Basis properties of the p, q-sine functions

We improve the currently known thresholds for basisness of the family of periodically dilated p,q-sine functions. Our findings rely on a Beurling decomposition of the corresponding change of coordinates in terms of shift operators of infinite multiplicity. We also determine refined bounds on the Riesz constant associated with this family. These results seal mathematical gaps in the existing literature on the subject.


Introduction
Let p, q > 1. Let F p,q : [0, 1] −→ [0, π p,q /2] be the integral where π p,q = 2F p,q (1). The p, q-sine functions, sin p,q : R −→ [− 1,1], are defined to be the inverses of F p,q , sin p,q (x) = F −1 p,q (x) for all x ∈ 0, π p,q 2 extended to R by the rules sin p,q (−x) = − sin p,q (x) and sin p,q π p,q 2 − x = sin p,q π p,q 2 + x , which make them periodic, continuous, odd with respect to 0 and even with respect to π p,q /2. These are natural generalizations of the sine function, indeed, sin 2,2 (x) = sin(x) and π 2,2 = π , and they are known to share a number of remarkable properties with their classical counterpart [1,2]. has received some attention recently [2][3][4][5], with a particular emphasis on the case p = q. In the latter instance, S is the set of eigenfunctions of the generalized eigenvalue problem for the onedimensional p-Laplacian subjected to Dirichlet boundary conditions [6,7], which is known to be of relevance in the theory of slow/fast diffusion processes, [8]. See also the related papers [9,10].
Set e n (x) = √ 2 sin(nπ x), so that {e n } ∞ n=1 is a Schauder basis of the Banach space L r ≡ L r (0, 1) for all r > 1. The family S is also a Schauder basis of L r if and only if the corresponding change of coordinates map, A : e n −→ s n , extends to a linear homeomorphism of L r . The Fourier coefficients of s n (x) associated with e k obey the relation s n (k) = Notions of 'nearness' between bases of Banach spaces are known to play a fundamental role in classical mathematical analysis [11, pp. 265-266], [12, §I.9] or [13, p. 71]. Unfortunately, the expansion (1.1) strongly suggests that S is not globally 'near' {e n } ∞ n=1 , for example, in the Krein-Lyusternik or the Paley-Wiener sense [12, p. 106]. Therefore, classical arguments, such as those involving the Paley-Wiener stability theorem, are unlikely to be directly applicable in the present context.
In fact, more rudimentary methods can be invoked in order to examine the invertibility of the change of coordinates map. From (1.1), it follows that In Binding et al. [5], it was claimed that the left-hand side of (1.2) held true for all p = q ≥ p 1 , where p 1 was determined to lie in the segment (1, 12 11 ). Hence, S would be a Schauder basis, whenever p = q ∈ (p 1 , ∞).
Further developments in this respect were recently reported by Bushell & Edmunds [4]. These authors cleverly fixed a gap originally published in [5, lemma 5] and observed that, as the lefthand side of (1.2) ceases to hold true whenever this allows q → 1 for p > 4/(12 − π 2 ). However, note that a direct substitution of p = q in (1.5) only leads to the suboptimal condition p > π 2 /4 − 1 ≈ 1.467401. In §2, we show that the family S is ω-linearly independent for all p, q > 1, see theorem 2.1. In §5, we establish conditions ensuring that A is a homeomorphism of L 2 in a neighbourhood of the region in the (p, q)-plane where ∞ j=3 |a j | = a 1 , see theorem 5.1 and also corollary 6.2. For this purpose, in §4, we find two further criteria which generalize (1.2) in the Hilbert space setting, see corollaries 4.3 and 4.4. In this case, the Riesz constant, characterizes how S deviates from being an orthonormal basis. These new statements yield upper bounds for r(S), which improve upon those obtained from the right-hand side of (1.2), even when the latter is applicable. The formulation of the alternatives to (1.2) presented below relies crucially on work developed in §3. From lemma 3.1, we compute explicitly the Wold decomposition of the isometries M j : they turn out to be shifts of infinite multiplicity. Hence, we can extract from the expansion (1.1) suitable components which are Toeplitz operators of scalar type acting on appropriate Hardy spaces. As the theory becomes quite technical for the case r = 2 and all the estimates analogous to those reported below would involve a dependence on the parameter r, we have chosen to restrict our attention with regards to these improvements only to the already interesting Hilbert space setting. Section 6 is concerned with particular details of the case of equal indices p = q, and it involves results on both the general case r > 1 and the specific case r = 2. Rather curiously, we have found another gap which renders incomplete the proof of invertibility of A for p 1 < p < 2 originally published in [5]. See remark 6.3. Moreover, the application of Bushell & Edmunds [4, theorem 4.5] only gets to a basisness threshold ofp 1 ≈ 1.198236 > 12 11 , wherep 1 is defined by the identity See also [2, remark 2.1]. In theorem 6.5, we show that S is indeed a Schauder basis of L r for p = q ∈ (p 3 , 6 5 ), where p 3 ≈ 1.087063 < 12 11 , see [14, problem 1]. As 6 5 >p 1 , basisness is now guaranteed for all p = q > p 3 (figure 3).
In §7, we report on our current knowledge of the different thresholds for invertibility of the change of coordinates map, both in the case of equal indices and otherwise. Based on the new criteria found in §4, we formulate a general test of invertibility for A which is amenable to analytical and numerical investigation. This test involves finding sharp bounds on the first few coefficients a k (p, q). See proposition 7.1. For the case of equal indices, this test indicates that S is a Riesz basis of L 2 for p = q > p 6 , where p 6 ≈ 1.043917 < p 2 .
All the numerical quantities reported in this paper are accurate up to the last digit shown, which is rounded to the nearest integer. In the online version of this manuscript, 1 we have included fully reproducible computer codes which can be employed to verify the calculations reported. Hence, mn=j f m a n = 0 ∀ j ∈ N.

