Abstract
Let \(\textbf{u}\) be a moment functional associated with the Hermite, Laguerre, or Jacobi classical orthogonal polynomials. We study approximation by polynomials in \(H^r(\textbf{u})\), the Sobolev space consisting of functions whose derivatives of consecutive orders up to r belong to the \(L^2\) space associated with \(\textbf{u}\). This requires the simultaneous approximation of a function f and its consecutive derivatives up to order \(N\leqslant r\). We explicitly construct orthogonal polynomials that achieve such simultaneous approximation and provide error estimates in terms of \(E_n(f^{(r)})\), the error of best approximation of \(f^{(r)}\) in \(L^{2}(\textbf{u})\).
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1 Introduction
In recent years, the study of orthogonal polynomials associated with Sobolev inner products (that is, inner products involving derivatives) have undergone intensive consideration. We refer the interested reader to the survey [28] for the latest presentation of the state of the art on Sobolev orthogonal polynomials. One of the most useful features of the Sobolev inner products is that they include terms “controlling” the behavior of the associated orthogonal polynomials on the boundary of the orthogonality domain. In particular, it is sometimes desirable for the corresponding orthogonal polynomials and their derivatives to have zeros on a set of points, for instance, on one or both end points of a closed interval [a, b]. This property is important in applications where, for a suitable function f, it is required that the partial sum of the corresponding Fourier orthogonal expansion and its derivatives up to some appropriate order \(m\geqslant 1\), coincide with the values \(f^{(j)}(a)\) and \(f^{(j)}(b)\), \(0\leqslant j \leqslant m\) (e.g., solving boundary-value problems using spectral methods based on representing the solution in terms of Sobolev orthogonal polynomials, [6, 7, 14, 18, 19, 35]). The Sobolev inner products mostly studied in the literature to deal with the above problem are mainly of the form (see [12, 25, 30, 31, 34])
where \(\mu _j\), \(0\leqslant j\leqslant m-1\), are discrete Borel measures supported on \(x=-1\) or \(x=1\), and \(\omega _{\alpha ,\beta }(x)=(1-x)^{\alpha }(1+x)^{\beta }\) is the Jacobi weight with \(\alpha ,\beta >-1\).
We note that orthogonal polynomials with respect to a weight function on [a, b] (or, in general, on a bounded or unbounded interval I) lack the nice property of the zeros mentioned above ([33, Section 6.2]), and, moreover, their approximation behavior in Sobolev spaces give much weaker results than optimal. In spite of this, the approximating properties of Sobolev orthogonal polynomials is still insufficiently understood.
The purpose of this paper is to consider simultaneous approximation of a function and its derivatives by polynomials on an interval in \(L^2\) norms associated with Jacobi, Laguerre, or Hermite weights. In order to explain our results, we need to introduce some notation.
For \(n\geqslant 0\), let \(\Pi _n\) be the linear space of polynomials of a real variable and real coefficients of degree at most n, and let \(\Pi =\bigcup _{n\geqslant 0}\Pi _n\).
Let \(\Pi ^*\) denote the algebraic dual space of \(\Pi \). That is, \(\Pi ^*\) is the linear space of linear functionals defined on \(\Pi \),
We denote by \(\langle \textbf{u}, p\rangle \) the image of the polynomials p under the moment functional \(\textbf{u}\).
Any linear functional \(\textbf{u}\) is completely defined by the values
and extended by linearity to all polynomials, where \(\mu _n\) is called the n-th moment of \(\textbf{u}\). Therefore, we refer to \(\textbf{u}\) as a moment functional. \(\textbf{u}\) is called positive definite if \(\langle \textbf{u}, p^2\rangle >0\) for every non-zero polynomial \(p\in \Pi \). If \(\textbf{u}\) is positive definite, then there exists an absolutely continuous measure \(d\mu \) supported in an infinite subset I or, equivalently, a real valued non-decreasing function \(\varphi (x)\) such that ([9, Theorem 6.3])
In this paper we assume that \(\textbf{u}\) can be expressed as an integral with respect to a postive weight function w(x) defined on an interval I, that is,
For a positive definite moment functional \(\textbf{u}\) associated with the weight function w(x), let \(L^2(\textbf{u})\) be the linear space of functions given by
where \(\mathcal {C}_f\) is the class of functions equivalent to f in the following sense:
As usual, we will not distinguish between f and \(\mathcal {C}_f\).
In \(L^2(\textbf{u})\) we can define the inner product
and the norm
The standard error of best approximation by polynomials of degree n is defined as
For \(r\geqslant 1\), let
and we define the norm on \(H^r(\textbf{u})\) as
In \(L^2(\textbf{u})\), the n-th partial sum of the Fourier orthogonal expansion \(S_nf\) in terms of the orthogonal polynomials associated with \(\textbf{u}\) satisfies
However, the approximating behavior of \(S_nf\) in a Sobolev space is weaker. Indeed, we show that there is a non-zero polynomial \(\phi \) of degree at most 2, such that for \(f\in H^r(\textbf{u})\) satisfying \(\phi ^{\frac{r-1}{2}}\,f^{(r)}\in L^2(\textbf{u})\),
where, hereon after, c denotes a generic positive constant independent of n and f, whose value may vary from line to line. On the other hand, we show that for \(f\in H^r(\textbf{u})\), there is a polynomial \(p_n\in \Pi _{2n}\) satisfying
where \(\nu _i\in \mathbb {R}\) are distinct and \(d_1+\cdots +d_s=r\), \(d_i\in \mathbb {N}\), such that
for the Jacobi case, and
for the Laguerre and Hermite case. Evidently, (1.2) is weaker than (1.3) and (1.4), and holds under more restrictive conditions. Estimates of the form
have been established for the Jacobi case in [34]. We point out that our estimates (1.3) and (1.4) are coarser than (1.5) since our functional approach encompasses all three positive definite classical moment functionals at once. Therefore, finer estimates may be deduced by considering each case separately.
