mathematics.md

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page	Mathematics	/mathematics/

• Mathematical notes
• Historical and bibliographic notes

Mathematical notes {#mathematical}

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

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

Historical and bibliographic notes {#historical}

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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