All things spectral filtering :)

Hankel matrices and the eigenbusiness

• [x] "Learning Linear Dynamical Systems via Spectral Filtering" - Hazan, Cyril, Karan

• [x] Hankel Operators and Applications (Lecture Notes) — Aleksey Kostenko

• tables of stuff and other resources

  • [ ] Handbook of Mathematical Functions — Abramowitz & Stegun, see chap 8
  • [ ] Handbook of Special Functions — Brychov
  • [ ] Table of Integral Transforms — the Bateman project
  • [ ] Handbook of Integral Equations
  • [ ] Hypergeometric Orthogonal Polynomials and Their q-Analogues — lists of orthogonal polynomials and their relations/diff eqs/asymptotics
  • [ ] Book of Laplace Transforms

• orthogonal polynomials and hypergeometrics

  • [ ] Intro to orthogonal polynomials — Koornwinder
  • [ ] Orthogonal Polynomials textbook — Gabor Szego
  • [ ] Advanced Determinant Calculus (basically the Hankel determninant cookbook) — Krattenthaler
  • [ ] https://dlmf.nist.gov/15 — great list of hypergeometric evaluations
  • [ ] Lectures on Algebraic Solutions of Hypergeometric Differential Equations — Matsuda. great list of hypergeometric transformations

• the Hilbert matrix, explicit diagonalizations, and generalizations

  • [x] "A remark on Hilbert's Matrix" — Grunbaum
  • [x] "On the Hilbert Matrix I" — Rosenblum
  • [x] "On the Hilbert Matrix II" — Rosenblum
  • [x] "Spectral analysis of the Hilbert matrix" — Stovicek
  • [ ] The Eigenfunctions of the Hilbert Matrix — Aleman, Montes-Rodriguez, Sarafoleanu
  • [ ] "A family of explicitly diagonalizable weighted Hankel matrices generalizing the Hilbert matrix" — Kalvoda, Stovicek
  • [ ] "New explicitly diagonalizable Hankel matrices related to the Stieltjes-Carlitz polynomials" — Stampach, Stovicek
  • [x] "On Positive Eigenvectors of Positive Infinite Matrices" — Tosio Kato
  • [ ] Quasi-Diagonalization of Hankel Operators — Yafaev
  • [ ] Quasi-Carleman Operators and Their Spectral Properties — Yafaev
  • [ ] Spectral and scattering theory for differential and Hankel operators (matrix elements are polynomial in $\ln(t)$) — Yafaev
  • [ ] A Commutator Method for the Diagonalization of Hankel Operators — Yafaev

• General Hankel matrices

  • [ ] "Eigenvalue estimates for (Positive Definite) Hankel matrices" — Zamarashkin, Tyrtyshnikov
  • [x] Hankel Operators and Applications (Lecture Notes) (Nehari theorem, Hardy spaces, and such)— Kostenko
  • [ ] Hankel Operators and their Applications (textbook) — Peller
  • [x] "On the Singular Values of Matrices with Displacement Structure" — Beckermann, Townsend
  • [ ] "A Non-Gaussian Limit for Linear Eigenvalue Statistics of Hankel Matrices" — Kumar, Maurya, Saha
  • [x] "On Hankel Matrices Commuting with Jacobi Matrices from the Askey Scheme" — Stampach, Stovicek
  • [x] Finite Sections of Moment Problems — Wilf
  • [x] "Small eigenvalues of large Hankel matrices" — Lawrence and Chen
    • asymptotics of bottom eigenvalue (basically $e^{-N}$)
    • roughly same result as Widom and Wilf's paper by the same name
  • [ ] Analytic Properties of Schmidt Pairs for a Hankel Operator and the Generalized Schur-Takagi Problem — Adamjan, Arov, Krein
    • shows that we can use the variational characterization for singular values/vectors by sweeping over finite-rank Hankel matrices (as opposed to finite-rank matrices)

• Sturm-Liouville

  • [ ] Sturm-Liouville Theory: Oscillation theory and the spectra of eigenvalues — TODO(evan): find the whole textbook from uchicago
  • [ ] Sturm-Liouville Problems textbook — Zettl
  • [ ] Rayleigh Quotient slides
  • [ ] 4.2: Properties of Sturm-Liouville Eigenvalue Problems — the SLP section of the great online PDE textbook by Russell Herman
  • [ ] Sturm-Liouville watered-down notes

Notes on the papers

On the Hilbert Matrix/Spectral Analysis of the Hilbert Matrix

• line of works that explicitly diagonalizes the Hilbert matrix
  • find commuting tridiagonal Jacobi operators
  • construct a unitary diagonalizing transform using Laguerre polynomials and the Cauchy kernel (more generally, askey hypergeometric functions and Hankel kernels)

On the Singular Values of Matrices with Displacement Structure