We give explicit expressions for the polynomial \(p_n\) in (1.3) and (1.4). In fact, it is the partial sum of the Fourier expansion in terms of orthogonal polynomials associated with a so-called discrete–continuous Sobolev inner product (see, for instance, [1, 12, 15, 17]). These polynomials are usually given in terms of classical orthogonal polynomials with non-standard parameters which require delicate extensions ([2, 3, 5, 14, 21, 23, 24, 29, 32]). We provide a more direct definition that holds for such non-standard parameters.
The paper is organized as follows. In the following section, we provide the basic facts about classical orthogonal polynomials needed to present our results. In Sect. 3, we discuss the approximation behavior of partial sums of Fourier expansions in terms of classical orthogonal polynomials, which give suboptimal results when used in Sobolev spaces. In Sect. 4, we present a discrete–continuous Sobolev inner product and construct associated orthogonal polynomials in terms of iterated integrals of classical orthogonal polynomials. The expressions of the Sobolev orthogonal polynomials provided there are suitable for studying Fourier expansions in terms of them. Simultaneous approximation by polynomials in \(H^r(\textbf{u})\) is studied in Sect. 5, and there we provide a more elaborate version of (1.3) and (1.4). A numerical example based on Laguerre polynomials is provided in the last section.
2 Classical orthogonal polynomials
In this section, we collect the basic facts about classical orthogonal polynomials that we will use throughout the work and which can be found in [13].
Let \(\textbf{u}\) be a moment functional. A sequence of polynomials \(\{P_n(x)\}_{n\geqslant 0}\) is called an orthogonal polynomial sequence (OPS) with respect to \(\textbf{u}\) if
- (1):
-
\(\deg \,P_n=n\),
- (2):
-
\(\langle \textbf{u}, P_n\,P_m\rangle = h_n\,\delta _{n,m}\), with \(h_n\ne 0\).
Here \(\delta _{n,m}\) denotes the Kronecker delta defined as
If there is an OPS associated with \(\textbf{u}\), then \(\textbf{u}\) is called quasi-definite. Positive definite moment functionals are quasi-definite.
Observe that an OPS \(\{P_n(x)\}_{n\geqslant 0}\) constitutes a basis for \(\Pi \). If for all \(n\geqslant 0\), the leading coefficient of \(P_n(x)\) is 1, then \(\{P_n(x)\}_{n\geqslant 0}\) is called a monic orthogonal polynomial sequence (MOPS).
Given a moment functional \(\textbf{u}\) and a polynomial q(x), we define the left multiplication of \(\textbf{u}\) by q(x) as the moment functional \(q\,\textbf{u}\) such that
and we define the distributional derivative \(D\textbf{u}\) by
Moreover, the product rule is satisfied, that is,
Definition 2.1
Let \(\textbf{u}\) be a quasi-definite moment functional, and let \(\{P_n(x)\}_{n\geqslant 0}\) be an OPS with respect to \(\textbf{u}\). Then \(\textbf{u}\) is classical if there are nonzero polynomials \(\phi (x)\) and \(\psi (x)\) with \(\deg \phi \leqslant 2\) and \(\deg \psi =1\), such that \(\textbf{u}\) satisfies the distributional Pearson equation
The sequence \(\{P_n(x)\}_{n\geqslant 0}\) is called a classical OPS.
The following characterizations of classical moment functionals and OPS will be useful in the sequel.
Theorem 2.2
Let \(\textbf{u}\) be a quasi-definite moment functional, and \(\{P_n(x)\}_{n\geqslant 0}\) its associated MOPS. The following statements are equivalent:
- 1.:
-
\(\textbf{u}\) is a classical moment functional.
- 2.:
-
There are nonzero polynomials \(\phi (x)\) and \(\psi (x)\) with \(\deg \phi \leqslant 2\) and \(\deg \psi =1\) such that, for \(n\geqslant 0\), \(P_n(x)\) satisfies
$$\begin{aligned} \phi (x)\,P_n''(x)+\psi (x)\,P_n'(x)\,=\,\lambda _n\,P_n(x), \end{aligned}$$(2.2)where \(\lambda _n=n\,(\frac{n-1}{2}\phi ''+\psi ')\).
- 3.:
-
There is a nonzero polynomial \(\phi (x)\) with \(\deg \phi \leqslant 2\), such that \(\left\{ \frac{P_{n+1}'(x)}{n+1} \right\} _{n\geqslant 0}\) is the MOPS associated with the moment functional \(\textbf{v}=\phi (x)\,\textbf{u}\).
- 4.:
-
There are real numbers \(\alpha _n\) and \(\beta _n\), \(n\geqslant 2\), such that
$$\begin{aligned} P_n(x)\,=\,\frac{P'_{n+1}(x)}{n+1}+\alpha _n\,\frac{P_n'(x)}{n}+\beta _n\,\frac{P'_{n-1}(x)}{n-1},\quad n\geqslant 2. \end{aligned}$$(2.3)
It is well known (see [4] as well as [22]) that, up to affine transformations of the independent variable, the only families of positive definite classical orthogonal polynomials are the Hermite, Laguerre, and Jacobi polynomials. The corresponding moment functionals admit an integral representation of the form
where \(I=\mathbb {R}\) and \(w(x)=e^{-x^2}\) in the Hermite case, \(I=(0,+\infty )\) and \(w(x)=x^{\alpha }e^{-x}\) with \(\alpha >-1\) in the Laguerre case, and \(I=(-1,1)\) and \(w(x)=(1-x)^{\alpha }(1+x)^{\beta }\) with \(\alpha ,\beta >-1\) in the Jacobi case. We note that in each case \(w(x)>0\) in I, and thus we say that w(x) is a weight function.