Linear independence
(2.1) We show that all f j = 0 by means of a double induction argument. Suppose that f 1 = 0. We prove that all a k = 0. Indeed, clearly a 1 = 0 from (2.1) with j = 1. Now, assume inductively that a j = 0 for all j = 1, . . . , k − 1. From (2.1), for j = k, we obtain Then, a k = 0 for all k ∈ N. As this would contradict the fact that A = 0, necessarily f 1 = 0.
Suppose now inductively that f 1 , . . . , f l−1 = 0 and f l = 0. We prove that again all a k = 0. First, a 1 = 0 from (2.1) with j = l, because f m a n = f l a 1 .
Second, assume by induction that a j = 0 for all j = 1, . . . , k − 1. From (2.1) for j = lk, we obtain 0 = f l a k + mn=lk m =l n =k f m a n = f l a k .
The latter equality is a consequence of the fact that, for mn = lk with m = l and n = k, either m < l (indices for the f m ) or n < k (indices for the a n ). Hence, a k = 0 for all k ∈ N. As this would again contradict the fact that A = 0, necessarily all f k = 0, so that f = 0.
The second assertion is shown as follows. Assume that A ∈ B(L 2 ). If f ∈ Ker A * , then f , Ag = 0 for all g ∈ L 2 , so f ⊥ Ran A which, in turns, means that f ⊥ s n for all n ∈ N. On the other hand, if the latter holds true for f , then f ⊥ Ae n for all n ∈ N, so A * f = 0, as required.
Therefore, S is a Riesz basis of L 2 if and only if A ∈ B(L 2 ) and RanA = L 2 . A simple example illustrates how a family of dilated periodic functions can break its property of being a Riesz basis.

The different components of the change of coordinates map
The fundamental decomposition of A given in (1.1) allows us to extract suitable components formed by Toeplitz operators of scalar type [15]. In order to identify these components, we begin by determining the Wold decomposition of the isometries M j , [15,16]. See remark 3.4.
The matrix of T(B) has the block representation [15, §5.9] The matrix associated with T(b) has exactly the same scalar form, replacing I by 1 ∈ B(C). Then, Hence,

Remark 3.4.
It is possible to characterize the change of coordinates A in terms of Dirichlet series, and recover some of the results here and below directly from this characterization. See for example the insightful paper [18] and the complete list of references provided in the addendum [19]. However, the full technology of Dirichlet series is not needed in the present context. A further development in this direction is reported elsewhere.

Invertibility and bounds on the Riesz constant
A proof of (1.2) can be achieved by applying corollary 3.3 assuming that Our next goal is to formulate concrete sufficient condition for the invertibility of A and corresponding bounds on r(S), which improve upon (1.2), whenever r = 2. For this purpose, we apply corollary 3.3 assuming that B has now the three-term expansion Optimal region of invertibility in lemma 4.1. The horizontal axis is α and the vertical axis is β.
Because sin p,q (x) > 0 for all x ∈ (0, π p,q ), then a 1 > 0. Below, we substitute α = a 3 /a 1 and β = a 9 /a 1 , then apply lemma 4.1 appropriately in order to determine the invertibility of A whenever pairs (p, q) lie in different regions of the (p, q)-plane. For this purpose, we establish the following hierarchy between a 1 and a j for j = 3, 9, whenever the latter are non-negative.
Proof. First, observe that sin p,q (π p,q x) is continuous, it increases for all x ∈ (0, 1 2 ) and it vanishes at x = 0.
Because a 1 > 0, (4.1) supersedes (1.2), only when the pair (p, q) is such that a 9 > 0. From this corollary, we see below that the change of coordinates is invertible in a neighbourhood of the threshold set by the condition (1.
Proof. The proof is similar to that of corollary 4.3.
We see in the following that corollary 4.3 is slightly more useful than corollary 4.4 in the context of the dilated p, q-sine functions. However, the latter is needed in the proof of the main theorem 5.1.

