Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010, Article ID 128746, 22 pages doi:10.1155/2010/128746 Research Article A Cohen Type Inequality for Fourier Expansions of Orthogonal Polynomials with a Nondiscrete Jacobi-Sobolev Inner Product ´ Bujar Xh Fejzullahu1 and Francisco Marcellan2 Department of Mathematics, Faculty of Mathematics and Natural Sciences, University of Prishtina, Mother Teresa 5, Prishtină 10000, Kosovo e Departamento de Matem´ ticas, Escuela Polit´ cnica Superior, Universidad Carlos III de Madrid, a e Avenida de la Universidad 30, 28911 Legan´ s, Spain e Correspondence should be addressed to Francisco Marcell´ n, pacomarc@ing.uc3m.es a Received May 2010; Accepted 24 August 2010 Academic Editor: J ozef Bana´ s ´ Copyright q 2010 B Xh Fejzullahu and F Marcell´ n This is an open access article distributed a under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited α,β Let {Qn x }n≥0 denote the sequence of polynomials orthogonal with respect to the non-discrete 1 Sobolev inner product f, g f x g x dμα,β x λ −1 f x g x dμα 1,β x , where λ > −1 α β − x x dx with α > −1, β > −1 In this paper, we prove a Cohen and dμα,β x α,β type inequality for the Fourier expansion in terms of the orthogonal polynomials {Qn x }n Necessary conditions for the norm convergence of such a Fourier expansion are given Finally, the failure of almost everywhere convergence of the Fourier expansion of a function in terms of the orthogonal polynomials associated with the above Sobolev inner product is proved Introduction Let dμα,β x − x α x β dx with α, β > −1 be the Jacobi measure supported on the interval −1, We say that f ∈ Lp dμα,β if f is measurable on −1, and f Lp dμα,β < ∞, where ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ f Lp dμα,β 1/p −1 f x p dμα,β x ⎪ ⎪ ⎪esssup f x , ⎪ ⎩ −1 0, are read as follows see 12, Theorem 8.21.8 : α,β Pn cos θ π −1/2 n−1/2 sin θ −α−1/2 cos θ −β−1/2 cos kθ γ O n−1 , 2.9 where k n α β /2, and γ − α 1/2 π/2 For α, β, μ > −1 and ≤ q ≤ ∞ see 12, page 391 Exercise 91 , as well as 10, 2.2 1−x μ α,β Pn x p 1/p dx ⎧ ⎪n−1/2 , ⎪ ⎪ ⎪ ⎨ ∼ n−1/2 log n ⎪ ⎪ ⎪ ⎪ α− 2μ /p ⎩n , if 2μ > pα − 1/p , if 2μ pα − if 2μ < pα − p , p , p 2.10 Asymptotics of Jacobi-Sobolev Orthogonal Polynomials α, β α, β Let us denote by Qn the monic Jacobi-Sobolev polynomial of degree n, that is, Qn x α, β−1 −1 α,β Qn x From 2.4 and 3, formula 2.7 see also 4, 14 in a more general hn framework , we have the following relation between the Jacobi-Sobolev and Jacobi monic orthogonal polynomials Proposition 3.1 For α, β > −1, α,β Pn x α,β an−1 α, β Pn−1 x α,β Qn α,β x dn−1 λ Qn−1 x , n ≥ 1, 3.1 where an−1 α, β is given in 2.5 and α,β Pn dn λ an α, β L2 dμα,β α,β Qn Sα , n ≥ 3.2 Journal of Inequalities and Applications Proposition 3.2 One gets: α,β Qn α,β S2 ∼ λn2 P α 1,β n−1 L2 dμα 3.3 1,β In particular, for dn λ defined in 3.2 one obtains 4λn2 dn λ ∼ 3.4 Proof We apply the same argument as in the proof of Theorem in 15 Using the extremal property α,β Pn L2 dμα,β inf L2 dμα,β P : deg P n, P monic , 3.5 we get the following: α,β Qn Qn α,β S2 α,β L2 dμα,β λ Qn α,β α,β L2 dμα ≥ Pn α 1,β λn2 Pn−1 L2 dμα,β 1,β L2 dμα 1,β 3.6 α,β On the other hand, from the extremal property of Qn α,β Qn α,β α,β S2 α,β ≤ Pn an−1 α, β Pn−1 α,β α,β Pn α,β ≤ Pn an−1 α, β Pn−1 L2 dμα,β , 2.4 , and 2.6 , we have α,β S2 L2 an−1 α, β α 1,β dμα,β α,β Since by 2.3 and 2.5 we have Pn 3.6 and 3.7 yield 3.3 α,β S2 L2 dμα,β λn2 Pn−1 α,β Pn−1 L2 dμα 3.7 1,β α 1,β λn2 Pn−1 L2 dμα,β ∼ P α 1,β n−1 L2 dμα 1,β L2 dμα 1,β and an α, β ∼ 1/2, then As a straightforward consequence of Propositions 3.1 and 3.2, using 2.1 we deduce the following Corollary 3.3 For α, β > −1, n α β α,β Pn x 2n α β n α α,β P x 2n α β n−1 α,β Qn x α,β dn−1 λ Qn−1 x , 3.8 Journal of Inequalities and Applications where n ≥ and α,β−1 dn λ hn dn λ α,β−1 hn−1 ∼ 2λn2 3.9 Corollary 3.4 For α > −1 and β > 0, α,β−1 Pn α,β x Qn α,β x dn−1 λ Qn−1 x , n ≥ 1, 3.10 and for α, β > −1, n α β α 1,β Pn−1 α,β x Qn α,β dn−1 λ Qn−1 x x n ≥ , 3.11 Proof The first statement follows from Proposition 3.1 and 2.4 The second one follows by taking derivatives in 3.10 and using 2.6 α,β Using 3.10 in a recursive way, the representation of the polynomials Qn α,β−1 x }∞ becomes of the elements of the sequence {Pn n n α,β Qn n −1 k bk x α,β−1 λ Pn−k x , in terms 3.12 k n where bk k j dn−j λ n λ and b0 λ n Proposition 3.5 There exists a constant c > such that the coefficients bk n bk k λ < c 1/n2 λ in 3.11 satisfy for all n ≥ and ≤ k ≤ n Proof From 3.9 , we have limn n dn λ Thus, there exist n0 ∈ N and a constant c > such that n dn λ < for all n ≥ n0 and n dn λ < c for n 1, , n0 − Therefore, for ≤ k ≤ n − n0 , n bk k dn−j λ < λ j 1 , n2k 3.13 and for n − n0 ≤ k ≤ n, n bk n−n0 k dn−j λ λ dn−j λ j ≤ j n−n0 1 c n2n−n0 k−n n0 3.14 ck−n n0 1 ≤ cn0 k n2k n2 Journal of Inequalities and Applications α,β Proposition 3.6 a For the polynomials Qn α,β Qn , one obtains ≤ cn−1/2 − x x −α/2−1/4 x −β/2 1/4 , 3.15 , 3.16 for x ∈ −1, , α ≥ −1/2, and β ≥ 1/2 α,β b For the polynomials Q n Qn α,β , one has the following estimate: ≤ cn1/2 − x x −α/2−3/4 x −β/2−1/4 for x ∈ −1, , α > −1, and β ≥ −1/2 Proof a Using 3.12 , we have the following: α,β Qn n ≤ cos θ n bk α,β−1 λ Pn−k cos θ 3.17 k From 2.7 , it is straightforward to prove that, for α, β ≥ −1/2 and k α,β n n−1/2 θ−α−1/2 π − θ n−k ≤c Pn−k cos θ 0, 1, , n − 1, −β−1/2 3.18 Thus, according to Proposition 3.5, α,β Qn cos θ n ≤ n bk α,β−1 λ Pn−k cos θ k n ≤ cbn λ cn−1/2 θ−α−1/2 π − θ −β 1/2 n−1 k ≤ cn−1/2 θ−α−1/2 π − θ −β 1/2 2k 3.19 On the other hand, from 3.11 , the proof of the