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Class Notes, 6211 and 6212 Ovidiu Costin Class notes, 6211 and 6212 Ovidiu Costin Contents 1 Fourier Series 6 1.1 Fejér’s theorem........................................ 12 1.2 Introduction to normed spaces and Hilbert spaces.................... 14 2 Measure theory 17 2.1 Measures............................................ 19 2.2 Measurable functions..................................... 20 2.3 Product s-algebras...................................... 20 2.4 More about measures..................................... 22 3 Construction of measures 24 3.1 Outer measures........................................ 24 3.2 Measures from pre-measures................................ 26 4 Borel measures on the real line 29 4.1 Push-forward of a measure................................. 32 4.2 Coin tosses and the Lebesgue measure.......................... 32 4.3 The Cantor set......................................... 33 4.4 Cantor’s function (a.k.a Devil’s staircase)......................... 34 5 Integration 34 5.1 Measurable functions (cont.)................................. 34 5.2 Simple functions....................................... 36 5.3 Integration of positive functions.............................. 38 6 Integration of complex-valued functions 40 7 The link with the Riemann integral 43 8 Some applications of the convergence theorems 44 9 Topologies on spaces of measurable functions 47 10 Product measures and integration on product spaces 49 11 The n−dimensional Lebesgue integral 53 11.1 Extensions of results from 1d................................ 53 1 Math 6211+6212, Real Analysis I+II 12 Polar coordinates 56 13 Signed measures 58 13.1 Two decomposition theorems................................ 59 14 The Lebesgue-Radon-Nikodym theorem 60 14.1 Complex measures...................................... 62 15 Differentiation 63 15.1 Lebesgue points........................................ 66 15.1.1 Differentiation of absolutely continuous measures................ 66 15.1.2 Nicely shrinking sets................................. 67 15.2 Metric density......................................... 68 16 Total variation, absolute continuity 69 16.1 NBV, AC............................................ 71 16.2 Lebesgue-Stieltjes integrals................................. 73 16.3 Types of measures on Rn .................................. 73 17 Semicontinuouos functions 74 17.1 Urysohn’s lemma....................................... 74 17.2 Locally compact Hausdorff spaces............................. 75 17.3 Support of a function..................................... 76 17.4 Partitions of unity....................................... 77 17.5 Continuous functions..................................... 77 17.6 The Stone-Weierstrass theorem............................... 78 18 Sequences and nets 80 18.1 Subnets............................................. 81 19 Tychonoff’s theorem 82 20 Arzelá-Ascoli’s theorem 84 21 Norms, seminorms and inner products 86 22 Hilbert spaces 87 22.1 Example: `2 .......................................... 87 22.2 Orthogonality......................................... 88 22.3 The Gram-Schmidt process................................. 89 22.4 A very short proof of Cauchy-Schwarz.......................... 89 22.5 Orthogonal projections.................................... 90 22.6 Bessel’s inequality, Parseval’s equality, orthonormal bases............... 91 23 Normed vector spaces 94 23.1 Functionals and linear operators.............................. 95 23.2 The Hahn-Banach theorem................................. 96 23.3 The Riesz representation theorem............................. 99 2/186 Math 6211+6212, Real Analysis I+II 23.4 Adjoints............................................ 100 24 Consequences of the Baire category theorem 100 25 Lp spaces 104 25.0.1 The space L¥ ..................................... 106 26 The dual of Lp 109 26.1 The dual of L¥ is not L1 ................................... 112 26.2 Inequalities in Lp spaces................................... 112 26.3 Weak Lp ............................................ 115 26.4 Lp interpolation theorems.................................. 117 26.5 Some applications....................................... 122 27 Radon measures 123 27.1 The Baire s-algebra...................................... 128 27.2 Regularity of Borel measures................................ 129 28 The dual of C0(X) 130 29 Fourier series, cont. 133 29.1 Pointwise convergence.................................... 134 30 The heat equation 135 30.1 Examples............................................ 137 30.2 The vibrating string...................................... 138 30.3 The Poincaré-Wirtinger inequality............................. 139 30.4 The Riemann-Lebesgue lemma (for L1(R))........................ 