In the sequel, we will need the explicit expression of the polynomials \(\phi (x)\) and \(\psi (x)\), and the parameters that appear in Theorem 2.2, as well as the square of the norms of the classical orthogonal polynomials, which we summarize in Table 1. We use [27] and [33] as reference.
Observe that from Theorem 2.2, if \(\textbf{u}\) is a classical moment functional satisfying (2.1), then \(\textbf{v}=\phi (x)\,\textbf{u}\) is a classical moment functional satisfying the Pearson equation
We point out that if we consider \(\phi (x)\) and \(\psi (x)\) as in Table 1, then a positive definite classical moment functional \(\textbf{u}\) still satisfies (2.1) with \(-\phi (x)\) and \(-\psi (x)\). But then \((-\phi )^k\textbf{u}\) is not necessarily a positive definite moment functional. In the sequel, we will only consider (2.1) with \(\phi (x)\) given in Table 1 to guarantee the positive definiteness of \(\textbf{v}=\phi \,\textbf{u}\).
Iterating this idea, we see that the high-order derivatives of classical orthogonal polynomials are again classical orthogonal polynomials of the same type.
Theorem 2.3
Let \(\textbf{u}\) be a classical moment functional satisfying (2.1), and \(\{P_n(x)\}_{n\geqslant 0}\) its corresponding MOPS. For \(k\geqslant 0\), let \(\textbf{v}_k=\phi ^k(x)\,\textbf{u}\) and \(\{Q_{n,k}(x)\}_{n\geqslant 0}\) be the sequence of polynomials given by
where \(p^{(k)}\) is the k-th derivative of p. Then, for each \(k\geqslant 0\), \(\{Q_{n,k}(x)\}_{n\geqslant 0}\) is an OPS associated with the moment functional \(\textbf{v}_k\), satisfying
where \(\psi _k(x)=\psi (x)+k\,\phi '(x)\). Therefore, \(\textbf{v}_k\) is a classical moment functional.
Clearly, the polynomials \(\{Q_{n,k}(x)\}_{n\geqslant 0}\) defined in (2.4) are not monic. Nevertheless, they satisfy the following property: for \(0\leqslant r \leqslant n\), we have
Moreover, we can provide a distributional formulation of the differential equation (2.2) satisfied by these polynomials.
Proposition 2.4
Let \(\textbf{u}\) be a classical functional satisfying (2.1), and \(\{P_n(x)\}_{n\geqslant 0}\) its corresponding MOPS. For \(k\geqslant 0\), let \(\textbf{v}_k=\phi ^k(x)\,\textbf{u}\), and \(\{Q_{n,k}(x)\}_{n\geqslant 0}\) be the sequence of polynomials defined in (2.4). Then,
where \(\lambda _{n,k}=(n+k)\,(\frac{n+k-1}{2}\phi ''+\psi ')-k\,(\frac{k-1}{2}\phi ''+\psi ')\).
Proof
For \(k\geqslant 0\), by Theorem 2.3, \(\{Q_{n,k}(x)\}_{n\geqslant 0}\) is a sequence of classical OPS and \(\textbf{v}_k\) satisfies the Pearson equation \(D(\phi \,\textbf{v}_k)=\psi _k\,\textbf{v}_k\) with \(\psi _k(x)=\psi (x)+k\,\phi '(x)\).
Moreover, from Theorem 2.2, we have that \(Q_{n,k}(x)\) satisfies the differential equation
with
Therefore, for \(n\geqslant 0\),
\(\square \)
Remark 2.5
We note that if \(\{P_n(x)\}_{n\geqslant 0}\) is a sequence of Hermite or Laguerre polynomials, then
- (1):
-
for \(k\geqslant 0\), \(\{-\lambda _{n,k}\}_{n\geqslant 0}\) is an increasing sequence of non-negative real numbers (see Table 1),
- (2):
-
for \(n\geqslant 0\) and \(0\leqslant i \leqslant k\), we have \(-\lambda _{n,k}\leqslant -\lambda _{n+i,k-i}\).
In the Jacobi case, (1) and (2) hold when \(\alpha +\beta +2k\geqslant 0\).
3 Fourier series in terms of classical orthogonal polynomials
Let \(\textbf{u}\) be a positive definite classical moment functional satisfying (2.1), and \(\{P_n(x)\}_{n\geqslant 0}\) its corresponding MOPS. For \(k\geqslant 0\), let \(\textbf{v}_k=\phi ^k(x)\,\textbf{u}\), and \(\{Q_{n,k}(x)\}_{n\geqslant 0}\) be the sequence of polynomials defined in (2.4). We note that \(\textbf{v}_0=\textbf{u}\), and \(\textbf{v}_k\) is a positive definite moment functional ([26, 27]).
For a function \(f\in L^2(\textbf{v}_k)\), we can define the Fourier coefficients of f as
We note that \(\left\langle \textbf{v}_k,f\,Q_{n,k} \right\rangle \) is finite by virtue of the Cauchy-Schwarz inequality.
For \(n,k\geqslant 0\), let \(S_{n,k}\) denote the projection operator \(S_{n,k}: L^2(\textbf{v}_k) \rightarrow \Pi _n\) defined as
This operator can be characterized in terms of \(E_{n,\textbf{v}_k}(f)\).
