The q-cosine Fourier transform and the q-heat equation

The aim of this work is to establish in great detail The q-Fourier analysis related to the q-cosine. The wise reader will note that the considered q-cosine coincides with the one given by T.H. Koornwinder and S.F. Swarttouw. Through the q-cosine product formula, we define and analyze the properties of the q-even translation and the q-convolution. Adopting the Titchmarsh approach, we study the q-cosine Fourier transform and its inverse formula. The second theme of this paper is an application of the q-Fourier analysis developed earlier. We extend the heat representation theory inaugurated by P.C. Rosenbloom and D.V. Widder to the q-analogue. We construct the q-solution source, the q-heat polynomials and solve the q-analytic Cauchy problem.


Introduction
During the last years, an intensive work was founded about the so-called q-basic theory. Taking account of the well-known Ramanujan works shown at the beginning of this century by Jackson ([9, 10]), many authors such as Askey, Gasper, Ismail, Rogers, Andrew, Koornwinder, and others (see references) have recently developed this topic.
The present article is devoted to the study of the q-analogue of the Fourier transforms and to showing how it plays a central role in solving the q-heat equation associated to the second q-derivative operator. The method used here differs from those given by T.H. Koornwinder and R.F. Swarttouw, who discovered a q-analogue of Hankel's Fourier-Bessel via some q-analogue orthogonality relations. We note that Ph. Feinsilver [4] gave a q-Harmonic Analysis for a q-Laplace transform with inversion formula.
Without entering into a dilemma through the analysis presented here, it seems that the point of view of T.H. Koornwinder and R.F. Swarttouw [12] is more suitable for harmonic analysis. We take as definition of the q-cosine the one given by the previous authors with a simple change and we prefer to write it as a series of functions denoted as b n (x; q 2 ). This q-cosine appears as an eigenfunction of the operator Δ q . Owing to a nice paper [12], we give a product formula written with the q-Jackson integral and we study the q-translation and the q-convolution. Next we define the q-analogue of the cosine Fourier transform with the purpose to find the transformation inverse. To this end, we prove the equivalent of the so-called Riemann-Lebesgue Lemma and discover that the Titchmarsh approach holds [15].
A motivation behind this work is to state some result about the q-heat equation associated to Δ q operator. We attempt to extend the heat representation theory studied in many cases ( [5,7,14], etc.). We define the q-heat polynomials and find that they are linked to the q-Hermite polynomials [13] and constitute with the q-associated functions a biorthogonal system. We conclude by solving the q-analytic Cauchy problem related to the q-heat equation.

Notations and preliminaries
We begin by recalling some q-elements of quantum analysis adapting the notation used in the book of Gasper and Rahman [6]. Let a and q be real numbers such that 0 < q < 1, the q-shift factorial is defined by A basic hypergeometric series is r ϕ s (a 1 , . . . , a r ; b 1 , . . . , b s ; q, z) = ∞ k=0 (a 1 , . . . , a r ; q) k (b 1 , . . . , b s , q; q) k (−1) k q ( k 2 ) 1+s−r z k .
A function f is q-regular at zero if lim n→∞ f (xq n ) = f (0) exists and is independent of x.
The q-derivative D q f of a function f is defined by The q-derivative at zero is defined by if it exists and does not depend on x.
We introduce the set The q-integral of Jackson is defined by The q-integration by parts is given for suitable functions f and g by The q-analogue of the Gamma function is defined as which tends to Γ (x) when q tends to 1 − .

q-Trigonometric functions
We define the q-cosine as where we have put b n x; q 2 = b n 1; In the same way, the q-sine is given by These q-trigonometric functions differ and should not be confused with the functions cos q and sin q considered in [6, p. 23]; but coincide with the one given in [12] and [15] with a minor change of variable. Furthermore, we have The following statements hold: Proof We only prove Part 2 since Part 1 is deduced from the definition of Δ q . The coefficients b n (1; q 2 ), defined by (6), can be written as where we have put q = e −t , t > 0.
As a consequence of the previous proposition, we can show that for λ ∈ C the function cos λx; q 2 = ∞ 0 (−1) n b n x; q 2 λ 2n , is the unique analytic solution of the q-differential equation with Proposition 3.2 For x ∈ R q and Log(1−q) Log(q) ∈ Z, we have 1.
Proof To prove Parts 1 and 2, we use the properties of 1 φ 1 given in [12] and their connection to the q-cosine. We obtain hence Parts 1 and 2 follow. A similar argument shows Parts 3 and 4.
Now we try to find a product formula for the q-cosine functions. We begin by proving the following result.