140 30.5 Hurwitz’s proof of the isoperimetric inequality..................... 141 31 Some conditions for pointwise convergence 141 31.1 Approximation to the identity................................ 144 32 The Poisson kernel 146 32.1 Several variables....................................... 147 33 The Fourier transform 149 33.1 The Schwartz space S .................................... 149 33.2 The Fourier inversion theorem, a direct approach.................... 155 34 Supplementary material: Some applications of the Fourier transform 156 34.1 The Schrödinger equation for a free particle in Rd .................... 156 34.2 The Airy equation....................................... 157 35 Convolutions and the Fourier transform 158 36 The Poisson summation formula 159 36.1 The Gibbs phenomenon................................... 161 3/186 Math 6211+6212, Real Analysis I+II 37 Applications to PDEs 162 37.1 The heat equation on the circle............................... 162 37.2 The heat equation on the line; smoothening by convolution.............. 162 37.3 General linear PDEs..................................... 164 37.4 Operators and symmetries.................................. 165 37.5 Supplementary material: Adjoints of linear operators.................. 167 37.6 Supplementary material: the Fourier transform of functions analytic in the lower half plane and the Laplace transform........................... 168 38 Distribution theory 169 ¥ n 38.1 The space of test functions D = Cc (R ), and its topology............... 170 38.2 Examples of distributions.................................. 171 38.3 Support of a distribution................................... 172 38.4 Extension of operators from functions to distributions................. 173 38.5 The Hadamard finite part.................................. 178 38.6 Green’s function........................................ 178 39 The dual of C¥(O) 179 39.0.1 Convolution of distributions............................ 180 40 The Fourier transform 180 41 Sobolev Spaces 183 42 Appendix A: inductive limits of Fréchet spaces 185 4/186 Math 6211+6212, Real Analysis I+II Part I: 6211 5/186 Math 6211+6212, Real Analysis I+II 1 Fourier Series Two landmark discoveries are typically credited in the development of analysis: Calculus (circ. 1665) and Fourier series, introduced by Joseph Fourier (1822). The latter mark the passage from finite-dimensional to infinite-dimensional mathematics. n n A choice of an orthonormal basis fekgk=1,...,n in R or C allows for a representation of vectors n n as strings of scalar components x = (xk)k=1,...,n and the inner product hx, yi as ∑ xkyk in R and k=1 n n ∑ xkyk in C where a + ib = a − ib. The natural generalization of the inner product and of the k=1 norm in the “continuum limit”, for two, say continuous, functions f , g : [a, b] ! C are Z b 2 h f , gi = f (t)g(t)dt; k f k2 = h f , f i a which are the Hilbert inner product and the Hilbert norm. With this generalization we may wonder, for a given orthonormal basis (ek)k2N, which func- tions can be represented by their, now infinite, set of components ( fk)k2N where fk = h f , eki, k 2 N (sometimes Z is a better choice than N). A possible choice of a basis are the monomials k−1 [ ] = [− ] x k2N which can be recombined to become an orthonormal set. If a, b 1, 1 , then the eks are the Legendre polynomials (Pk)k+12N: 1 1 1 P (x) = 1, P (x) = x, P (x) = 3x2 − 1 , P (x) = 5x3 − 3x , P (x) = 35x4 − 30x2 + 3 , ··· 0 1 2 2 3 2 4 8 and in general, 1 n n2 ( ) = ( − )n−k( + )k Pn x n ∑ x 1 x 1 2 k=0 k These polynomials satisfy the orthogonality condition Z 1 2 Pm(x)Pn(x) dx = dmn −1 2n + 1 ¥ In infinite dimensions the question of which functions can be written as ∑k=0 ckPk is, in this naive formulation, not well posed. By an infinite sum we must mean some form of a limit. This could be a uniform limit, a pointwise limit, a limit in the sense of Cèsaro averages, or, with Cn in mind, a limit in the “distance” sense, i.e. in the sense of integral means of order two: N lim k f − fkPkk2 = 0 N!¥ ∑ k=0 Each of these definitions leads to quite different answers, as we shall see in due course. The last one can only be satisfactorily answered after replacing Riemann integrals with the much more general and well-behaved Lebesgue integration, which in turn requires measure theory that we will study in Chapter 2. You probably noted that in Cn a good choice of the basis often simplifies the analysis. This is even more so in infinite dimensional (Hilbert) spaces. A very important orthonormal set (in 6/186 Math 6211+6212, Real Analysis I+II 2 [ ] 2pikx 2pikx the Hilbert space L ) on 0, 1 is e k2Z; (finite) linear combinations of e are called trig polynomials. Series of the form 2pikx ∑ fke k2Z are called Fourier series. Exercise: Check orthonormality of this set. If f is represented
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