Riesz basis properties beyond the applicability of (1.2)
Our first goal in this section is to establish that the change of coordinates map associated with the family S is invertible beyond the region of applicability of (1.2). We begin by recalling a calculation which was performed in the proof of [3, proposition 4.1] and which will be invoked several times below. Let a(t) be the inverse function of sin p,q (π p,q t). Then, Indeed, integrating by parts twice and changing the variable of integration to t = sin p,q (π p,q x) yields a j (p, q) = √ 2 1 0 sin p,q (π p,q x) sin(jπ x) dx Theorem 5.1. Let r = 2. Suppose that the pair (p,q) is such that the following two conditions are satisfied (a) a 3 (p,q), a 9 (p,q) > 0 (b) ∞ j=3 |a j (p,q)| = a 1 (p,q).
Then, there exists a neighbourhood (p,q) ∈ N ⊂ (1, ∞) 2 , such that the change of coordinates A is invertible for all (p, q) ∈ N .
Proof. From the dominated convergence theorem, it follows that each a j (p, q) is a continuous function of the parameters p and q. Therefore, by virtue of (5.1) and a further application of the dominated convergence theorem, j∈F |a j | is also continuous in the parameters p and q. Here, F can be any fixed set of indices, but below in this proof we need to consider only F = N \ {1, 9} for the first possibility and F = N \ {1, 3, 9} for the second possibility. Writeã j = a j (p,q). The hypothesis implies (ã 3 /ã 1 ,ã 9 /ã 1 ) ∈ T, because 0 <ã 3 a 1 +ã 9 a 1 < 1.
As β > α/(4 − α), Thus, Thus, once again by continuity of all quantities involved, there exists a neighbourhood (p,q) ∈ N 3 ⊂ (1, ∞) 2 such that the left-hand side and hence the right-hand side of (4.2) hold true for all (p, q) ∈ N 3 . The conclusion follows by defining either We now examine other further consequences of the corollaries 4.3 and 4.4.

The case of equal indices
We now consider in closer detail the particular case p = q < 2. Our analysis requires setting various sharp upper and lower bounds on the coefficients a j (p, p) for j = 1, 3, 5, 7, 9. This is our first goal. Proof. All the stated bounds are determined by integrating a suitable approximation of sin p,p (π p,p x). Each one requires a different set of quadrature points, but the general structure of the arguments in all cases is similar. Without further mention, in the following, we repeatedly use  For reference, we also show sin p 6 ,p 6 (π p 6 ,p 6 x), sin(3π x), sin 4/3,4/3 (π 4/3,4/3 x) and sin 2,2 (π x) = sin(π x).
Let p be as in the hypothesis. Then, similar to the previous case (a), sin p,p (π p,p x j ) > y j j = 1, 2, 3.

Remark 6.3.
In Binding et al. [5], it was claimed that the hypothesis of (1.2) held true whenever p = q ≥ p 1 for a suitable 1 < p 1 < 12 11 . The argument supporting this claim [5, §4] was separated into two cases: p ≥ 2 and 12 11 ≤ p < 2. With our definition 3 of the Fourier coefficients, in the latter case, it was claimed that |a j | was bounded above by As it turns, there is a missing power 2 in the term π 12/11,12/11 for this claim to be true. This corresponds to taking second derivatives of sin p,p (π p,p t) and it can be seen by applying the Cauchy-Schwartz inequality in (5.1). The missing factor is crucial in the argument and renders the proof of Binding et al. [5, theorem 1] incomplete in the latter case.
Then, I 0 + I 1 < 0 and I 3 + I 4 < 0, so The following result fixes the proof of the claim made in [5, §4 and claim 2] and improves the threshold of invertibility determined in [4, theorem 4.5]. Theorem 6.5. There exists 1 < p 3 < 6 5 such that The family S is a Schauder basis of L r (0, 1) for all p 3 < p = q < 6 5 and r > 1.
Hence, the first statement is ensured as a consequence of the intermediate value theorem. From (5.1), it follows that j ∈{1,3,5,7} for all p 3 < p < 6 5 . Lemma 6.1 guarantees positivity of a j for j = 3, 5, 7. Then, by rearranging this inequality, the second statement becomes a direct consequence of (1.2).