Proposition 3.3
For reals x and y, y = 0, we have cos x, q 2 cos y, q 2 k+s cos q s y, q 2 .
Note that this formula can be expressed in terms of 1 ϕ 1 as follows Proof To show (11) and (12), we begin by expanding the q-cosines in series absolutely and uniformly convergent on every compact of R. From the product rule of series and the fact that we obtain for y = 0 On the other hand, we have We deduce (11) after the interchange of summation order. To prove (12), we write cos x; q 2 cos y; q 2 = I + J, In I , we make the change k − s into k and use the equality Now we make the change k + s into k in J and use the equalities This identity is easily deduced from [11]. Then we obtain We add these sums to find that (12) holds.
Remark 3.4 (1) If we replace y by q y , x by q x , and assume the proposition the hypothesis, we obtain from (12) that the following integral representation holds (2) The product formula (11) leads to cos x; q 2 cos y; q 2 = ∞ n=0 b n x; q 2 Δ n q cos y; q 2 .

q-Translation and q-convolution
We define, for x and y in R q , the measure where δ u denotes the unit mass supported at u, and (2) d q μ (x,y) is of bounded variation. (3) Proof For n, m ∈ Z, the relation (2.3) from [12] leads to D q n , q m ; q s = D q m , q n ; q s+m−n .
We obtain Part 1 after the change s − n + m by s. To prove Part 2, we suppose | x y | ≤ 1; from the formulas (2.4) in [12] we have Finally, from (2.8) in [12], we can show that Part 3 is true.
We introduce the q-translation which generalizes the even translation given by Let f be a function with support in R q , the q-translation is defined for x and y in R q by From the previous proposition and the q-product formula (12), we have (ii) Δ q T q,y f = T q,y Δ q f, Δ q,;y T q,y f = T q,y Δ q,y f. From the relation we can write the q-translation of a function f as and have in the limit when q tends to 1 − the classical even translation cited before. Now we denote by L 1 q (R q ) the space of functions f defined on R q such that Then we are able to define the q-convolution by where f and g are two functions in L 1 q (R q ). We can show that this space is an algebra.

q-Analogue of Fourier-cosine
In this section, we suppose Log(1−q) Log(q) ∈ Z. The q-analogue of Fourier transform is defined for λ ∈ R q by where f is a function in L 1 q (R q ). This definition is the same (after a minor change) as that given by T.H. Koornwinder and R.F. Swarttouw (see [12]).

Proposition 5.1 For f, g ∈ L 1
q (R q ), the following properties hold: (2) F q (f q g) = F q (f )F q (g).
Proof Part 1. The inequality (21) follows from Proposition 3.2 and the identity Part 2 is a direct consequence of the q-product formula (12). Part 3 is obtained after the exchange of the integration order and taking into account the invariability of the q-integral by the q-translation. Now we focus our attention on the inversion of the linear map F q . We proceed by looking at the q-analogue of the Riemman-Lebesgue Lemma, the localization theorem, and we show that the Titchmarsh approach holds in the q-theory.

Proposition 5.2 Let f be a function in L
Proof To prove this, first we have from Proposition 3.2 And for λ ∈ R q we have lim λ→∞ f (x) cos λx; q 2 = 0, λ∈ R q , so the result is true.

