 D t f, (D t f ) x (n 2) = t 2 D t (f x (n 2) ).
Operators that exhibit modulation invariance (on top of translation and dilation invariances) are harder to analyze; they typically require rather refined time-frequency analysis. An example of such an operator is the Carleson operator, studied in connection with pointwise a.e. convergence of Fourier series of a function on L 2 (T): Cf (x) = sup f (n)e 2πinx, x T; N Z n N note that C commutes with both translations and modulations, i.e. Cτ y = τ y C for all y T, and CΛ k = Λ k C for all k Z. Most of the operators we will encounter in this course will not be modulation invariant.