Mathematical notes

Strong emphasis on function spaces and structures in analysis. Detailed proofs of theorems.

\(C[0,1]\): the Faber-Schauder basis, the Riesz representation theorem, and the Borel \(\sigma\)-algebraPDF
\(C^k\) spaces and spaces of test functionsPDF
\(L^0\), convergence in measure, equi-integrability, the Vitali convergence theorem, and the de la Vallée-Poussin criterionPDF
\(L^1\) norms of products of sinesPDF
\(L^p\) norms of a sine sumPDF
\(L^p\) norms of trigonometric polynomialsPDF
A series of secantsPDF
Abstract Fourier series and Parseval's identityPDF
Alternating multilinear formsPDF
Banach algebrasPDF
Bernoulli polynomialsPDF
Chebyshev polynomialsPDF
Compact operators on Banach spacesPDF
Convolution semigroups, canonical processes, and Brownian motionPDF
Cyclotomic polynomialsPDF
Decomposition of the spectrum of a bounded linear operatorPDF
Denjoy's theorem on circle diffeomorphismsPDF
Diophantine vectorsPDF
Explicit construction of the \(p\)-adic numbersPDF
Fatou's theorem, Bergman spaces, and Hardy spaces on the circlePDF
Fréchet derivatives and Gâteaux derivativesPDF
Functions of bounded variation and a theorem of KhinchinPDF
Functions of bounded variation and differentiabilityPDF
Gaussian integralsPDF
Gaussian measures and Bochner's theoremPDF
Gaussian measures, Hermite polynomials, and the Ornstein-Uhlenbeck semigroupPDF
Germs of smooth functionsPDF
Gibbs measures and the Ising modelPDF
Gradients and Hessians in Hilbert spacesPDF
Haar wavelets and multiresolution analysisPDF
Hamiltonian flows, cotangent lifts, and momentum mapsPDF
Harmonic analysis on the \(p\)-adic numbersPDF
Harmonic polynomials and the spherical LaplacianPDF
Hensel's lemma, valuations, and \(p\)-adic numbersPDF
Hermite functionsPDF
Infinite product measuresPDF
Integral operatorsPDF
Jointly measurable and progressively measurable stochastic processesPDF
Khinchin's inequality and Etemadi's inequalityPDF
Laguerre polynomials and Perron-Frobenius operatorsPDF
Locally compact abelian groupsPDF
Lévy's inequality, Rademacher sums, and Kahane's inequalityPDF
Markov kernels, convolution semigroups, and projective families of probability measuresPDF
Martingales, Lévy's continuity theorem, and the martingale central limit theoremPDF
Meager sets of periodic functionsPDF
Measure theory and Perron-Frobenius operators for continued fractionsPDF
Newton's identities and the pentagonal number theoremPDF
Nonholomorphic Eisenstein series, the Kronecker limit formula, and the hyperbolic LaplacianPDF
Norms of trigonometric polynomialsPDF
Notes on the KAM theoremPDF
Orbital stability for NLSPDF
Orthonormal bases for product measuresPDF
Oscillatory integralsPDF
Polish spaces and Baire spacesPDF
Positive definite functions, completely monotone functions, the Bernstein-Widder theorem, and Schoenberg's theoremPDF
Projection-valued measures and spectral integralsPDF
Proof of the pentagonal number theoremPDF
Rademacher functionsPDF
Ramanujan's sumPDF
Random trigonometric polynomialsPDF
Real reproducing kernel Hilbert spacesPDF
Regulated functions and the regulated integralPDF
Schwartz functions, Hermite functions, and the Hermite operatorPDF
Semicontinuous functions and convexityPDF
Singular integral operators and the Riesz transformPDF
Sobolev spaces in one dimension and absolutely continuous functionsPDF
Some fundamental results on modular formsPDF
Spectral theory, Volterra integral operators and the Sturm-Liouville theoremPDF
Stationary phase, Laplace's method, and the Fourier transform for Gaussian integralsPDF
Subdifferentials of convex functionsPDF
Subgaussian random variables, Hoeffding's inequality, and Cramér's large deviation theoremPDF
Tauber's theorem and Karamata's proof of the Hardy-Littlewood tauberian theoremPDF
Test functions, distributions, and Sobolev's lemmaPDF
The \(C^\infty\) Urysohn lemmaPDF
The Banach algebra of functions of bounded variation and the pointwise Helly selection theoremPDF