The thresholds for invertibility and the regions of improvement
If sharp bounds on the first few Fourier coefficients a j (p, q) are at hand, the approach employed above for the proof of theorem 6.5 can also be combined with the criteria (4.1) or (4.2). A natural question is whether this would lead to a positive answer to the question of invertibility for A, whenever ∞ k=3 a j ≥ a 1 .
In the case of (4.1), we see below that this is indeed the case. The key statement is summarized as follows.
(b) a 3 (a 1 + a 9 ) > 4a 9 a 1 . If Proof. Assume that the hypotheses are satisfied. The combination of (5.1) and (7.1) gives Then, and so the conclusion follows from (4.1).
We now discuss the connection between the different statements established in the previous sections with those of the papers [3][4][5]. For this purpose, we consider various accurate approximations of a j and a j . These approximations are based on the next explicit formulae Here, I is the incomplete beta function, B is the beta function and Γ is the gamma function. Moreover, by considering exactly the steps described in [4]    log cot π x 2π p,q 2 F 1 threshold for all r > 1 a j < 0 for some j for p 2 ≈ 1.043989. The condition a 3 (p, p)(a 1 (p, p) + a 9 (p, p)) > 4a 9 (p, p)a 1 (p, p) is fulfilled for all p 4 < p < 12 11 , where p 4 ≈ 1.038537. The Fourier coefficients a j (p, p) ≥ 0 for all 1 ≤ j ≤ 35 whenever 1 < p < 12 11 . Remarkably, we need to get to k = 35, for a numerical verification of the conditions of proposition 7.1 allowing p < p 2 . Indeed, we remark the following.
This indicates that the threshold for invertibility of A in the Hilbert space setting for p = q is at least p 6 . Now, we examine the general case. The graphs shown in figures 4 and 5 correspond to regions in the (p, q)-plane near (p, q) = (1, 1). Curves on figure 4 that are in red (online version) are relevant only to the Hilbert space setting r = 2. Black curves (online version) pertain to r > 1. Figure 4a and a blowup shown in figure 4b have two solid (black) lines. One that shows the limit of applicability of theorem 5.2(a) and one that shows the limit of applicability of the result of [3]. The dashed line indicates where (1.3) occurs. To the left of that curve, (1.2) is not applicable. There are two filled regions of different colours in (figure 4a), which indicate where a 3 (a 1 + a 9 ) < 4a 1 a 9 and where a j < 0 for j = 3, 9. Proposition 7.1 is not applicable in the union of these regions. We also show the lines where a 3 = 0 and a 9 = 0. The latter forms part of the boundary of this union. The solid red line corresponding to the limit of applicability of theorem 5.2(b) is also included in figure 4a,d. To the right of that line, in the white area, we know that A is invertible for r = 2. The blowup in figure 4b clearly shows the gap between theorem 5.2(a),(b) in this r = 2 setting.
Certainly, p = q = 2 is a point of intersection for all curves where a j = 0 for j > 1. These curves are shown in figure 4c also for j = 5 and j = 7. In this figure, we also include the boundary of the region where a 3 (a 1 + a 9 ) < 4a 1 a 9 and the region where a j < 0 now for j = 3, 5, 7, 9. Note that the curves for a 7 = 0 and a 9 = 0 form part of the boundary of the latter. Comparing figure 4a  and figure 4c, the new line that cuts the p axis at p ≈ 1. proposition 7.1 for k = 7 is applicable (for p to the right of this line). The gap between the two red lines (case r = 2) indicates that proposition 7.1 can significantly improve the threshold for basisness with respect to a direct application of theorem 5.2(b).
As we increase k, the boundary of the corresponding region moves to the left, see the blowups in figure 4d,e. The two further curves in red located very close to the vertical axis, correspond to the precise value of the parameter k where proposition 7.1 allows a proof of invertibility for the change of coordinates which includes the break made by (1.3). For k < 35, the region does not include the dashed black line, for k = 35, it does include this line. The region shown in blue indicates a possible place where corollary 4.3 may still apply, but further investigation in this respect is needed. Figure 5 concerns the statement of theorem 5.2(c). The small wedge shown in green is the only place where the former is applicable. As it turns, it appears that the conditions of corollary 4.4 prevent it to be useful for determining invertibility of A in a neighbourhood of (p, q) = (1, 1). However, in the region shown in green, the upper bound on the Riesz constant consequence of (4.2) is sharper than that obtained from (1.2).
Ethics statement. No part of this research has been conducted on humans or animals. Data accessibility. All the numerical quantities reported in this paper are accurate up to the last digit shown, which is rounded to the nearest integer. In the online version available at http://arxiv.org/abs/1405.7337, we have included fully reproducible computer codes which can be employed to verify the calculations reported.