Fourier Series of Gegenbauer-Sobolev Polynomials

We study the partial sum operator for a Sobolev-type inner product related to the classical Gegenbauer polynomials. A complete characterization of the partial sum operator in an appropriate Sobolev space is given. Moreover, we analyze the convergence of the partial sum operators.

Given 1 < p < ∞, we say that f ∈ L p (dµ α ) if f is a measurable function in [−1, 1] and Let us define the measure µ α,M := µ α +M (δ 1 +δ −1 ). We consider the space W α p , with 1 < p < ∞, as the set of equivalence classes, with respect to the (semi)norm in L p (µ α,M ), of measurable This paper is a contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applications (OPSFA14). The full collection is available at https://www.emis.de/journals/SIGMA/OPSFA2017.html functions defined on [−1, 1] such that there exists an element in the class f for which f ′ (1) and f ′ (−1) are defined, and The main target of this paper is the study of the uniform boundedness of the operators G n . In fact, we will prove the following characterization.
There exists a constant C, independent of n and f , such that The uniform boundedness of the partial sum operators for Gegenbauer polynomials in L p (dµ α ) was given by Pollard [11] who extended it to the Jacobi setting in [12]. A general result including weights for Jacobi expansions can be seen in [9]. In [3], by applying the boundedness with weights of the Hilbert transform, the authors did a complete study of the boundedness of the partial sum operators related to generalized Jacobi weights. The same authors studied the generalized Jacobi weights with mass points on the interval [−1, 1] (see [4]). The uniform boundedness with weights of the partial sum operator for the generalized Jacobi polynomials has been used to proved, using an idea dating back to J. Marcinkiewicz, some results related to interpolating polynomials (see [15,16] and the references in [10]).
It would be natural to consider our problem for the Jacobi weight instead of the Gegenbauer one. This extension requires some results about the corresponding Jacobi-Sobolev polynomials that are unavailable in the literature at this moment. We hope to develop these tools in a forthcoming paper to obtain a complete characterization in that case as well.
As far as we know, a complete characterization of the uniform boundedness of the partial sums in the Sobolev setting is completely new. In [6], the authors observed that the main obstacle to analyze this problem is the lack of Christoffel-Darboux formula for Sobolev orthogonal polynomials. As a consequence of this fact, except for certain particular cases, the convergence of Fourier expansions in Sobolev orthogonal polynomials has not been resolved. For example, the particular case of the Fourier series associated to the Jacobi-Sobolev polynomials defined by the inner product was treated in [5] but, unfortunately, the given results are not completely satisfactory.
Our proof of Theorem 1.1 relies on some results about multipliers and transplantation operators for Jacobi expansions proved by Muckenhoupt and other authors in the eighties of the last century (see [8] and the references therein).
From a standard argument, the uniform boundedness of the operator G n will imply the convergence for functions in the class W α p if the polynomials form a dense class. However, the reverse implication is not true because the space W α p is not complete. The density of the polynomials is contained in the next result.
Theorem 1.2. The set of polynomials is dense in the space W α p . That is, given f ∈ W α p , for all ε > 0 there exists a polynomial q n of degree n such that Now, from Theorems 1.1 and 1.2, we deduce the convergence for functions in the class W α p of the partial sums G n .
In Section 2 we present the necessary definitions and results concerning to the Gegenbauer and Gegenbauer-Sobolev polynomials. Section 3 and Section 4 are devoted to prove Theorem 1.1 and Theorem 1.2, respectively.

