linear operators

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1: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

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§1.18(iii) Linear Operators on a Hilbert Space

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Bounded and Unbounded Linear Operators

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$x, y$	real variables.
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$\mathcal{L}$	linear operator defined on a manifold $\mathcal{M}$
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3: 1.3 Determinants, Linear Operators, and Spectral Expansions

§1.3 Determinants, Linear Operators, and Spectral Expansions

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§1.3(iv) Matrices as Linear Operators

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Linear Operators in Finite Dimensional Vector Spaces

►Square matices can be seen as linear operators because

\mathbf{A}(\alpha\mathbf{a}+\beta\mathbf{b})=\alpha\mathbf{A}\mathbf{a}+\beta% \mathbf{A}\mathbf{b}

for all

\alpha,\beta\in\mathbb{C}

and

\mathbf{a},\mathbf{b}\in\mathbf{E}_{n}

, the space of all

n

-dimensional vectors. …

4: Bibliography D

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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.

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External Links:: ISSN 0021-9991, MathReview, ZentralBlatt
Referenced by:: §22.11.
Encodings:: BibTeX

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N. Dunford and J. T. Schwartz (1988) Linear operators. Part II. Wiley Classics Library, John Wiley & Sons, Inc., New York.

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Notes:: Spectral theory. Selfadjoint operators in Hilbert space, With the assistance of William G. Bade and Robert G. Bartle, Reprint of the 1963 original, A Wiley-Interscience Publication
External Links:: ISBN 0-471-60847-5, MathReview Entry
Referenced by:: (1.18.68), §1.18(ix), §1.18(ix), §1.18(ix), §1.18(x).
Encodings:: BibTeX

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… ►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:

Section 1.2

New subsections, 1.2(v) Matrices, Vectors, Scalar Products, and Norms and 1.2(vi) Square Matrices, with Equations (1.2.27)–(1.2.77).
Section 1.3

The title of this section was changed from “Determinants” to “Determinants, Linear Operators, and Spectral Expansions”. An extra paragraph just below (1.3.7). New subsection, 1.3(iv) Matrices as Linear Operators, with Equations (1.3.20), (1.3.21).
Section 1.4

In 1.4(v), three new paragraphs on Stieltjes integrals/measure, with Equations (1.4.23_1)–(1.4.23_3).
Section 1.8

In Subsection 1.8(i), the title of the paragraph “Bessel’s Inequality” was changed to “Parseval’s Formula”. We give the relation between the real and the complex coefficients, and include more general versions of Parseval’s Formula, Equations (1.8.6_1), (1.8.6_2). The title of Subsection 1.8(iv) was changed from “Transformations” to “Poisson’s Summation Formula”, and we added an extra remark just below (1.8.14).
Section 1.10

New subsection, 1.10(xi) Generating Functions, with Equations (1.10.26)–(1.10.29).
Section 1.13

New subsection, 1.13(viii) Eigenvalues and Eigenfunctions: Sturm-Liouville and Liouville forms, with Equations (1.13.26)–(1.13.31).
Section 1.14(i)

Another form of Parseval’s formula, (1.14.7_5).
Section 1.16

We include several extra remarks and Equations (1.16.3_5), (1.16.9_5). New subsection, 1.16(ix) References for Section 1.16.
Section 1.17

Two extra paragraphs in Subsection 1.17(ii) Integral Representations, with Equations (1.17.12_1), (1.17.12_2); Subsection 1.17(iv) Mathematical Definitions is almost completely rewritten.
Section 1.18

An entire new section, 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions, including new subsections, 1.18(i)–1.18(x), and several equations, (1.18.1)–(1.18.71).

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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

\Delta

is the forward difference operator (§3.6(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-az)(1-bz)R_{n-1}(z;qa,qb,c,d\,|\,q)

, where

R_{n}(z)

is given in (18.28.1_5). …

9: 18.19 Hahn Class: Definitions

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Hahn class (or linear lattice class). These are OP’s $p_{n}(x)$ where the role of $\frac{\mathrm{d}}{\mathrm{d}x}$ is played by $\Delta_{x}$ or $\nabla_{x}$ or $\delta_{x}$ (see §18.1(i) for the definition of these operators). The Hahn class consists of four discrete and two continuous families.

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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. …