The basis property of generalized Jacobian elliptic functions

The Jacobian elliptic functions are generalized to functions including the generalized trigonometric functions. The paper deals with the basis property of the sequence of generalized Jacobian elliptic functions in any Lebesgue space. In particular, it is shown that the sequence of the classical Jacobian elliptic functions is a basis in any Lebesgue space if the modulus $k$ satisfies $0 \le k \le 0.99$.


Introduction
The Jacobian elliptic function sn(x, k) and the complete elliptic integral of the first kind K(k) play important roles in expressing exact solutions of, for example, the pendulum equation u ′′ + λ sin u = 0, a typical bistable equation u ′′ + λu(1 − u 2 ) = 0, and so on. Now we will propose new generalization of sn(x, k) and K(k). For constants p, q ∈ (1, ∞) and k ∈ [0, 1), we define a generalized Jacobian elliptic function sn pq (x, k) : [0, K pq (k)] → [0, 1] with a modulus k as where p ′ = p/(p − 1) and We extend the domain of sn pq (x, k) to R so that we obtain a 4K pq (k)-period function like the sine function, and call the extended function sn pq (x, k) again. Then, sn 22 (x, k) = sn(x, k) and K 22 (k) = K(k) when p = q = 2; and sn pq (x, 0) = sin pq x and K pq (0) = π pq /2 when k = 0, where sin pq x is the generalized trigonometric function and π pq is the half period of sin pq x, which will be introduced in Section 3 below. Therefore, sn pq (x, k) is also generalization of both sn(x, k) and sin pq x.
In the previous paper [22], the author proposed another generalization of sn(x, k) and K(k), and applied them to bifurcation problems for p-Laplacian. As we will see in Section 3, sn pq (x, k) and K pq (k) above are defined in a slightly different way from those in [22], but sn pq (x, k) also satisfies the following equation involving p-Laplacian nevertheless.
While the generalization of K(k) of [22] converges to a finite value as k → 1 when p > 2, the K pq (k) diverges to ∞ as k → 1 for any p > 1. In this sense, sn pq (x, k) has closer properties to sn(x, k) than the function defined in [22]. In the present paper, we will show the basis property of functions f n (x, k) = sn pq (2nK pq (k)x, k), n = 1, 2, . . . , (1.1) which means that the family of these functions is a basis in Banach spaces.
Here, a sequence {ϕ n } in a Banach space X is called a basis for X if for every u ∈ X there exists a unique sequence of scalars {α n } such that u = ∞ n=1 α n ϕ n in the strong sense. In general, when we try to find an approximation of a given function by a family of functions {ϕ n }, it is desirable that {ϕ n } is a basis which approximates to the function with convergence of higher order as possible. Concerning this, for example, we have known an interesting study [3] of Boulton and Lord. They study the best index q for which {sin q (nπ q x)} approximates well to the solution of p-Poisson problem, where sin q x = sin qq x and π q = π qq . The basis property is quite fundamental to such a stimulating problem.
When p = q = 2, the sequence (1.1) is the family of Jacobian elliptic functions {sn(2nK(k)x, k)}. In this case, Craven [4] proves that if the modulus k satisfies 0 ≤ k ≤ 0.99, then the sequence is complete in L 2 (0, 1). Since the sequence is not orthogonal, we have no guarantee of its basis property.
We will give another corollary of Theorem 1.1, whose conditions are verified easier than (1.2) and (1.3). Corollary 1.4. Let p, q ∈ (1, ∞) and r = max{p ′ , q}. If then {f n (x, k)} forms a Riesz basis of L 2 (0, 1) and a Schauder basis of L α (0, 1) for any α ∈ (1, ∞) when The paper is organized as follows. In Section 2 we give a summary of general properties of bases in Banach spaces. In Section 3 we recall the generalized trigonometric functions and introduce new generalization of Jacobian elliptic functions. In Section 4 we observe properties of the generalized Jacobian elliptic function sn pq (x, k) and its quarter period K pq (k). To show that the sequence (1.1) is a basis in L α (0, 1) for any α ∈ (0, 1), we depend on the strategy of Binding et al. [1] and Edmunds et al. [10]. Our main device is a linear mapping T of L α (0, 1), satisfying T e n = f n , where e n = sin (nπx), and decomposing into a linear combination of certain isometries. In Section 5 we show that T is a bounded operator for p ∈ (1, ∞). Section 6 is devoted to the proof of boundedness of the inverse for the ranges (1.2) and (1.3).