Theorem 3.1
For \(n,k\geqslant 0\) and \(f\in L^2(\textbf{v}_k)\), \(S_{n,k}f(x)\) is the unique polynomial in \(\Pi _n\) satisfying
Proof
Since \(\{Q_{n,k}(x)\}_{n\geqslant 0}\) is a basis of \(\Pi _n\), we can write any polynomial \(p\in \Pi _n\) as
for some real coefficients \(c_j\), \(0\leqslant j\leqslant n\). We compute
To minimize \(\Vert f-p\Vert _{\textbf{v}_k}^2\) we impose \(\dfrac{\partial }{\partial \,c_j}\Vert f-p\Vert _{\textbf{v}_k}^2=0\) for \(0\leqslant j \leqslant n\). Since \(\Vert f-p\Vert _{\textbf{v}_k}^2\) is a convex function in the variables \(c_j\), and taking into account that the Hessian matrix is positive definite, we deduce that the minimum of \(\Vert f-p\Vert _{\textbf{v}_k}^2\) is unique and is attained when \(c_j=\widehat{f}_{j,k}\). \(\square \)
It is well known (see [10, Chapter II, §4 – §8]) that for the Hermite, Laguerre, and Jacobi polynomials, we have
where \(\textbf{v}_k\) is the corresponding functional in each case. This means that we are allowed to write the Fourier expansion of \(f\in L^2(\textbf{v}_k)\),
where the equality holds for almost every point in the support I. By orthogonality, we obtain the Parseval identity
It is possible to pass down relation (2.3) to the Fourier coefficients in the expansion of a function with respect to classical orthogonal polynomials.
Lemma 3.2
Let \(\textbf{u}\) be a classical moment functional and \(\{P_n(x)\}_{n\geqslant 0}\) be its associated MOPS. For \(n,k\geqslant 0\) and \(f\in L^2(\textbf{v}_{k})\cap L^2(\textbf{v}_{k+1})\),
where \(\alpha _{n+k+1}\) and \(\beta _{n+k+2}\) are the real numbers appearing in (2.3).
Proof
Differentiating (2.3) and using (2.4), we get \(\{Q_{n,k}(x)\}_{n\geqslant 0}\) satisfies
Then, we compute
Using (3.1), we obtain
and the desired result follows by dividing both sides by \(h_{n,k+1}\). \(\square \)
The following lemma will be useful often in the sequel.
Lemma 3.3
For each \(1\leqslant r\leqslant k\) and \(f\in L^2(\textbf{v}_{k-r})\) such that \(f^{(r)}\in L^2(\textbf{v}_k)\),
Proof
Using \(Q'_{n+1,k-1}(x)=Q_{n,k}(x)\) and Proposition 2.4, we get
Then, iterating the result above \(r-1\) more times, we obtain
where \(\lambda _{n,k}=(n+k)\,(\frac{n+k-1}{2}\phi ''+\psi ')-k\,(\frac{k-1}{2}\phi ''+\psi ')\).
Setting \(f(x)=Q_{n+r,k-r}(x)\) in (3.3), we get
Since \(Q_{n+r,k-r}^{(r)}(x)=Q_{n,k}(x)\), we have
Using (3.3) and the equation above, we obtain
\(\square \)
It is possible to describe the behavior of the partial sum \(S_{n,k}f\) under differentiation.
Proposition 3.4
Let \(n,k\geqslant 0\) and \(0\leqslant r \leqslant n\). If \(f\in L^2(\textbf{v}_k)\) such that \(f^{(r)}\in L^2(\textbf{v}_{k+r})\), then
Proof
Using (2.5), we get
From (3.2), we have \(\widehat{f}_{m+r,k}=\widehat{f^{(r)}}_{m,k+r}\) and therefore \(\left( S_{n,k}\,f\right) ^{(r)}=S_{n-r,k+r}f^{(r)}\).
Now, we focus our attention on the approximation behavior of \(S_{n,k}f\). First, we study the approximation in the Sobolev space
Hence, for a function \(f\in L^2(\textbf{v}_k)\), we study the error of approximation for \(f^{(m)}\) in \(L^2(\textbf{v}_{k+m})\).
Theorem 3.5
Let \(n,k\geqslant 0\), \(r\geqslant 1\). For \(f\in W_2^r(\textbf{v}_k)\),
or, equivalently,
holds for \(k\geqslant 0\) in the Hermite case, for \(\alpha +k>-1\) in the Laguerre case, and for \(\alpha +k,\beta +k\geqslant 0\) in the Jacobi case.
Proof
First, we prove (3.6) for \(m=0\). From the Parseval identity, (3.2) and the fact that \(r\le n\), we have
and using (3.4), we get
By Remark 2.5, we have
Therefore,
holds for \(k\geqslant 0\). Moreover, in any case \((-\lambda _{j-r,k+r})^r\) is an increasing sequence in j. Hence,
and, thus, (3.6) holds for \(m=0\).
Now, let \(m\geqslant 1\). From (3.5)
If we bound \(E_{n-m,\textbf{v}_{k+m}}\left( f^{(m)}\right) \) using (3.7) with r replaced with \(r-m\), then we get
which finishes the proof.\(\square \)
In order to estimate the error of approximation of \(S_{n,k}\) in \(H^r(\textbf{v}_k)\) defined in (1.1), we need the following lemma.
Lemma 3.6
Let \(\textbf{u}\) be a classical moment functional and \(\{P_n(x)\}_{n\geqslant 0}\) be its associated MOPS. For \(n,k\geqslant 0\),
with \(d_{n,n}^{(k)}=1\),
and
where \(\alpha _{n+k}\) and \(\beta _{n+k}\) are the real numbers appearing in (2.3).