case b can be done in a similar way Proposition 3.7 Let α, β > −1, then α,β Qn Qn x α,β ⎧ ⎪cnα , ⎪ ⎪ ⎪ ⎪ ⎨ ≤ cnβ−1 , ⎪ ⎪ ⎪ ⎪ ⎪ −1/2 ⎩cn , x ≤ for x ∈ 0, , α ≥ − , for x ∈ −1, , β ≥ , 1 for x ∈ −1, , α ≤ − , β ≤ , 2 ⎧ ⎪cnα , for x ∈ 0, , α > −1, ⎨ ⎪cnβ , ⎩ for x ∈ −1, , β ≥ − 3.20 Journal of Inequalities and Applications Proof Taking into account that the Jacobi polynomials satisfy the following see 12, paragraph below Theorem 7.32.1 : α,β Pn x ⎧ ⎪cnα , ⎪ ⎪ ⎪ ⎪ ⎨ ≤ cnβ , ⎪ ⎪ ⎪ ⎪ ⎪ −1/2 ⎩cn , for x ∈ 0, , α ≥ − , for x ∈ −1, , β ≥ − , 1 for x ∈ −1, , α ≤ − , β ≤ − , 2 3.21 for n ≥ 1, thus, for ≤ j ≤ n − 1, α,β Pn−j x ⎧ ⎪c n − j ⎪ ⎪ ⎪ n ⎪ ⎪ ⎪ ⎪ ⎨ n−j ≤ c ⎪ n ⎪ ⎪ ⎪ ⎪ ⎪ n−j ⎪ ⎪c ⎩ n α nα , for x ∈ 0, , α ≥ − , nβ , for x ∈ −1, , β ≥ − , β −1/2 n−1/2 , 3.22 1 for x ∈ −1, , α ≤ − , β ≤ − 2 As a consequence, the statement follows from the latter estimates and arguments similar to those we used in the proof of Proposition 3.6 Corollary 3.8 For α ≥ −1/2 and β ≥ 1/2, α,β Qn cos θ ≤ cA n, α, β − 1, θ , 3.23 cos θ ≤ cA n, α 3.24 and for α > −1 and β ≥ −1/2, Qn α,β 1, β, θ , where A n, α, β, θ ⎧ ⎪n−1/2 θ−α−1/2 π − θ ⎪ ⎪ ⎪ ⎨ nα , ⎪ ⎪ ⎪ ⎪ β ⎩n , −β−1/2 , c c ≤θ≤π− , n n c if ≤ θ ≤ , n c if π − ≤ θ ≤ π n if 3.25 Proof The inequality nα ≤ cn−1/2 θ−α−1/2 3.26 Journal of Inequalities and Applications holds for θ ∈ 0, c/n , as well as nβ ≤ cn−1/2 π − θ −β−1/2 3.27 holds for θ ∈ π − c/n, π Therefore, the statement follows from Propositions 3.6 and 3.7 α,β Next, we show that the Jacobi-Sobolev polynomial Qn x attains its maximum in −1, at the end points To be more precise, consider the following Proposition 3.9 a For α ≥ −1/2, β ≥ 1/2, and q α,β max Qn where a α,β if q α, and a −1 if q β − b For α > −1, β ≥ −1/2, and q max{α max Qn α,β where b if q α 1, and b −1 if q Qn α,β x α,β−1 Pn 3.28 b α,β ∼ nq , 3.29 β β − 1, the the proof can be done in a similar Proof a We will prove only the case q α If q way From 3.9 , 3.10 , and Proposition 3.7, Qn ∼ nq , 1, β}, x −1≤x≤1 a Qn x −1≤x≤1 max{α, β − 1}, α,β α,β−1 x − dn−1 λ Qn−1 x Pn x − O nα−2 3.30 Now, from 12, Theorem 7.32.1 and Proposition 3.7, the result follows Taking into account 2.6 , the case b can be proved in a similar way α,β Next, we deduce a Mehler-Heine type formula for Qn α,β and Qn Proposition 3.10 Let α, β > −1 Uniformly on compact subsets of C, one gets a α,β lim n−α Qn n→∞ cos z n z −α Jα z , 3.31 b lim n−α−2 Qn α,β cos n→∞ Proof a Multiplying in 3.8 by n Vn z −α z n z − α Jα z 3.32 , we obtain Yn z Dn−1 λ Yn−1 z , 3.33 10 Journal of Inequalities and Applications where Vn z n −α n α β / 2n α α,β β Pn cos z/n × cos z/n , Yn z n −α Qn cos z/n and Dn−1 λ according to 3.9 Using the above relation in a recursive way, we obtain α,β n n −1 k Bk Yn z n α / 2n dn−1 λ n/ n α,β α β Pn−1 α ∼ c/n2 λ Vn−k z , 3.34 k n where Bk k j