The Bernstein and Nikolsky inequalities for trigonometric polynomialsPDF
The Berry-Esseen theoremPDF
The Bochner-Minlos theoremPDF
The Dirac delta distribution and Green's functionsPDF
The Dunford-Pettis theoremPDF
The Euler equations in fluid mechanicsPDF
The Fourier transform of holomorphic functionsPDF
The Fourier transform of spherical surface measure and radial functionsPDF
The Gelfand transform, positive linear functionals, and positive-definite functionsPDF
The Glivenko-Cantelli theoremPDF
The Gottschalk-Hedlund theorem, cocycles, and small divisorsPDF
The Hamilton-Jacobi equationPDF
The Heisenberg group and Hermite functionsPDF
The Hilbert transform on \(\mathbb{R}\)PDF
The Kolmogorov continuity theorem, Hölder continuity, and the Kolmogorov-Chentsov theoremPDF
The Kolmogorov extension theoremPDF
The Legendre transformPDF
The Lindeberg central limit theoremPDF
The Poincaré-Dulac normal form theorem for formal vector fieldsPDF
The Polya-Vinogradov inequalityPDF
The Pontryagin duals of \(\mathbb{Q}/\mathbb{Z}\) and \(\mathbb{Q}\) and the adelesPDF
The Schwartz space and the Fourier transformPDF
The Segal-Bargmann transform and the Segal-Bargmann spacePDF
The Stone-Čech compactification of Tychonoff spacesPDF
The Voronoi summation formulaPDF
The Wiener algebra and Wiener's lemmaPDF
The Wiener-Pitt tauberian theoremPDF
The cross-polytope, the ball, and the cubePDF
The heat kernel on \(\mathbb{R}^n\)PDF
The heat kernel on the torusPDF
The inclusion map from the integers to the reals and universal properties of the floor and ceiling functionsPDF
The inhomogeneous heat equation on \(\mathbb{T}\)PDF
The law of the iterated logarithmPDF
The narrow topology on the set of Borel probability measures on a metrizable spacePDF
The nonlinear Schrödinger equation is HamiltonianPDF
The one-dimensional periodic Schrödinger equationPDF
The profinite completion of the integers, the \(p\)-adic integers, and Prüfer \(p\)-groupsPDF
The singular value decomposition of compact operators on Hilbert spacesPDF
The spectra of the unilateral shift and its adjointPDF
The spectrum of a self-adjoint operator is a compact subset of \(\mathbb{R}\)PDF
The symmetric difference metricPDF
The theorem of F. and M. RieszPDF
The uniform metric on product spacesPDF
The weak and strong laws of large numbersPDF
The weak topology of locally convex spaces and the weak-* topology of their dualsPDF
Topological spaces and neighborhood filtersPDF
Total variation, absolute continuity, and the Borel \(\sigma\)-algebra of \(C(I)\)PDF
Trace class operators and Hilbert-Schmidt operatorsPDF
Unbounded operators in a Hilbert space and the Trotter product formulaPDF
Uncertainty principles and compressed sensingPDF
Unordered sums in Hilbert spacesPDF
Vinogradov's estimate for exponential sums over primesPDF
Vitali coverings on the real linePDF
Watson's lemma and Laplace's methodPDF
Weak symplectic forms and differential calculus in Banach spacesPDF
What is a Costas array?PDF
What is the domain of a solution of an ODE?PDF
Wiener measure and Donsker's theoremPDF
Zygmund's Fourier restriction theorem and Bernstein's inequalityPDF

Historical and bibliographic notes

Early instances of the martingale PDF
Notes on the history of Liouville's theorem PDF
Summable series and the Riemann rearrangement theorem PDF
The Euler-Maclaurin summation formula PDF
Bibliography for the history of the Jacobian PDF
The logarithmic integral PDF
Bibliography for the history of induction in mathematics PDF
Pell's equation and side and diagonal numbers PDF
Greek numbers PDF
Ancient Greek weights and measures PDF
Ancient balance scales PDF
The great year, calendars, and the incommensurability of celestial rotations PDF
Book IV of Euclid's Elements and ancient Greek mosaics PDF
Plato's theory of forms and the axiom of foundation PDF
Predicates PDF
Amphibolia PDF
Zeno of Elea, locomotion, infinity, and time PDF
What is a wave? PDF
Bibliography for the history of resonance PDF
Numbers and fractions in Greek papyri PDF
Minoan weights PDF