Proof
This result follows from iterating (3.1).\(\square \)
It is obvious that, for Hermite polynomials, \(Q_{n,k}(x)=Q_{n,0}(x)\) for \(n,k\geqslant 0\). In the Laguerre case, it is straightforward to deduce from Table 1 the explicit expression for the coefficients in Lemma 3.6. For Jacobi polynomials, we can recover the expressions obtained in [34]. Indeed, for the Laguerre polynomials with parameter \(\alpha >-1\), we obtain \(d_{n,j}^{(k)}=(-1)^{n+j}\). We also recall here the coefficients for Jacobi polynomials with parameters \(\alpha ,\beta >-1\),
where
Our main effort lies in establishing the following result, which states the error of simultaneous approximation of a function and its derivatives by the projection operator \(S_{n,k}\).
Theorem 3.7
Let \(k+r\geqslant 1\), and \(f\in H^r(\textbf{v}_{k})\) such that \(f'\in W_2^r(\textbf{v}_{k+1})\). Then, for \(n\geqslant r\),
holds for \(k\geqslant 1\) in the Hermite case, for \(\alpha +k>0\) in the Laguerre case, and for \(\alpha +k,\, \beta +k\geqslant 1\) in the Jacobi case.
Proof
We shall consider each case separately.
First, let \(\textbf{u}\) be the moment functional associated with the Hermite polynomials. In this case, \(\textbf{u}=\textbf{v}_{k}\) for \(k\geqslant 0\), and thus \(\Vert f\Vert _{\textbf{u}}=\Vert f\Vert _{\textbf{v}_k}\). It follows from (3.6) and the fact that in this case \(\lambda _{n,k}\) is independent of the second subindex, that
For the Laguerre case, we start by considering the case \(m=0\). It is not difficult to modify the proof of Theorem 3.5 to deduce (3.8) since, by Remark 2.5, we have
Now, we consider the case \(m=1\). By triangle inequality
The first term on the right side is \(E_{n-1,\textbf{v}_k}(f')\). In order to bound the second term, consider the Fourier expansion of \(f'\). Since \(f\in H^r(\textbf{v}_k)\) and, thus, \(f'\in L^2(\textbf{v}_k)\),
Moreover, since \(f'\in W_2^r(\textbf{v}_{k+1})\), which means that \(f'\in L^2(\textbf{v}_{k+1})\), we can also write
where the second equality follows from (3.2) and Lemma 3.6, and the third equality follows from interchanging the order of the summations. Comparing the two expressions for \(f'\), we deduce
Using this, together with
we obtain
Therefore,
Using (3.4), we have
Moreover, by Table 1, and the fact that \(-\lambda _{j+i,k-i}\leqslant -\lambda _{n+i,k-i}\) for \(0\leqslant j \leqslant n-1\), and \(1\leqslant i \leqslant k\), we can estimate
Therefore, using (3.6), we obtain
and, thus,
By (3.6), we get
This proves (3.8) in the Laguerre case for \(m=1\) and \(r\geqslant 1\).
The case \(m\geqslant 2\) follows inductively. With this in mind, we start by proving the following estimate
Since
then we estimate as follows
where we have used (3.6) and (3.9). This establishes (3.10) for \(m\geqslant 2\).
Assuming that (3.8) has been established for a fixed m, we prove that it also holds for \(m+1\). By triangle inequality,
The second term on the right side is bounded in (3.10). The first term on the right side can be bounded, by induction with r replaced with \(r-1\), by a bound that is less than the above. This completes the proof of the Laguerre case.
The Jacobi case has already been established in [34] in the form
Here, we recast the important points of the proof in [34] in terms of \(\lambda _{n,k}\) and \(E_{n-r,\textbf{v}_{k+r-1}}(f^{(r)})\). Since in the Jacobi case \(-\lambda _{n,k}\sim n^2\), then for \(m=1\) and \(r=1\), we obtain
Then, (3.8) is established for \(m=1\) and \(r\geqslant 1\) by using (3.6). Moreover, for \(m\geqslant 1\), from (2.18) in [34] we have
Then (3.8) follows from (3.6) and by induction as in the Laguerre case.\(\square \)
4 Discrete–continuous Sobolev orthogonal polynomials
This section is devoted to studying sequences of orthogonal polynomials with respect to a discrete–continuous Sobolev inner product associated with a positive definite classical moment functional \(\textbf{u}\) and a set of s distinct real numbers
More concretely, the inner product that we consider is of the form
where
with \(d_1+\cdots +d_s=N\), \(d_i\in \mathbb {N}\), and \(\Lambda \) is a \(N\times N\) positive definite symmetric matrix.
In this section, we present two main results:
- 1.:
-
For a given sequence of polynomials (defined below in (4.6)), there is a matrix \(\Lambda \) such that the sequence of polynomials is orthogonal with respect to (4.1).
- 2.:
-
For a given matrix \(\Lambda \), there is a sequence of orthogonal polynomials associated with (4.1).
In order to prove these results, we need to introduce a basis of polynomials and its dual basis associated with the set \(\{\nu _i:\ 1\leqslant i \leqslant s\}\).
We define the following polynomial associated with (4.1),
Hence, \(\deg \,\varphi =N\). In the sequel, we fix \(\theta \in [\nu _1,\, \nu _s]\). We introduce the basis of \(\Pi \),
and its associated dual basis;
such that \(\langle \varvec{\sigma }_{i,j},(x-\theta )^m\,\varphi ^n\rangle = \delta _{i,m}\delta _{j,n}\). Notice that for every \(p\in \Pi \),
where \(a_{m,n}=0\) if \(m+nN>\deg p\).