Properties of Bases
In this section we will give a summary of properties of bases in Banach spaces. For details, we can refer to Gohberg and Kreȋn [12], Higgins [13], and Singer [21].
A sequence {x n } in an infinite dimensional Banach space X is called a basis of X if for every x ∈ X there exists a unique sequence of scalars {α n } such that x = ∞ i=1 α i x i (i.e., such that lim n→∞ x − n i=1 α i x i = 0). A basis {x n } of a topological linear space U is said to be a Schauder basis of U, if all coefficient functionals f n , n = 1, 2, . . ., are continuous on U.
i.e., there exists a constant c > 0 such that we have for all finite sequences of scalars α 1 , . . . , α n .
i.e., there exists a constant C > 0 such that we have for all finite sequences of scalars α 1 , . . . , α n . (c) a Riesz basis, if it is both a Bessel basis and a Hilbert basis, i.e., there exist two constants c > 0 and C > 0 such that we have for all finite sequences of scalars α 1 , . . . , α n . Example 2.3. In the space X = L p (−π, π), p ∈ (1, ∞), the sequence {x n }, where is a bounded Bessel basis if p ≥ 2 and a bounded Hilbert basis if 1 < p ≤ 2.
In particular, it is a Riesz basis if p = 2.
We call two sequences {φ n } and {ψ n } in Banach space X equivalent if there exists a linear homeomorphism (i.e., bounded, linear and invertible operator) T on X such that ψ n = T (φ n ) for every n. Note that by 'invertible' we mean that T −1 exists and is bounded on all of X.

Generalized Functions
This section is devoted to the definitions of two kinds of generalized functions.
For any constants p, q ∈ (1, ∞), we define π pq by where B and Γ are the Beta and Gamma functions, respectively. Then, for any x ∈ [0, π pq /2] we define sin pq x by We extend the domain of sin pq x to [0, π pq ] by sin pq x = sin pq (π pq − x), and furthermore, to the whole of R by sin pq (x + π pq ) = − sin pq x, so that sin pq x has 2π pq -periodicity. We can see that π 22 = π and sin 22 x = sin x. Moreover, the function y = sin pq x satisfies that y, We agree that π p and sin p x denote π pp and sin pp x when p = q, respectively. In that case, we can also refer to [5,6,7,8,11,15].
Using sin pq x, for x ∈ [0, π pq /2] we also define Clearly, cos pq x is a decreasing function in x from [0, π pq /2] onto [0, 1]. We extend the domain of cos pq x to [−π pq /2, π pq /2] by cos pq x = cos pq (−x), and furthermore, to the whole of R in the same way as sin pq x. Then, cos pq x has 2π pq -periodicity. We can see that cos 22 x = cos x. An analogue of tan x is obtained by defining for those values of x at which cos pq x = 0. This means that tan pq x is defined for all x ∈ R except for the points (k + 1/2)π pq (k ∈ Z). We denote by cos p x and tan p x as for the case sin p x. The functions sin p x and cos p x are useful for Prüfer transformation of half-linear differential equations. For this, see [6,7,11,19].
It is useful to collect formulae for case p = r ′ and q = r for some r ∈ (1, ∞).
We can find many other properties of these functions in [10,14].
Remark 3.1. There are some different definitions of cos pq x from (3.1). For example, Drábek and Manásevich [9] define cos pq x by and so (3.5) gives cos p pq x + sin q pq x = 1, which is slightly different from (3.2). The fact that sin pq x satisfies (3.5) is essential, independently of the definition of cos pq x.
Proof. Putting 1 − t r = s in the definition of π r ′ r , we have It suffices to show that tB(t, t) is decreasing on (0, 1 Clearly, the right-hand side is decreasing in t (note that log v < 0), so that tB(t, t) is also decreasing on (0, 1). Furthermore, since lim t→1 tB(t, t) = B(1, 1) = 1 and lim t→0 log tB(t, t) = log 2, we obtain the values of limits.
Remark 3.3. We can find another proof of Proposition 3.2 in [10, Lemma 2.4], in which they use the fact that the area of r-circle |x| r + |y| r = 1 is π r ′ r (see also [16]).
We also state the case p = r ′ and q = r for some r ∈ (1, ∞).
As mentioned in Introduction, the author [22] has introduced another generalized Jacobian elliptic functions, which also include both the Jacobian elliptic functions and the generalized trigonometric functions. However, we should note that the definitions above of K pq (k) and sn pq (x, k) are slightly different from those of [22], in which the common index of 1 − k q t q to (3.7) and (3.8) is not 1/p ′ but 1/p. On account of the index, K pq (k) has similar asymptotic behavior near k = 1 as K(k), indeed, lim k→1 K pq (k) = ∞ for any p, q ∈ (1, ∞).
To observe the convergence properties of generalized Jacobian elliptic functions as k → 1, we will prepare generalized hyperbolic functions, for which similar definitions are seen in [15].
For x ∈ [0, ∞), we define sinh pq x by and extend its domain to R by sinh pq x = − sinh pq (−x). Using sinh pq x, for x ∈ [0, ∞), we define and extend its domain to R by cosh pq x = cosh pq (−x). The function tanh pq x is defined by We agree that sinh p x, cosh p x and tanh p x denote sinh pp x, cosh pp x and tanh pp x when p = q, respectively. Putting p = q and t p = s p /(1 − s p ) in (3.9), we have Then, it is easy to prove the following properties: for any p, q ∈ (1, ∞) and all x ∈ R, 4 Properties of sn pq (x, k) and K pq (k) In this section we observe some properties of generalized Jacobian elliptic function sn pq (x, k) and its quarter period K pq (k). The function y = sn pq (x, k) satisfies that sn pq (0, k) = 0, sn pq (K pq (k), k) = 1, 0 < sn pq (x, k) < 1 for x ∈ (0, K pq (k)), y ∈ C 1 [0, K pq (k)], and When p > 2, we see that y ′′ ∈ L 1 (0, K pq (k)) and To obtain the estimate of K r ′ r (k) in Lemma 4.7 below, we state Tchebycheff's integral inequality in [18,20].
If one of the functions f or g is nonincreasing and the other nondecreasing, then the inequality in (4.2) is reversed.
Concerning the following Lemmas 4.2-4.6, we can refer to [2,10,14] for the corresponding results of sin pq x and π pq , which are in case k = 0 of sn pq (x, k) and K pq (k). Lemma 4.7 extends an estimate of K(k) by Qi and Huang [20,Eq.(10)] to that of K r ′ r (k) for any r ∈ (1, ∞).

