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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 |