Let
For a differentiable function f(x), the Hermite interpolation polynomial \(\mathcal {H}_{\varphi }f(x)\) defined as ([11])
is the unique polynomial of degree at most \(N-1\) such that
We will need to extend the polynomials defined in (2.4) to negative values of the parameter k. Let \(\{Q_{n,0}(x)\}_{n\geqslant 0}\) be the polynomials defined in (2.4) associated with the classical moment functional \(\textbf{u}\). For \(N\geqslant 1\), we define recursively the polynomial
Observe that \(\deg Q_{n,-N}=n\), and \(Q_{n,-N}^{(r)}(x)=Q_{n-r,-N+r}(x)\) for \(0\leqslant r \leqslant n\). Moreover,
In this way, by Taylor’s theorem, we have the following alternative expression for \(Q_{n,-N}(x)\) with \(1\leqslant N \leqslant n\),
We are ready to state our first result.
Theorem 4.1
There is a \(N \times N\) positive definite symmetric matrix \(\Lambda \) such that the set of polynomials \(\{q_n(x)\}_{n\geqslant 0}\) with
is orthogonal with respect to (4.1). Moreover,
Proof
First, we prove the existence of a \(N\times N\) non-singular matrix M such that, for all polynomials p(x) and q(x),
Then we show that \(\{q_n(x)\}_{n\geqslant 0}\) is orthogonal with respect to (4.1) with \(\Lambda =(M^{-1})^{\top }\,M^{-1}\) which is evidently a positive definite symmetric matrix ([20]).
Using the bases \(\mathfrak {B}_{\theta ,\varphi }\) and \(\mathfrak {B}'_{\theta ,\varphi }\) introduced in (4.3) and (4.4), for each \(1\leqslant i \leqslant s\), we write
where
For \(n\geqslant 1\), the polynomial \(\varphi (x)^n\) can be represented as
Then,
For \(0\leqslant j \leqslant d_i-1\), it follows that
which means that
Given any polynomial p(x), for each \(1\leqslant i \leqslant s\), we apply both sides of the above distributional equations to p(x) and obtain
Define the \(d_i\times N\) matrices
For each \(1\leqslant i \leqslant s\), we have
where \(P_i\) is the vector defined in (4.2), and, as a consequence,
We now show that the \(N\times N\) matrix M is non-singular. By construction, (4.9) has at least one solution \( (\langle \varvec{\sigma }_{0,0},p\rangle , \ldots , \langle \varvec{\sigma }_{N-1,0},p\rangle )^{\top }\). Suppose that \( (g_0,\ldots ,g_{N-1})^{\top }\) is another solution of (4.9). Define the polynomials
Then, for each \(1\leqslant i \leqslant s\),
Let \(h(x) = f(x)-g(x)\). Clearly \(\deg h\leqslant N-1\). On the other hand,
This implies that \(\nu _i\) is a zero of h(t) with multiplicity at least \(d_i\). But this is true for \(1\leqslant i \leqslant s\). It follows that \(\deg h\geqslant N\). Therefore, \(h(x) =0\), or, equivalently, \(g_m=\langle \varvec{\sigma }_{m,0}, p\rangle \) for \(0\leqslant m \leqslant N-1\). Hence, M is a non-singular symmetric matrix.
Now, we prove the orthogonality of \(\{q_n(x)\}_{n\geqslant 0}\) with respect to (4.1) with \(\Lambda =(M^{-1})^{\top }\,M^{-1} \). For \(0\leqslant n \leqslant N-1\), \(q_n^{(N)}(x)=0\). Then, by (4.7),
If \(0\leqslant m \leqslant N-1\), then it is clear that \((q_n,q_m)=\delta _{n,m}\). For \(m\geqslant N\), by (4.5), \((q_n,q_m)=0\). Furthermore, for \(n,m\geqslant N\),
\(\square \)
Remark 4.2
It is important to note that (4.9) is equivalent to
where the matrix M was defined in (4.8)–(4.9). Furthermore, let \((\cdot ,\cdot )_{\Pi _{N-1}}\) denote the restriction of (4.1) to \(\Pi _{N-1}\). Then \(\Lambda \) is the Gram matrix of \((\cdot ,\cdot )_{\Pi _{N-1}}\) with respect to the basis \(\{\ell _{i,j}(x):\ 0\leqslant j \leqslant d_i-1, \ 1\leqslant i \leqslant s \}\) and, thus,
is the Gram matrix of \((\cdot ,\cdot )_{\Pi _{N-1}}\) with respect to the basis \(\{(x-\theta )^m:\ 0\leqslant m \leqslant N-1 \}\).
Now we deal with the case when \(\Lambda \) in (4.1) is a prescribed \(N\times N\) positive definite symmetric matrix. Recall that in this case there is a unique lower triangular matrix \(\Xi \) with positive real numbers in the main diagonal, such that \(\Lambda =\Xi \,\Xi ^{\top }\) ([20]). In this way,
Since the factor \(M^{\top }\,\Xi \) is a non-singular matrix, we conclude that G is a positive definite matrix ([20]). We denote by \(L^{-1}\) the unique lower triangular matrix with positive real numbers in the main diagonal such that \(G=L^{-1}\,(L^{-1})^{\top }\).
In this way, we can state our second main result in terms of \(\Lambda \) and \(L^{-1}\).
