Decay of (p,q)-Fourier coefficients

Abstract

We show that essentially the speed of decay of the Fourier sine coefficients of a function in a Lebesgue space is comparable to that of the corresponding coefficients with respect to the basis formed by the generalized sine functions sin p,q .

Basis and regularity properties of ( p , q )-trigonometric functions and the decay of ( p , q )-Fourier coefficients

“…Theorem 6.1 shows that if f ∈ L r (I) and (1.6) holds, then b k decays at the same rate α as a k for α ∈ (0, 3] improving that of [13, theorem 2.4], where α is restricted to the interval (0, 2). In proposition 6.3, we show that results can be achieved in the context of Lorentz sequence spaces complementing those of theorem 2.3 in [13]. Our paper proves that when f ∈ L r (I) and (1.6) holds, while…”

“…where Id is the identity operator defined on L r (I) and M j are the linear isometries defined on page 3 of the Introduction. Consequently, using an argument similar to that in the proof of lemma 2.2 in [13], the series representations of f stated in (1.1) imply that b k = m,n∈N mn=k a n h m . Moreover, using |τ 2j+1 | (2j + 1) −γ and as the constants h 2j+1 converge to zero as j → ∞, there exists a constant k > 0 such that, for all j ∈ N,…”

“…Theorem 6.1 constitutes improvements of those stated in [13, theorem 2.4]. To prove sufficiency, we need to mention the following results of [13]. For any j ∈ N, define…”

“…the rate of the decay experiences a loss of sharpness (see, proposition 6.3 below). Work of this kind appears as one of the main results in [13] (theorem 2.3), where it is shown that the Fourier sine coefficients a k and their (p, q)-counterparts b k decay at the same rate when they are considered as elements of Lorentz sequence spaces l u,∞ for u ∈ (1, ∞).…”

Basis properties of the p , q -sine functions

“…In the latter instance, is the set of eigenfunctions of the generalized eigenvalue problem for the one-dimensional 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 ].…”