Definitions and auxiliary results
Let {R α n } n≥0 be the sequence of Gegenbauer polynomials given by the Rodrigues formula If we call {B α n } n ≥ 0 the sequence of orthogonal polynomials with respect to (1.1), the following relation between R α n and B α n was proved in [1] B α n (x) = a n (n + 2α + 1) 4 (−n) 4 2 6 (α + 2)(α + 3)(α + 1) 4 1 − x 2 2 R α+4 n−4 (x) where (a) n is the shifted factorial (or Pochhammer symbol), defined by (a) n = Γ(a+n) Γ(a) , and Here and elsewhere we use the convention that R γ n ≡ 0 if n < 0. In [7] it was proved the identity Γ(2α + 2)n! (2n + 2α + 1)Γ(n + 2α + 1) 4Ó. Ciaurri and J. Mínguez . Now, denoting by {P α n } n≥0 the sequence of orthonormal Gegenbauer polynomials, given by P α n = β n,α R α n with where A n,4 = a n (n + 2α and A n,0 = c n λn,α βn,α . We consider the notations for some constants d j and D j that will be different in each occurrence of the g(n, J) and h(n, α), respectively. With the previous notation, by using that for any J ∈ N, we deduce in an easy way that and β n,α = (n + 1) α+1/2 g(n, J), for any j ∈ N. Then, using that (2.4), the result follows.
The following results, that we will use in the proof of Theorem 1.1, can be found in [7]. The notation appearing in Lemma 2.3, f (n) ≈ g(n), indicates the existence of positive constants C and D such that Cf (n) ≤ g(n) ≤ Df (n) for n large enough.  Let S γ n f be the n-th partial sum of Fourier expansion in terms of orthonormal Gegenbauer polynomials, From the main result in [8] we can deduce the following result Lemma 2.4. Let γ > −1 and 1 < p < ∞. There exists a constant C, independent of n and f , such that if and only if a + (γ + 1)

6Ó. Ciaurri and J. Mínguez
Let d be an integer number. We define the transplantation operator The operator T β,γ d is well defined, for example, for functions f having a finite expansion in terms of the Gegenbauer polynomials P γ n . The following result plays a crucial role in our work. It is essentially a special case of a general weighted transplantation theorem due to Muckenhoupt, see [9, Theorem 1.6].
The last tool that we will need for the proof of Theorem 1.1 is related to the boundedness of a specific multiplier for Gegenbauer expansions. We define the operator Lemma 2.6. Let γ > −1 and 1 < p < ∞. If |2b + 1| < p and This lemma is a particular case of [9, Theorem 1.10] because the multiplier 1/(k + 1) belongs to the class M (1, 1) there defined.

Proof of Theorem 1.1
Taking the kernel it is easy to see that Recall that The necessity of the condition (1.2) is a consequence of [2, Theorem 1] and its sufficiency will be obtained from two following propositions. Proposition 3.1. Let α > −1/2 and 1 < p < ∞. If (1.2) holds, then where C is a constant independent of n and f .
Proof of Proposition 3.1. From Minkowski's inequality, we know that First, it will be proved that Using (2.2), we have A k,j A k,m P α+j k−j (x)P α+m k−m (y).
By using a standard duality argument, to deduce (3.2) it is enough to prove for m ≤ j. (k + 1) −s P α+j k−j (x)P α+m k−m (y), s = 0, 1, with C a constant independent of n, we deduce 1 − (·) 2 j/2 P α+j n−j L p (dµα) ≤ C for p < 4(α + 1)/(2α + 1). In this way, applying Hölder inequality, for each p verifying (1.2). It is easy to check that for a constant K α , with j = 4, 2, 0 and g(x) = 1 − x 2 −j/2 f (x). Then, if p satisfies (1.2), from Lemma 2.4, with a = j(1/2 − 1/p) and γ = α + j, we deduce Now, for m < j, we can check that for a constant C α , whith h(x) = 1 − x 2 −m/2 f (x). So, using Lemma 2.5 with β = α + j, γ = α + m, and b = α − p(α + 1/2)/2, we have where in the last step we have used Lemma 2.4 as we have done for M j,j,0 n . To analyze the operators M j,m,1 n we observe the identities . Then the boundedness of these operators follows as in the previous cases but using moreover the estimate which can be deduced from Lemma 2.6 taking b = α − p(α + 1/2)/2 and γ = α + j under the assumption (1.2). In this way the proof of (3.2) is completed.
To finish the proof of (3.1), we are going to prove the estimates
Proof of Proposition 3.2. We are going to show the estimates and The analysis of |G n f (−1)|, for M > 0, and |(G n f ) ′ (−1)|, for N > 0, are completely similar and the details will be omitted. It is clear that If M > 0, from Lemmas 2.2 and 2.3 it is obtained that Then, applying Hölder inequality, we have 10Ó. Ciaurri and J. Mínguez because the last integral converges if q < 4(α + 1)/(2α + 1), which is equivalent to p > 4(α + 1)/(2α + 3). On the other hand, using again Lemma 2.3 we deduce the bounds On the other hand, givenε there exists a polynomial q n of degree n such that Then, f − q n p Wp < ε and the proof is completed.