Duhamel solutions of non-homogeneous 𝑞²-analogue wave equations

Rubin, Richard

doi:10.1090/S0002-9939-06-08525-X

Duhamel solutions of non-homogeneous $q^2$-analogue wave equations
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Proc. Amer. Math. Soc. 135 (2007), 777-785 Request permission

Abstract:

$q$-analogue non-homogeneous wave equations are solved by a Duhamel solution strategy using constructions with $q$-analogue Fourier multipliers to compensate for the dependence of the analogue differential Leibnitz rule on the parity of the functions involved.

References

J. Bustoz and J. L. Cardoso, Basic analog of Fourier series on a $q$-linear grid, J. Approx. Theory 112 (2001), no. 1, 134–157. MR 1857606, DOI 10.1006/jath.2001.3599
M. Fichtmüller and W. Weich, The limit transition $q\to 1$ of the $q$-Fourier transform, J. Math. Phys. 37 (1996), no. 9, 4683–4689. MR 1408114, DOI 10.1063/1.531647
George Gasper and Mizan Rahman, Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, vol. 35, Cambridge University Press, Cambridge, 1990. With a foreword by Richard Askey. MR 1052153
G. H. Hardy, J. E. Littlewood, G. Polya, Inequalities, Cambridge Univ. Press, Cambridge, 1964.
Yitzhak Katznelson, An introduction to harmonic analysis, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 0248482
Tom H. Koornwinder and René F. Swarttouw, On $q$-analogues of the Fourier and Hankel transforms, Trans. Amer. Math. Soc. 333 (1992), no. 1, 445–461. MR 1069750, DOI 10.1090/S0002-9947-1992-1069750-0
Richard L. Rubin, A $q^2$-analogue operator for $q^2$-analogue Fourier analysis, J. Math. Anal. Appl. 212 (1997), no. 2, 571–582. MR 1464898, DOI 10.1006/jmaa.1997.5547
Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, N.J., 1971. MR 0304972