Proposition 5.3 We have the identity
Proof This is a consequence of (2.8) in [12].
Proof Indeed, the first hypothesis shows that for an arbitrary ε > 0 we have for large The second hypothesis and Proposition 3.2 show that Since from Proposition 3.2 we have that sin(λx; q 2 ) tends to zero as λ tends to ∞, the proposition is then a direct consequence.

Theorem 5.5 (The q-cosine Fourier integral theorem)
then

q-Heat equation and q-heat polynomials
In this section, the two q-analogues of the elementary exponential functions are crucial and they are defined by and e x; q 2 = 1 These functions satisfy the identity e x; q 2 E −x; q 2 = 1, and have as limit, when q tends to 1 − , the classical exponential function. Now we purpose to give the q-analogue of the heat equation associated to the second derivative operator (even in x) We consider as q-heat equation associated to the second q-derivative operator the partial q-difference equation We take as the initial condition

q-Solution source
To find the solution source related to the q-heat equation, we apply the Fourier method with the adapted q-Fourier cosine studied before. Putting Eq. (28) becomes and, taking into account conditions (29), we obtain The problem consists in finding the function which has e(−λ 2 t; q 2 ) as its q-Fouriercosine transform. For this end, we need the following lemma.
Lemma 6.1 For n = 0, 1, 2, . . . and t > 0, we have Proof From (26) we find Secondly, the use of the well-known Ramanujan [8] identity leads to the result after minor computation.
As an immediate consequence we are now able to define the q-source solution associated to the q-heat equation (28) by In the same manner as in the classical heat equation theory, we put G x, y, t; q 2 = T y,q G x, t; q 2 , with T y,q being the q-translation studied in Sect. 4. Through this approach we show that the solution of the q-Cauchy problem (28) and (29) can been written in the form of It is natural to ask how other properties such as the positivity of G(x, t; q 2 ) and the existence of the q-semigroup can be established.
6.2 q-Heat polynomials Proposition 6.3 It is easy to see that, for x ∈ R and t > 0, the analytic function λ → e −λ 2 t; q 2 cos λx; q 2 , is a solution of (28) and it has the expansion e −λ 2 t, q 2 cos λx, with the functions b n being given by (6). From Proposition 3.1 we deduce immediately the following properties: We note that formula (34) can be inverted:
Proof We have ∞ 0 G x, y, t, q 2 b n y; q 2 d q y = ∞ 0 T q,x G y, t, q 2 b n y; q 2 d q y In [14], the authors defined the so-called associated functions by the Appell transform. We extend this notion by defining for t > 0 the q-associated functions of v 2n by w 2n (x, t; q) = (−1) n Δ n q,y G x, y, t; q 2 y=0 .

Convergence of n≥0 α n v 2n (x, t; q)
Now we establish the following estimates that will be needed later Lemma 6.6 For n = 0, 1, . . . and 0 < Proof Indeed, the first inequality is a consequence of b 0 (1; q 2 ) = 1 and the hypothesis, and the second follows from where K is a constant depending δ. Theorem 6.10 Let (α n ) be a sequence of real or complex numbers such that lim n→∞ n e q −2(n−1) |α n | 1/n = 1 σ < +∞.
Then the series n≥0 α n v 2n (x, t; q), converges in the strip and converges uniformly in any region of this strip.
To prove the theorem, we adopt the same approach as in [14] by taking account of the q-equivalent estimation (41).
Remark If we write u(x, t) as the sum of the previous series, then this function satisfies the q-heat equation (28) u(x; t) is an analytic function of two variables x and t in the strip S σ given by (42) and satisfies the q-heat equation (28). Furthermore, the coefficients α n are given by α n = Δ n q u(x, t) (x,t)=(0,0) .
Proof To show this, we note that the theorem gives that u(x, t) is analytic in the whole strip S σ . Now for a fixed integer p the series n≥0 α n+p v 2n (x, t; q) converges uniformly in any compact region of S σ . To prove (44), it suffices to see that for integers n and p we have Δ n q,x v 2p (x, t; q) (0,0) = δ n,p , where δ n,p is the Kronecker symbol.
Finally the following statement is established. Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.