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linear operators


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1: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
§1.18(iii) Linear Operators on a Hilbert Space
Bounded and Unbounded Linear Operators
2: 1.1 Special Notation
x , y real variables.
linear operator defined on a manifold
3: 1.3 Determinants, Linear Operators, and Spectral Expansions
§1.3 Determinants, Linear Operators, and Spectral Expansions
§1.3(iv) Matrices as Linear Operators
Linear Operators in Finite Dimensional Vector Spaces
Square matices can be seen as linear operators because 𝐀 ( α 𝐚 + β 𝐛 ) = α 𝐀 𝐚 + β 𝐀 𝐛 for all α , β and 𝐚 , 𝐛 𝐄 n , the space of all n -dimensional vectors. …
4: Bibliography D
  • B. Deconinck and J. N. Kutz (2006) Computing spectra of linear operators using the Floquet-Fourier-Hill method. J. Comput. Phys. 219 (1), pp. 296–321.
  • N. Dunford and J. T. Schwartz (1988) Linear operators. Part II. Wiley Classics Library, John Wiley & Sons, Inc., New York.
  • 5: Errata
    This especially included updated information on matrix analysis, measure theory, spectral analysis, and a new section on linear second order differential operators and eigenfunction expansions. … The specific updates to Chapter 1 include the addition of an entirely new subsection §1.18 entitled “Linear Second Order Differential Operators and Eigenfunction Expansions” which is a survey of the formal spectral analysis of second order differential operators. The spectral theory of these operators, based on Sturm-Liouville and Liouville normal forms, distribution theory, is now discussed more completely, including linear algebra, matrices, matrices as linear operators, orthonormal expansions, Stieltjes integrals/measures, generating functions. …
  • Chapter 1 Additions

    The following additions were made in Chapter 1:

  • 6: DLMF Project News
    error generating summary
    7: 2.9 Difference Equations
    Many special functions that depend on parameters satisfy a three-term linear recurrence relation …or equivalently the second-order homogeneous linear difference equation …in which Δ is the forward difference operator3.6(i)). … This situation is analogous to second-order homogeneous linear differential equations with an irregular singularity of rank 1 at infinity (§2.7(ii)). … For an introduction to, and references for, the general asymptotic theory of linear difference equations of arbitrary order, see Wimp (1984, Appendix B). …
    8: 18.38 Mathematical Applications
    The Dunkl type operator is a q -difference-reflection operator acting on Laurent polynomials and its eigenfunctions, the nonsymmetric Askey–Wilson polynomials, are linear combinations of the symmetric Laurent polynomial R n ( z ; a , b , c , d | q ) and the ‘anti-symmetric’ Laurent polynomial z 1 ( 1 a z ) ( 1 b z ) R n 1 ( z ; q a , q b , c , d | q ) , where R n ( z ) is given in (18.28.1_5). …
    9: 18.19 Hahn Class: Definitions
  • 1.

    Hahn class (or linear lattice class). These are OP’s p n ( x ) where the role of d d x is played by Δ x or x or δ x (see §18.1(i) for the definition of these operators). The Hahn class consists of four discrete and two continuous families.

  • 10: William P. Reinhardt
    He has recently carried out research on non-linear dynamics of Bose–Einstein condensates that served to motivate his interest in elliptic functions. Older work on the scattering theory of the atomic Coulomb problem led to the discovery of new classes of orthogonal polynomials relating to the spectral theory of Schrödinger operators, and new uses of old ones: this work was strongly motivated by his original ownership of a 1964 hard copy printing of the original AMS 55 NBS Handbook of Mathematical Functions. …