Dn−j λ Proposition 3.5, we have n λ and B0 n Bk λ Moreover, by using the same argument as in k λ < c 1/n2 |Yn z | ≤ for every n ≥ and ≤ k ≤ n Thus, n n λ |Vn−k z | Bk 3.35 k On the other hand, from 2.8 , we have that {Vn z }∞ is uniformly bounded on n compact subsets of C Thus, for a fixed compact set K ⊂ C, there exists a constant C, depending only on K, such that when z ∈ K, n ≥ |Vn z | < C, 3.36 Thus, the sequence {Yn z }∞ is uniformly bounded on K ⊂ C As a conclusion, n Yn z O n−2 , Vn z z ∈ K, 3.37 and using 2.8 , we obtain the result b Since we have uniform convergence in 3.31 , taking derivatives and using some properties of Bessel functions, we obtain 3.32 α,β on −1, Now, we give the inner strong asymptotics of Qn Proposition 3.11 Let θ ∈ α,β Qn ,π − π −1/2 n−1/2 cos θ and sin θ > For α ≥ −1/2, β ≥ 1/2, one has −α−1/2 cos θ −β 1/2 cos k1 θ γ O n−1 , 3.38 and for α > −1, β ≥ −1/2, one has Qn α,β π cos θ −1/2 n α β n−1 −1/2 sin θ −α−3/2 cos θ −β−1/2 cos k1 θ γ1 O n−1 , 3.39 where k1 n α β /2, γ − α 1/2 π/2, and γ1 − α 3/2 π/2 Journal of Inequalities and Applications 11 x }∞ is uniformly bounded on Proof From Proposition 3.6 a , the sequence {n1/2 Qn n 1/2 compact subsets of −1, Multiplication by n in 3.10 yields α,β α,β n1/2 Qn α,β−1 n n−1 n−1 x − dn−1 λ n1/2 Pn x 1/2 α,β 3.40 Qn−1 x Since n n−1 dn−1 λ , n2 O 3.41 we have α,β n1/2 Qn α,β−1 n1/2 Pn x O n−2 x 3.42 Now, 3.38 follows from 2.9 Concerning 3.39 , it can be obtained in a similar way by using Proposition 3.6 b 3.11 and Next, we obtain an estimate for the Sobolev norms of the Jacobi-Sobolev polynomials Proposition 3.12 For α > −1/2, α ≥ β ≥ −1/2, and ≤ p ≤ ∞, one has α,β Qn Notice that if p assume ≤ p < ∞ α,β Sp ⎧ 4α ⎪n1/2 , ⎪ if ⎪ ⎪ ⎪ 2α ⎪ ⎪ ⎨ 4α ∼ n1/2 log n, if ⎪ 2α ⎪ ⎪ ⎪ ⎪ ⎪ α 2− 2α /p 4α ⎪ ⎩n , if 2α > p, p, < p 3.43 ∞, then we have Proposition 3.9 b Thus, in the proof we will Proof In order to establish the upper bound in 3.38 , it is enough to prove that α,β Qn α 1,β α,β Sp ≤ cn Pn Lp dμα 3.44 1,β Using 3.8 in a recurrence way and then Minkowski’s inequality, we obtain α,β Qn Lp dμα,β ≤c n n bk k λ n α,β Pn−k Lp dμα,β c n bk k λ α,β Pn−k−1 Lp dμα,β 3.45 12 Journal of Inequalities and Applications On the other hand, for α, β > −1 and k 1/2 n−k 0, 1, , n, 2.10 implies α,β Pn−k α,β Lp dμα,β ≤ cn1/2 Pn Lp dμα,β 3.46 Thus, α,β Pn−k Lp dμα,β n α,β Pn n−k ≤ Lp dμα,β ≤ k ≤ n − , 3.47 On the other hand, from Proposition 3.5, n n bk α,β n λ Pn−k Lp k dμα,β ≤ cbn n−1 n bk λ α,β Pn−k λ Lp dμα,β k ≤c α,β Pn−1 n−1 Lp dμα,β i 3.48 α,β ≤ c Pn 2k Lp dμα,β Thus, α,β Qn α,β Lp dμα,β ≤ c Pn α 1,β Lp dμα,β ≤ cn Pn Lp dμα 3.49 1,β In the same way as above, we conclude that Qn α,β Lp dμα ≤ cn 1,β n n bk λ α 1,β Pn−k−1 k α 1,β Lp dμα 1,β ≤ cn Pn Lp dμα 3.50 1,β Thus, 3.44 follows from 3.49 and 3.50 In order