which are also orthogonal

SearchAdvancedHelp

(0.002 seconds)

11—20 of 28 matching pages

11: 18.38 Mathematical Applications

… ►

Quadrature

… ►also the case $\beta =0$ of (18.14.26), was used in de Branges’ proof of the long-standing Bieberbach conjecture concerning univalent functions on the unit disk in the complex plane. … ►See also §14.30. … ►

Riemann–Hilbert Problems

… ►

Group Representations

…

12: Errata

… ►We have significantly expanded the section on associated orthogonal polynomials, including expanded properties of associated Laguerre, Hermite, Meixner–Pollaczek, and corecursive orthogonal and numerator and denominator orthogonal polynomials. …We also discuss non-classical Laguerre polynomials and give much more details and examples on exceptional orthogonal polynomials. We have also completely expanded our discussion on applications of orthogonal polynomials in the physical sciences, and also methods of computation for orthogonal polynomials. … ►

At the end of the subsection the text which originally stated “then the measure in (18.28.10) is uniquely determined” has been updated to be “then the measure in (18.28.10) is the unique orthogonality measure”.

… ►

The Legendre polynomial $P_n$ was mistakenly identified as the associated Legendre function $P_n$ in §§10.54, 10.59, 10.60, 18.18, 18.41, 34.3 (and was thus also affected by the bug reported below). These symbols now link correctly to their definitions. Reported by Roy Hughes on 2022-05-23

…

13: 18.17 Integrals

§18.17 Integrals

… ►

§18.17(ii) Integral Representations for Products

… ►

§18.17(v) Fourier Transforms

… ►

§18.17(vi) Laplace Transforms

… ►

§18.17(vii) Mellin Transforms

…

14: 18.36 Miscellaneous Polynomials

… ►

§18.36(iv) Orthogonal Matrix Polynomials

… ►These results are proven in Everitt et al. (2004), via construction of a self-adjoint Sturm–Liouville operator which generates the $L_n^{(-k)}(x)$ polynomials, self-adjointness implying both orthogonality and completeness. … ►

§18.36(vi) Exceptional Orthogonal Polynomials

… ►EOP’s are non-classical in that not only are certain polynomial orders missing, but, also, not all EOP polynomial zeros are within the integration range of their generating measure, and EOP-orthogonality properties do not allow development of Gaussian-type quadratures. … ►resulting in orthogonality; …

15: 18.15 Asymptotic Approximations

§18.15 Asymptotic Approximations

… ►Also, … ►See also Deaño et al. (2013). … ►See also Geronimo et al. (2004). … ►See also Dunster (1999), Atia et al. (2014) and Temme (2015, Chapter 32).

16: 18.33 Polynomials Orthogonal on the Unit Circle

§18.33 Polynomials Orthogonal on the Unit Circle

►

§18.33(i) Definition

… ►Also denote … ►

§18.33(iii) Connection with OP’s on the Line

… ►

Recurrence Relations

…

17: 18.25 Wilson Class: Definitions

§18.25 Wilson Class: Definitions

… ►For the Wilson class OP’s $p_n(x)$ with $x=\lambda (y)$: if the $y$-orthogonality set is $\{0,1,\mathrm{\dots },N\}$, then the role of the differentiation operator $d/dx$ in the Jacobi, Laguerre, and Hermite cases is played by the operator ${\mathrm{\Delta }}_y$ followed by division by ${\mathrm{\Delta }}_y(\lambda (y))$, or by the operator ${\nabla }_y$ followed by division by ${\nabla }_y(\lambda (y))$. Alternatively if the $y$-orthogonality interval is $(0,\mathrm{\infty })$, then the role of $d/dx$ is played by the operator ${\delta }_y$ followed by division by ${\delta }_y(\lambda (y))$. … ►Under certain conditions on their parameters the orthogonality range for the Wilson polynomials and continuous dual Hahn polynomials is $(0,\mathrm{\infty })\cup S$, where $S$ is a specific finite set, e. … ►

§18.25(ii) Weights and Standardizations: Continuous Cases

…

18: 18.39 Applications in the Physical Sciences

… ►Also presented are the analytic solutions for the $L^2$, bound state, eigenfunctions and eigenvalues of the Morse oscillator which also has analytically known non-normalizable continuum eigenfunctions, thus providing an example of a mixed spectrum. … ►which is the quantum superposition principle. … ►This is not the orthogonality of Table 18.8.1, as the co-ordinate arguments depend, independently on $p$ and $q$. … ►which is Bohr’s quantization for the Coulomb bound state energies (18.39.31). … ►which corresponds to the exact results, in terms of Whittaker functions, of §§33.2 and 33.14, in the sense that projections onto the functions ${\varphi }_{n,l}(sr)/r$, the functions bi-orthogonal to ${\varphi }_{n,l}(sr)$, are identical. …

19: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►This insures the vanishing of the boundary terms in (1.18.26), and also is a choice which indicates that $𝒟(T)=𝒟(T^{\ast })$, as $f(x)$ and $g(x)$ satisfy the same boundary conditions and thus define the same domains. … ►Other choices of boundary conditions, identical for $f(x)$ and $g(x)$, and which also lead to the vanishing of the boundary terms in (1.18.26), each lead to a distinct self adjoint extension of $T$. … ►Orthogonality and normalization may then be chosen such that analogous to (1.18.19) and (1.18.20), we have … ►Then orthogonality and normalization relations are … ►This work is well overviewed by Coddington and Levinson (1955, Ch. 9), and then applied in detail by Titchmarsh (1946), Titchmarsh (1962a), Titchmarsh (1958), and Levitan and Sargsjan (1975) which also connects the Weyl theory to the relevant functional analysis. …

20: 18.16 Zeros

… ►

Inequalities

… ►

Asymptotic Behavior

… ►See also (18.16.2), (18.16.3) or (3.5.23)–(3.5.25). … ►Also, … ►

§18.16(vii) Discriminants

…