harmonic analysis

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1: Donald St. P. Richards

… ►Richards has published numerous papers on special functions of matrix argument, harmonic analysis, multivariate statistical analysis, probability inequalities, and applied probability. He is editor of the book Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications, published by the American Mathematical Society in 1992, and coeditor of Representation Theory and Harmonic Analysis: A Conference in Honor of R. A. Kunze (with T. …

2: 15.17 Mathematical Applications

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§15.17(iii) Group Representations

►For harmonic analysis it is more natural to represent hypergeometric functions as a Jacobi function (§15.9(ii)). …Harmonic analysis can be developed for the Jacobi transform either as a generalization of the Fourier-cosine transform (§1.14(ii)) or as a specialization of a group Fourier transform. …

3: Tom H. Koornwinder

… ►Koornwinder has published numerous papers on special functions, harmonic analysis, Lie groups, quantum groups, computer algebra, and their interrelations, including an interpretation of Askey–Wilson polynomials on quantum SU(2), and a five-parameter extension (the Macdonald–Koornwinder polynomials) of Macdonald’s polynomials for root systems BC. …

4: Bibliography T

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Harmonic Analysis on Symmetric Spaces and Applications. II. Springer-Verlag, Berlin.

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External Links:

ISBN 3-540-96663-3, MathReview (Stephen Gelbart), ZentralBlatt

Referenced by:

§35.1, §35.5(ii), Notations K.

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Representation Theory and Harmonic Analysis. Contemporary Mathematics, Vol. 191, American Mathematical Society, Providence, RI.

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Notes:

Papers from the conference held in honor of Ray A. Kunze at the AMS Special Session, Cincinnati, Ohio, January 12–14, 1994

External Links:

ISBN 0-8218-0310-7, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

Profile Donald St. P. Richards.

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5: Bibliography W

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Harmonic Analysis. In Studies in Real and Complex Analysis, I. I. Hirschman (Ed.), Studies in Mathematics, Vol. 3, pp. 124–178.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§1.15(iii), §1.15(iv), §1.15(v).

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On the functions associated with the parabolic cylinder in harmonic analysis. Proc. London Math. Soc. 35, pp. 417–427.

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External Links:

Document, ZentralBlatt

Referenced by:

§12.1, §12.5(i), §12.5(ii), §12.9(i).

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6: Bibliography H

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Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains. Translations of Mathematical Monographs, Vol. 6, American Mathematical Society, Providence, RI.

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Notes:

Reprinted in 1979.

External Links:

ISBN 0-8218-1556-3, MathReview, ZentralBlatt

Referenced by:

§35.4(i).

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7: Bibliography M

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Spherical Harmonics. An Elementary Treatise on Harmonic Functions with Applications. 3rd edition, International Series of Monographs in Pure and Applied Mathematics, Vol. 98, Pergamon Press, Oxford.

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External Links:

MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

Ch.14.

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New ladder operators for a rational extension of the harmonic oscillator and superintegrability of some two-dimensional systems. J. Math. Phys. 54 (10), pp. Paper 102102, 12 pp..

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External Links:

MathReview Entry

Referenced by:

§18.36(vi).

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Connection between quantum systems involving the fourth Painlevé transcendent and $k$-step rational extensions of the harmonic oscillator related to Hermite exceptional orthogonal polynomial. J. Math. Phys. 57 (5), pp. Paper 052101, 15 pp..

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External Links:

ISSN 0022-2488, Document, Link, MathReview Entry

Referenced by:

§18.38(iii).

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On the evaluation of Bessel functions. Appl. Comput. Harmon. Anal. 1 (1), pp. 116–135.

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External Links:

ISSN 1063-5203, Document, MathReview, ZentralBlatt

Referenced by:

§10.77(iii).

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A Course in Real Analysis. Academic Press Inc., San Diego, CA.

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External Links:

ISBN 0-12-742830-5, MathReview (Sherif T. El-Helaly), ZentralBlatt

Referenced by:

§1.4(v), §18.2(i).

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8: Bibliography L

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Asymptotic Analysis. Mathematical Centre Tracts, Mathematisch Centrum, Amsterdam.