to prove the lower bound in relation 3.43 , we will need the following Proposition 3.13 For α > −1 and ≤ p < ∞, one has Qn ⎧ ⎪n1/2 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ α,β Lp dμα 1,β ≥ c n1/2 log n, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ α 2− 2α /p ⎪ ⎩n , 4α 2α 4α if 2α 4α if 2α if > p, p, < p 3.51 Journal of Inequalities and Applications 13 Proof We will use a technique similar to 12, Theorem 7.34 According to 3.11 , π/2 θ2α Qn α,β p cos θ ω/n θ2α dθ > ω ≥ cn−2α−4 Qn α,β t2α Qn ∼ cnp α −2α−4 ω t p cos θ α,β 2α t dθ cos − α p t n dt 3.52 p Jα t dt |Jα t |p dt cnp α −2α−4 ω t2α 3−p α On the other hand, Stempak’s lemma see 16, Lemma 2.1 , for γ > −1 − pα and ≤ p < ∞, implies ω tγ |Jα t |p dt ∼ ⎧ ⎪c, ⎨ if γ < ⎪ ⎩c log ω, if γ p − 1, p − 3.53 Thus, for α / 2α ≤ p and ω large enough, 3.51 follows Finally, from 3.39 we obtain the following: π/2 θ2α α,β Qn cos θ p π/2 dθ > θ2α π/4 Qn α,β cos θ p dθ ∼ np/2 3.54 For the proof of Proposition 3.12, from 3.51 , for α > −1 and ≤ p < ∞, we get α,β Qn α,β Sp ⎧ 4α ⎪n1/2 , ⎪ if ⎪ ⎪ ⎪ 2α ⎪ ⎪ ⎨ 4α ≥ c n1/2 log n, if ⎪ 2α ⎪ ⎪ ⎪ ⎪ ⎪ α 2− 2α /p 4α ⎪ ⎩n , if 2α > p, p, < p 3.55 Thus, using 3.44 and 3.55 , the statement follows A Cohen Type Inequality for Jacobi-Sobolev Expansions α,β For f ∈ S1 , its Fourier expansion in terms of Jacobi-Sobolev polynomials is ∞ α,β f k Qk k x , 4.1 14 Journal of Inequalities and Applications where −1 α,β f k Qk α,β f, Qk α,β S2 , k 4.2 0, 1, The Ces` ro means of order δ of the expansion 4.1 is defined by see 17, pages 76a 77 , δ Cn−k n δ σn f x k δ where Ck k δ k δ Cn α,β f k Qk 4.3 x , α,β For a function f ∈ Sp and a fixed sequence {ck,n }n , n ∈ N ∪ {0}, of real numbers k with cn−1,n α,β o n2 cn,n , we define the operators Tn by α,β Tn n α,β ck,n f k Qk f 4.4 k Let q0 4α / 2α and let p0 be the conjugate of q0 Now, we can state our main result Theorem 4.1 For α > −1/2 and α α,β Tn Sα p ≥ β ≥ −1/2, one has ⎧ ⎪n 2α /p− 2α /2 , ⎪ ⎪ ⎨ ≥ c|cn,n | log n 2α / 4α , ⎪ ⎪ ⎪ ⎩ 2α /2− 2α /p , n if ≤ p < p0 , if p p0 , p For ck,n α,β Sp −→ ∞, 4.5 if q0 < p ≤ ∞ Corollary 4.2 Let α, β, p0 , q0 , and p be as in Theorem 4.1 For ck,n outside the interval p0 , q0 , one has σn q0 , 1, k n −→ ∞ 0, , n, and for p 4.6 δ δ Cn−k /Cn , ≤ k ≤ n, Theorem 4.1 yields the following Corollary 4.3 For α > −1/2 and α ≥ β ≥ −1/2, one has 2α 2α − , p if ≤ p < p0 , 2α 2α 0 α 1/2 − 2α /p ≥ β − 1/2 − 2β /p On the other hand, from 2.6 , 4.9 , and 12, formula 4.5.4 , one has α,β−1,j gn j α j,β−1 j − x Pn x −2j − x2 j−1 α j,β−1 j xPn 4j n α j − x2 2n α β 2j − x j−1 4j n 1 − x2 2n α β 2j − 2j − x2 j−1 α j,β−1 j Pn n x α β α−1 j,β−1 j Pn j−1 x 2j α j,β j x 4.23 x α−1 j,β−1 j Pn n j − x Pn α β x 2j j α j,β j − x Pn x 18 Journal of Inequalities and Applications 3/2 − 2α From 2.10 , for j > max{α j−1 − x2 for α ≥ β and j > α 5/2 − 2α ≥ β and j > α α−1 j,β−1 j Pn j−1 4.24 1,β α j,β−1 j Pn 3/2 − 2α ∼ n−1/2 , 4.25 ∼ n−1/2 Lp dμα 4.26 1,β /p, j ≥ β and j > α ∼ n−1/2 , Lp dμα α j,β j − x Pn Thus, for α /p}, /p, − x2 and for α 3/2 − 2β /p, β 5/2 − 2α Lp dμα 1,β /p, α,β−1,j gn Lp dμα ≤ cn1/2 4.27 1,β By using 4.22 and 4.27 , we find from 4.21 that α,β−1,j gn for α α,β Sp ≤ cn1/2 , 4.28 ≥ β and j > α 5/2 − 2α /p Now, we can prove our main result Proof of Theorem 4.1 By duality, it is enough to assume that q0 ≤ p ≤ ∞ From 4.11 , 4.20 , and 4.28 , one has α,β Tn α,β Sp ≥ −1 α,β−1,j gn α,β Tn α,β Sp α,β−1,j ≥ cn−1/2 cn,n gn n α,β−1,j gn α,β Sp α,β Qn α,β Sp 4.29 −1/2 − cn α,β−1,j cn−1,n gn α,β ∼ cn−1/2 |c1 cn,n | Qn α,β Sp n−1 1− α,β Qn−1 c2 cn−1,n c1 n2 cn,n Now from Proposition 3.12, the statement of the theorem follows α,β Sp Journal of Inequalities and Applications 19 Necessary Conditions for the Norm Convergence The problem of the convergence in the norm of partial sums of the Fourier expansions in terms of Jacobi polynomials has been discussed by many authors See, for instance, 18–20 and the references therein α,β Let qn be the Jacobi-Sobolev orthonormal polynomials, that is, α,β qn −1 α,β x Qn α,β Qn α,β S2 5.1 x α,β For f ∈ S1 , the Fourier expansion in terms of Jacobi-Sobolev orthonormal polynomials is ∞ α,β f k qk x , 5.2 k where α,β f k , f, qk k 0, 1, 5.3 Let Sn f be the nth partial sum of the expansion 5.2 as follows: n α,β Sn f, x x f k qk 5.4 k Theorem 5.1 Let α > −1/2, α that ≥ β ≥ −1/2, and < p < ∞ If there exists a constant c > such Sn f α,β Sp ≤c f α,β Sp , 5.5 α,β for every f ∈ Sp , then p ∈ p0 , q0 Proof For the proof, we apply the same argument as in 19 Assume that 5.5 holds, then α,β f, qn α,β qn x Sn f − Sn−1 f α,β Sp α,β Sp ≤ 2c f α,β Sp 5.6 Therefore, α,β qn where p is the conjugate of q x α,β α,β Sp qn x Sα q < ∞, 5.7 20 Journal of Inequalities and Applications On the other hand, from 3.43 we obtain the Sobolev norms of Jacobi-Sobolev orthonormal polynomials as follows: α,β qn α,β Sp ⎧ ⎪c, ⎪ ⎪ ⎨ ∼ log n, ⎪ ⎪ ⎪ ⎩ α 3/2− 2α n if p < q0 , if p /p q0 , 5.8 , if p > q0 , for α > −1/2, α ≥ β ≥ −1/2, and ≤ p ≤ ∞ Now, from 5.8 it follows that the inequality 5.7 holds if and only if p ∈ p0 , q0 The proof of Theorem 5.1 is complete Divergence Almost Everywhere For λ and α β 0, Pollard 21 showed that for each p < 4/3 there exists a function f ∈ Lp dx such that its Fourier expansion 4.27 diverges almost everywhere on −1, Later on, Meaney 22 extended the result to p 4/3 Furthermore, he proved that this is a special case of a divergence result for the Fourier expansion in terms of Jacobi polynomials The failure of almost everywhere convergence of the Fourier expansions associated with systems of orthogonal polynomials on −1, and Bessel systems has been discussed