Theorem 4.3
For a \(N\times N\) positive definite symmetric matrix \(\Lambda \), the sequence of polynomials \(\{\widetilde{q}_n(x)\}_{n\geqslant 0}\) given by
and,
is orthogonal with respect to (4.1) with \(\Lambda \). Moreover,
Proof
Consider the matrix
Note that by (4.11), \(H=L\,G\,L^{\top }\). Therefore, H is the identity matrix of order N. It follows that
For \(n\geqslant N\), it is clear that \((\widetilde{q}_n,\widetilde{q}_m)=h_{n-N,0}\,\delta _{n,m}\), \(m\geqslant 0\).\(\square \)
Remark 4.4
It is important to note that L and \(L^{-1}\) are both lower triangular matrices with non-zero real numbers in the main diagonal. Consequently, for \(\widetilde{q}_n(x)\) defined in (4.11), we have \(\deg \widetilde{q}_n = n\).
5 Fourier series in terms of Sobolev orthogonal polynomials
In this section, we study the approximation behavior of the Fourier series associated with the two sequences of orthogonal polynomials (4.6) and (4.11) introduced in the previous section.
Let \(\textbf{u}\) be a positive definite classical moment functional. For \(f\in H^{N}(\textbf{u})\), we define the Fourier orthogonal expansion of f with respect to the orthogonal basis (4.6) as,
and with respect to the orthogonal basis (4.11) as,
For \(n\geqslant 0\), let \(\mathcal {S}_{n}^N\) and \(\widetilde{\mathcal {S}}_{n}^N\) denote the projection operators \(\mathcal {S}_{n}^N: H^{N}(\textbf{u})\rightarrow \Pi _n\) and \(\widetilde{\mathcal {S}}_{n}^N: H^{N}(\textbf{u}) \rightarrow \Pi _n\) defined as
For \(N=0\), the operators \(\mathcal {S}_{n}^0f(x)=\widetilde{\mathcal {S}}_{n}^0f(x)=S_{n,0}f(x)\) is the partial sum of the usual classical expansion in orthogonal polynomials. These operators satisfy several simple properties and can be written, in particular, in terms of the partial sum \(S_{n-N,0}f\).
Lemma 5.1
For \(f\in H^{N}(\textbf{u})\),
- (1):
-
\(\mathfrak {f}_n^N=f^{(n)}(\theta )/n!\) if \(0\leqslant n \leqslant N-1\), and \(\mathfrak {f}_n^N=\widehat{f^{(N)}}_{n-N,0}\) if \(n\geqslant N\);
- (2):
-
\(\left( \mathcal {S}_{n}^Nf\right) ^{(N)}=S_{n-N,0}f^{(N)}\) if \(n\geqslant N\);
- (3):
-
for \(n\geqslant N\),
$$ \left( \mathcal {S}_{n}^Nf\right) ^{(j)}(\nu _i)=f^{(j)}(\nu _i), \quad 0\leqslant j \leqslant d_i-1, \quad 1\leqslant i \leqslant s. $$
Proof
As in the proof of Theorem 4.1, it is easy to see that if \(0\leqslant n \leqslant N-1\), \((f,q_n)=f^{(n)}(\theta )/n!\) and \((q_n,q_n)=1\), whereas if \(n\geqslant N\),
and \((q_n,q_n)=h_{n-N,0}\). In this way, (1) is proved. The statement (2) follows easily from (1).
where M is the matrix defined in (4.9). Therefore,
Moreover, by (4.6), if \(n\geqslant N\), \(q_n^{(j)}(\nu _i)=0\), \(0\leqslant j \leqslant d_i-1\), \(1\leqslant i \leqslant s\), and, thus
This proves (3).\(\square \)
Lemma 5.2
For \(f\in H^{N}(\textbf{u})\),
- (1):
-
\(\widetilde{\mathfrak {f}}_n^N=\widehat{f^{(N)}}_{n-N,0}\) if \(n\geqslant N\), and
$$\begin{aligned} \left( \widetilde{\mathfrak {f}}_0^N,\ldots , \widetilde{\mathfrak {f}}_{N-1}^N \right) = \left( f(\theta ),f'(\theta ),\ldots , \frac{f^{(N-1)}(\theta )}{(N-1)!} \right) \,L^{-1}; \end{aligned}$$ - (2):
-
\(\left( \widetilde{\mathcal {S}}_{n}^Nf\right) ^{(N)}=S_{n-N,0}f^{(N)}\) if \(n\geqslant N\);
- (3):
-
for \(n\geqslant N\),
$$\begin{aligned} \left( \widetilde{\mathcal {S}}_{n}^Nf\right) ^{(j)}(\nu _i)=f^{(j)}(\nu _i), \quad 0\leqslant j \leqslant d_i-1, \quad 1\leqslant i \leqslant s. \end{aligned}$$
Proof
It is easy to see that (1) follows from (4.11) and that (2) follows from (1). The proof of (3) is as in Lemma 5.1 after noticing that by (1) and (4.11), we have
\(\square \)
Now we consider the simultaneous approximation by polynomials. First, we establish estimates for derivatives of order at least N.
Theorem 5.3
Let \(r\geqslant N+1\) and \(f\in H^r(\textbf{u})\) such that \(f^{(r)}\in L^2(\textbf{v}_{r-N-1})\). Then
holds always in the Hermite case, for \(\alpha >-1\) in the Laguerre case, and for \(\alpha , \beta \geqslant 0\) in the Jacobi case.