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External Links:

MathReview (C. Comstock), ZentralBlatt

Referenced by:

Ch.2.

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Computations of spheroidal harmonics with complex arguments: A review with an algorithm. Phys. Rev. E 58 (5), pp. 6792–6806.

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Notes:

Mathematica package for eigenvalues and solutions of the spheroidal wave equations

External Links:

Code(SpheroidF), Document

Referenced by:

2nd item, 3rd item.

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Microwave specific attenuation by oblate spheroidal raindrops: An exact analysis of TCS’s in terms of spheroidal wave functions. J. Electromagn. Waves Appl. 12 (6), pp. 709–711.

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External Links:

ISSN 0920-5071, Document, MathReview

Referenced by:

§30.14(v).

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The spectral analysis of three families of exceptional Laguerre polynomials. J. Approx. Theory 202, pp. 5–41.

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External Links:

ISSN 0021-9045, Document, Link, MathReview (Edmundo J. Huertas)

Referenced by:

§18.36(vi).

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An Introduction to Fourier Analysis and Generalised Functions. Cambridge Monographs on Mechanics and Applied Mathematics, Cambridge University Press, New York.

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External Links:

ISBN 0521091284, MathReview (G. Temple), ZentralBlatt

Referenced by:

§1.16(ix), §1.16(v).

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9: Bibliography F

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Theory and Computation of Spheroidal Harmonics with General Arguments. Master’s Thesis, The University of Western Australia, Department of Physics.

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Notes:

Contains Mathematica package for spheroidal wave functions of complex arguments with complex parameters.

External Links:

FullText

Referenced by:

§30.12, 1st item, 2nd item.

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Analysis on Symmetric Cones. Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford-New York.

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External Links:

ISBN 0-19-853477-9, MathReview (Hong Ming Ding), ZentralBlatt

Referenced by:

§35.1, §35.4(i), §35.7(i), §35.7(iii), §35.8(i), §35.8(iv), §35.8(v), §35.8(v), Ch.35, Notations J, Notations K.

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Singularity analysis of generating functions. SIAM J. Discrete Math. 3 (2), pp. 216–240.

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ISSN 0895-4801, Document, MathReview (E. Rodney Canfield), ZentralBlatt

Referenced by:

§2.10(iv).

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Application of semiclassical scattering analysis. Ann. Physics 7 (3), pp. 287–322.

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External Links:

ISSN 0003-4916, Document, MathReview, ZentralBlatt

Referenced by:

§9.16.

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Chebyshev Polynomials in Numerical Analysis. Oxford University Press, London.

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Notes:

Reprinted in 1972 with corrections.

External Links:

MathReview (G. N. Lance), ZentralBlatt

Referenced by:

§3.11(ii).

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10: 10.73 Physical Applications

… ►Bessel functions first appear in the investigation of a physical problem in Daniel Bernoulli’s analysis of the small oscillations of a uniform heavy flexible chain. … ►Consequently, Bessel functions $J_n\left(x\right)$, and modified Bessel functions $I_n\left(x\right)$, are central to the analysis of microwave and optical transmission in waveguides, including coaxial and fiber. … ►Bessel functions enter in the study of the scattering of light and other electromagnetic radiation, not only from cylindrical surfaces but also in the statistical analysis involved in scattering from rough surfaces. … ►With the spherical harmonic $Y_{\mathrm{\ell },m}\left(\theta ,\varphi \right)$ defined as in §14.30(i), the solutions are of the form $f=g_{\mathrm{\ell }}(k\rho )Y_{\mathrm{\ell },m}\left(\theta ,\varphi \right)$ with $g_{\mathrm{\ell }}={𝗃}_{\mathrm{\ell }}$, ${𝗒}_{\mathrm{\ell }}$, ${𝗁}_{\mathrm{\ell }}^{(1)}$, or ${𝗁}_{\mathrm{\ell }}^{(2)}$, depending on the boundary conditions. … ►The analysis of the current distribution in circular conductors leads to the Kelvin functions $\mathrm{ber}x$, $\mathrm{bei}x$, $\ker x$, and $\mathrm{kei}x$. …