in 16, 23 If the sequence {Sn f }n≥0 is uniformly bounded on a set, say E, of positive measure in −1, , then α,β f n qn x α,β S∞ ,E n ∈ N, x ∈ E < c, 6.1 Therefore, f n qn α,β x < c, n ∈ N, 6.2 almost everywhere on E From Egorov’s Theorem, it follows that there is a subset E1 ⊂ E of positive measure such that f n qn α,β < c, x 6.3 uniformly for x ∈ E1 On the other hand, from 3.39 f n cos k1 θ γ1 O n−1 < c, 6.4 uniformly for cos θ ∈ E1 Using the Cantor-Lebesgue Theorem, as described in 24, Section 1.5 , see also 17, page 316 , we obtain f n < c 6.5 Journal of Inequalities and Applications 21 α,β ≥ β ≥ −1/2 There is an f ∈ Sp , ≤ p ≤ p0 , whose Fourier Theorem 6.1 Let α > −1/2 and α α,β expansion 5.2 diverges almost everywhere on −1, in the norm of S∞ Proof Consider the linear functionals Tn f α,β f n f, qn , 6.6 α,β on Sp , ≤ p ≤ p0 By using 1, Theorem 3.8 , we have Tn α,β qn α,β Sp , q0 ≤ p ≤ ∞ 6.7 ∞ 6.8 Thus, from 5.8 , sup Tn n α,β As a consequence of the Banach-Steinhaus theorem, there exists f ∈ Sp , ≤ p ≤ p0 , such that sup Tn f n ∞ 6.9 Since this result contradicts 6.5 , then for this f the Fourier series diverges almost α,β everywhere on −1, in the norm of S∞ Acknowledgments The authors thank the careful revision of the paper by the referees Their remarks and suggestions have contributed to improve the presentation The work of the second author F Marcell´ n has been supported by Direcci on General de Investigacion, Ministerio de Ciencia a ´ ´ e Innovacion of Spain, Grant no MTM2009-12740-C03-01 ´ References R A Adams, Sobolev Spaces, vol of Pure and Applied Mathematics, Academic Press, New York, NY, USA, 1975 H G Meijer, “Determination of all coherent pairs,” Journal of Approximation Theory, vol 89, no 3, pp 321–343, 1997 D H Kim, K H Kwon, F Marcell´ n, and G J Yoon, “Zeros of Jacobi-Sobolev orthogonal a polynomials,” International Mathematical Journal, vol 4, no 5, pp 413–422, 2003 H G Meijer and M G de Bruin, “Zeros of Sobolev orthogonal polynomials following from coherent pairs,” Journal of Computational and Applied Mathematics, vol 139, no 2, pp 253–274, 2002 P J Cohen, “On a conjecture of 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Journal of Inequalities and Applications B Dreseler and P M Soardi, ? ?A Cohen- type inequality for Jacobi expansions and divergence of Fourier series on compact symmetric spaces,” Journal of Approximation... case of a divergence result for the Fourier expansion in terms of Jacobi polynomials The failure of almost everywhere convergence of the Fourier expansions associated with systems of orthogonal polynomials. .. terms of Jacobi-Sobolev polynomials, the well-known Cohen type inequality in the framework of Approximation Theory A Cohen type inequality has been established in other contexts, for example,