Proof
For \(m\geqslant N\), we have
Then, the result follows from Remark 2.5 and Theorem 3.7. \(\square \)
In order to handle the case of derivatives of lower order, we need to introduce the following subspace
Observe that for \(f,g\in H_{0}^{N}(\textbf{u})\),
This means that \(\left\langle \textbf{u}, f^{(N)}\,g^{(N)} \right\rangle \) is an inner product on \(H_{0}^{N}(\textbf{u})\). By the Riesz Representation Theorem, for each \(h\in H_{0}^{N}(\textbf{u})\), there is a unique \(g\in H_{0}^{N}(\textbf{u})\) such that
which is equivalent to
Moreover, we have the following identity for \(v\in H^N(\textbf{u})\),
Theorem 5.4
Let \(r\geqslant N+1\) and \(f\in H^r(\textbf{u})\) such that \(f^{(r)}\in L^2(\textbf{v}_{r-N-1})\). Then,
always holds in the Hermite case, for \(\alpha >-1\) in the Laguerre case, and for \(\alpha ,\beta \geqslant 0\) in the Jacobi case.
Proof
To prove the case \(m=0\), we use the well-known duality argument, the so-called Aubin–Nitsche technique, based on the identity
Let \(g\in H_{0}^{N}(\textbf{u})\) be defined as in (5.1). Applying (5.2) with \(v= f-\mathcal {S}_{n}^Nf\), we obtain
By the Cauchy-Schwarz inequality, we have
and
Observe that
Using the Parseval identity and (3.4), we get
It follows from (5.4) that
Therefore,
By (5.3) and Theorem 5.3, we get
We handle the intermediate case \(1\leqslant m \leqslant N-1\) by following the argument found in [8] (see also [16]): Let T be a function from \([0,\infty )\) to \(L^2(\textbf{u})\):
Suppose that T has N continuous derivatives and T and \(T^{(N)}\) are bounded on \([0,\infty )\). Then by [8, Theorem I], we have
where \(\Vert T\Vert _{\infty }=\sup _{t\in [0,\infty )}\Vert T(t)\Vert _{\textbf{u}}\). In particular, consider the translation operator \(A_t\) defined by \(A_tf(x)=f(t+x),\) then \(A_t\) can be represented as \(A_t=e^{t\frac{d}{dx}}\). Notice that \(\Vert A_tf\Vert _{\textbf{u}}\leqslant \Vert f\Vert _{\textbf{u}}\) for all \(t\in [0,\infty )\) and \(f\in L^2(\textbf{u})\). Taking into account the above, the set \(\{A_t\}_{t\geqslant 0}\) is a contraction semigroup of operators (with infinitesimal generator \(\frac{d}{dx}\)) on \(L^2(\textbf{u})\). Therefore, for \(v\in H^N(\textbf{u})\), if \(T(t):=A_t\,v\), then T is N times continuously differentiable and
where we have used
Setting \(v=f-\mathcal {S}_{n}^Nf\) and using (5.6), we obtain
The result follows from Theorem 5.3 and (5.5).
Remark 5.5
Theorems 5.3 and 5.4 with \(\widetilde{\mathcal {S}}_n^N\) can be proved with identical proofs.
6 A numerical example
For \(\alpha >-1\), let \(L_n^{\alpha }(x)\) denote the n-th monic Laguerre polynomial defined as ([33, Chapter V])
These polynomials are orthogonal with respect to the positive definite moment functional \(\textbf{u}_{\alpha }\) defined by
Moreover, we have
By Theorem 2.3, the derivatives
are orthogonal with respect to \(\textbf{u}_{\alpha +k}\,=\,x^k\,\textbf{u}_{\alpha }\); that is,
Consider the numbers \(\nu _1=0\), \(\nu _2=1\), \(\theta \,=\,0\), and the polynomial
The Hermite interpolation polynomial (associated with \(\varphi (x)\)) for a function f is given by
Then, by Theorem 4.1, there is a \(3 \times 3\) positive definite symmetric matrix \(\Lambda \) such that the set of polynomials \(\{q_n(x)\}_{n\geqslant 0}\) with
where
is orthogonal with respect to the discrete–continuous Sobolev inner product
Indeed, computing the matrix M in (4.9) and using the fact that \(\Lambda =(M^{-1})^{\top }\,M^{-1}\), we get
Using (6.1) and integration by parts, we obtain
Observe that \(Q_{n,-3}^{\alpha +3}(0)\,=\,(Q_{n,-3}^{\alpha +3})'(0)\,=\,(Q_{n,-3}^{\alpha +3})''(0)\,=\,0\), and consequently,
Moreover,
Now, consider the function
For \(\alpha = 0\), Table 2 shows the values of
where \(\mathcal {S}_n^N\) denotes the projection operator with respect to the Sobolev orthogonal polynomials (6.2). In this case, \(\lambda _{n,k}=-n\) (see Table 1) which happens to be independent of k. Since f belongs to \(C^{\infty }\), we choose \(r=8\) and observe that the ratio of the above errors seems to decay as n grows for \(m=0\) and \(m=1\). For \(m=2\), it is likely that the ratio of the errors behaves similarly to the other two cases; however, it seems that it does so much slower.
Figure 1 depicts the graphs of \(f,\,f',\) and \(f''\) together with the graphs of \(\mathcal {S}^N_nf\), \((\mathcal {S}^N_nf)'\), and \((\mathcal {S}^N_nf)''\) for \(n=15,20,25\). We note that the projection operator \(\mathcal {S}^N_{25}f\) seems to provide an adequately close simultaneous approximation on the interval shown. We should also remark that \(f(0)=\mathcal {S}^N_nf(0)\), \(f'(0)=(\mathcal {S}_n^Nf)'(0)\), and \(f(1)=\mathcal {S}^N_nf(1)\) for all values of n.
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García-Ardila, J.C., Marriaga, M.E. Approximation by polynomials in Sobolev spaces associated with classical moment functionals. Numer Algor 95, 285–318 (2024). https://doi.org/10.1007/s11075-023-01572-3
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DOI: https://doi.org/10.1007/s11075-023-01572-3