Properties of the Function sn pq (x, k)
Since f n (x, k) = f 1 (nx, k), it suffices to observe properties of f 1 (x, k) = sn pq (2K pq (k)x, k) in order to study those of (1.1).
Proof. First we will show that sn pq (2K pq (k)x, k) is decreasing in p ∈ (1, ∞). Let 1 < p < r < ∞. Putting and Since g(t) is increasing in t ∈ (0, 1) when p < r, it is easy to see that which means that G(t) < 0, i.e, f ′ (t) < 0 for each t ∈ (0, 1]. Thus, Therefore we conclude that so that sn pq (2K pq (k)x, k) is decreasing in p > 1. The assertions for q and k are proved in a similar way. It is enough to replace f (t) by (1 < q < r) and sn −1 pq (t, l) sn −1 pq (t, k) (0 ≤ k < l < 1), respectively, and to replace g(t) by respectively.
Proof. The assertion on p immediately follows from The remaining parts also follow from the form of K pq (k). Lemma 4.5. Let p, q ∈ (1, ∞) and k ∈ [0, 1). Then Proof. Putting 1 − t q = x p ′ in (3.7), we have The integration of the right-hand side is equal to K q ′ p ′ (k q p ′ ).
Proof. Case p ′ ≤ q follows from only Lemma 4.2. Case p ′ > q is also proved similarly after using Lemma 4.5.
Moreover, putting sin r ′ r θ = t, we have Similarly, putting cos r ′ r θ = t and t r = 1−k r k r s r 1−s r , we obtain Thus, we accomplished the first and second inequalities of (4.3). Finally, from the equality above, we obtain the third inequality of (4.3) The graphs of terms of (4.3) for r = 2 can be shown in Figure 1.

The Operator T
Let α ∈ (1, ∞) be an arbitrary number. In this section, we will make the functions correspond to the sine series e n (x) = sin(nπx) ∈ L α (0, 1), n = 1, 2, . . . , which form a basis of L α (0, 1). Figure 1: The graphs of terms of (4.3) for r = 2. The black line and the gray lines indicate K r ′ r (k) and the others, respectively.
Proof. By Example 2.3, any function in L α (−1, 1) has a unique sine-cosine series representation. For any f ∈ L α (0, 1), we can thus represent its odd extension to L α (−1, 1) uniquely in a sine series, so the e n form a basis of L α (0, 1). Since {e n } and {f n } are equivalent, according to Proposition 2.4 the same is true for the f n . It follows from Proposition 2.1 that they form a Schauder basis of L α (0, 1). The argument for a Riesz basis when α = 2 is similar and follows from Proposition 2.5.
In the remainder of this section we define T as a linear combination of certain isometries of L α (0, 1). Then we show that T is a bounded operator satisfying T e n = f n , n = 1, 2, . . ., for all p, q ∈ (1, ∞).
The functions f n have Fourier sine series expansions An argument involving symmetry with respect to the middlepoint x = 1/2 easily shows that f 1 (l) = 0 whenever l is even. On account of this property, we can show f n (l) by using f 1 (l) as follows. In what follows we will often denote f 1 (m) by τ m . We first find a bound on |τ m | which will be crucial in the definition of T below. Since τ m = 0 if m is even, we may assume that m is odd. Integration by parts ensures that where the integrals exist because f ′′ 1 ∈ L 1 (0, 1). In fact (4.1) shows that In order to construct the linear operator T , we next define isometries M m of the Banach space L α (0, 1) by M m g(x) := g * (mx), m = 1, 2, . . ., where g * is its successive antiperiodic extension of g over R + by g * = g on [0, 1], and Notice that M m e n = e mn . Proof.
Proof of Corollary 1.3. Let 1 < p ′ ≤ q < ∞. Then r = q, and it suffices to show that (1.2) is satisfied. Since we have the inequality tB(t, t) ≤ 2 in the proof of Proposition 3.2, we obtain so that (1.2) holds.
Proof of Corollary 1.4. Suppose that q and r satisfy (1.4). Since tB(t, t) ≤ 2 as the proof of Corollary 1.3, we obtain