orthogonality properties
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31—40 of 46 matching pages
31: 35.4 Partitions and Zonal Polynomials
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►For any partition , the zonal polynomial
is defined by the properties
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►See Muirhead (1982, pp. 68–72) for the definition and properties of the Haar measure
.
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§35.4(ii) Properties
… ►Orthogonal Invariance
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35.4.5
.
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32: Bibliography B
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Orthogonal Polynomials in Coding Theory and Algebraic Combinatorics.
In Orthogonal Polynomials (Columbus, OH, 1989),
NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Vol. 294, pp. 25–53.
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Normalization integrals of orthogonal Heun functions.
J. Math. Phys. 38 (7), pp. 3692–3699.
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Integrable Hamiltonian systems and the Painlevé property.
Phys. Rev. A (3) 25 (3), pp. 1257–1264.
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Padé-type Approximation and General Orthogonal Polynomials.
International Series of Numerical Mathematics, Vol. 50, Birkhäuser Verlag, Basel.
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Discrete Cosine and Sine Transforms. General Properties, Fast Algorithms and Integer Approximations.
Elsevier/Academic Press, Amsterdam.
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33: 30.15 Signal Analysis
34: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
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, corresponding to distinct eigenvalues, are orthogonal: i.
…If an eigenvalue has multiplicity , the eigenfunctions may always be orthogonalized in this degenerate sub-space.
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►Orthogonality and normalization may then be chosen such that analogous to (1.18.19) and (1.18.20), we have
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►Then orthogonality and normalization relations are
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►Note that eigenfunctions for distinct (necessarily real) eigenvalues of a self-adjoint operator are mutually orthogonal.
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35: 1.2 Elementary Algebra
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►All of the above are defined for , or square matrices of order n, note that matrix multiplication is not necessarily commutative; see §1.2(vi) for special properties of square matrices.
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►The scalar product has properties
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►Two vectors and are orthogonal if
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►Square matrices (said to be of order
) dominate the use of matrices in the DLMF, and they have many special properties.
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Special Properties and Definitions Relating to Square Matrices
…36: 14.30 Spherical and Spheroidal Harmonics
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§14.30(ii) Basic Properties
►Most mathematical properties of can be derived directly from (14.30.1) and the properties of the Ferrers function of the first kind given earlier in this chapter. … ►Orthogonality
… ►where and . …37: 30.4 Functions of the First Kind
38: Bibliography N
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The resurgence properties of the large order asymptotics of the Anger-Weber function I.
J. Class. Anal. 4 (1), pp. 1–39.
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The resurgence properties of the large order asymptotics of the Anger-Weber function II.
J. Class. Anal. 4 (2), pp. 121–147.
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The resurgence properties of the incomplete gamma function II.
Stud. Appl. Math. 135 (1), pp. 86–116.
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The resurgence properties of the incomplete gamma function, I.
Anal. Appl. (Singap.) 14 (5), pp. 631–677.
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Orthogonal polynomials.
Mem. Amer. Math. Soc. 18 (213), pp. v+185 pp..
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39: Bibliography L
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Monotonicity properties of zeros of generalized Airy functions.
Z. Angew. Math. Phys. 39 (2), pp. 267–271.
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Monotonicity and convexity properties of zeros of Bessel functions.
SIAM J. Math. Anal. 8 (1), pp. 171–178.
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Higher monotonicity properties of certain Sturm-Liouville functions. III.
Canad. J. Math. 22, pp. 1238–1265.
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Higher monotonicity properties of certain Sturm-Liouville functions. IV.
Canad. J. Math. 24, pp. 349–368.
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Higher monotonicity properties of certain Sturm-Liouville functions..
Acta Math. 109, pp. 55–73.
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40: Bibliography C
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Work Group of Computational Mathematics, University of Kassel, Germany.
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Orthogonalization Procedures and the Localization of Wannier Functions.
Phys. Rev. 105, pp. 102–103.
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Some reformulated properties of Jacobian elliptic functions.
J. Math. Anal. Appl. 323 (1), pp. 522–529.
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Stability properties of disk polynomials.
Numer. Algorithms.
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Properties of generalized Freud polynomials.
J. Approx. Theory 225, pp. 148–175.
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