By Sergei Suslov

It was once with the e-book of Norbert Wiener's e-book ''The Fourier In tegral and sure of Its purposes" [165] in 1933 by means of Cambridge Univer sity Press that the mathematical neighborhood got here to achieve that there's another method of the examine of c1assical Fourier research, particularly, in the course of the thought of c1assical orthogonal polynomials. Little may he be aware of at the moment that this little inspiration of his may support bring in a brand new and exiting department of c1assical research referred to as q-Fourier research. makes an attempt at discovering q-analogs of Fourier and different comparable transforms have been made by way of different authors, however it took the mathematical perception and instincts of none different then Richard Askey, the grand grasp of precise features and Orthogonal Polynomials, to work out the traditional connection among orthogonal polynomials and a scientific concept of q-Fourier research. The paper that he wrote in 1993 with N. M. Atakishiyev and S. ok Suslov, entitled "An Analog of the Fourier remodel for a q-Harmonic Oscillator" [13], was once most likely the 1st major ebook during this sector. The Poisson k~rnel for the contin uous q-Hermite polynomials performs a task of the q-exponential functionality for the analog of the Fourier vital lower than considerationj see additionally [14] for an extension of the q-Fourier remodel to the overall case of Askey-Wilson polynomials. (Another very important component of the q-Fourier research, that merits thorough research, is the speculation of q-Fourier series.

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**Extra resources for An Introduction to Basic Fourier Series**

**Example text**

_ (_ aq l/4 ei8, _ aq l/4 e-i8;ql/2)00 cq (x, a) - (2. 5) from Appendix A. Eq. 32) shows that the function (qa 2; q2) cq (x; a) is an entire function in a and z when ei8 = qZ. 8). 4. 4. Basie Trigonometrie Functions The basic cosine Cq (x, 1/j w) and basic sine Sq (x,1/jw) functions can be introduced by the following analog of the Eu1er formula Eq (x, 1/j iw) = Cq (x, 1/j w) + iSq (x, 1/j w). 3) -qw jq 00 -q _qeiO+irp,_qeiB-irp,_qeirp-iB,_qe-iB-irp 2) ( x 4c,oa _q, q3/2, _q3/2 j q,-w . 1), + 1/) and sinw (x + 1/), respectively.

4) Verify the following identities 1 a (I-')'Y (I-') ="2 'Y(21-') , + (a 2 - 1) 'Y2 (I-') = a (21-'), a (I-' + 11) a (21-') - (a 2 - 1) 'Y (I-' + 11) 'Y (21-') = a (I-' - 11), a (I-' + 11) 'Y (21-') - 'Y (I-' + 11) a (21-') = 'Y (I-' - 11), a (-I-') = a (I-') , 'Y ( -I-') = -'Y (I-'), a (I-' - 1) 'Y (I-') - a (11 - 1) 'Y (11) = a (I-' + 11 - 1) 'Y (I-' - 11) , "Y(I-' - 1) 'Y (I-') - 'Y (11 - 1) 'Y (11) = 'Y (I-' + 11 - 1) 'Y (I-' - 11) , a 2 (I-') where the functions a (I-') and 'Y (I-') are defined in the previous exercise.

For example, Iwi < 1 only. C. 8) j q2, q) 24 2. 5). 7) are entire functions in z when ei8 = qZ. 8). Eqs. 11) determine the large w-asymptorics of the basie eosine Cq (x;w) and basie sine Bq (x;w) funetions, respectively. 4. BASIC TRIGONOMETRIC FUNCTIONS + )-1 e-i9w(q2w2e-2i9;q2) 00 q2n+1/4(_qe-2i9;q) ( _. 00 " . 6). 13) are not in terms of the usual inverse power sequenee {( xw) -n } but are sums of two expansions in terms of the "inverse generalized powers" (q2 w2e±2i9;q2):1; cf. [54]. 7) in the entire complex w-plane.