Approximation with Bernstein-Szeg¨ o polynomials
Volker H¨ osel∗1 , Rupert Lasser+ +
GSF-National Research Center for Environment and Health Institute of Biomathematics and Biometry Ingolst¨adter Landstraße 1, 85764 Neuherberg, Germany ∗
Munich University of Technology Centre of Mathematics 85748 Garching, Germany
E-mail:
[email protected],
[email protected]
Abstract We present approximation kernels for orthogonal expansions with respect to Bernstein-Szeg¨o polynomials. The construction is derived from known results for Chebyshev polynomials of the first kind and does not pose any restrictions on the Bernstein-Szeg¨o polynomials.
Keywords: Orthogonal polynomials, Approximation kernels 2000 Mathematics Subject Classification: 41A10
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Corresponding author
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Introduction
The convergence of orthogonal series in certain function spaces B is a long established research topic. Of special interest for applications is the possibility to approximate a function f ∈ B by weighted generalized Fourier expansions AN (f )(t) :=
N X
aN,k fˆ(k) pk (t).
k=0
Here, (pk )k∈N0 is a sequence of polynomials on the real line with deg pk = k orthonormal with respect to a probability measure µ. The Fourier coefficients fˆ(k) of f ∈ B are given by Z ˆ f (k) := f (t) pk (t) dµ(t). R
The related approximation problem for Fourier series is a classical field of research, and many proposals for appropriate weights aN,k exist, compare e.g. [3, 7] and [14]. In the context of orthogonal polynomials, where µ has compact support, the investigations have been focussed on the class of Jacobi polynomials. Many results can be found in Pollard [9, 10, 11], Askey and Hirschman [1], Bavinck [2], Chanillo and Muckenhoupt [4], Lasser and Obermaier [5], Yadav [13]. This list of references is by no means complete. A survey on this issue is provided by Lubinsky [8]. The Chebyshev polynomials of the first kind show up as the real part when studying even functions by classical Fourier series on the one hand and appear as a special case of Jacobi polynomials on the other hand. The purpose of this paper is to investigate approximation processes for BernsteinSzeg¨o polynomials. From different points of view this class of orthogonal polynomials is very close to the Chebyshev polynomials of the first kind. Given a real polynomial %(x) strict positive on [−1, 1], the Bernstein-Szeg¨o polynomials p%n (x) are those polynomials of degree n that are orthonormal (with posic √% dx on [−1, 1]. (c% tive leading coefficient) with respect to dπ% (x) = %(x) 1 − x2 is a constant such that π% is a probability measure.) There exists a polynomial r P H(z) = αk z k , z ∈ C, with real coefficients αk having no zeros for |z| ≤ 1 such k=0
that %(cos t) = |H(eit )|2 , t ∈ R [12, Theorem 1.2.2]. If degree(%) = r then the orthonormal polynomials p%n (cos t) are up to normalization equal to Re(eint H(eit ) ) for r < 2n [12, Theorem 2.6]. It is convenient to renorm the Bernstein-Szeg¨o polynomials p%n (x) by Tn% (x) = p%n (x)/p%n (1). In particular Tn% (1) = 1 is valid, and Z 1 −1 2 % % h (n) := (Tn (x)) dπ% (x) = (p%n (1))2 . −1
2
Notify that h% (n) is for all n ≥ n0 , where n0 is the first integer with r < 2n0 . Moreover, if αk is the real coefficient of z k in H(z) we get for r < 2n Tn% (cos t)
=
r X
αk cos(n − k)t.
k=0
In the sequel %(x) will be a polynomial strictly positive on [−1, 1]. Henceforth B will denote one of the Banach spaces C([−1, 1]) or Lp ([−1, 1], π% ), 1 ≤ p < ∞. It is easy to check that Lp ([−1, 1], π% ) = Lp ([−1, 1], π1 ), where dπ1 (x) = 1 √ 1 dx is the Chebyshev orthogonalization measure on [−1, 1], since the Lp -norm π 1−x2 with respect to π% is equivalent to that with respect to π1 . For the polynomials Tn1 (x) we write Tn (x), also h(n) for h1 (n). Note that h(0) = 1, h(n) = 2 for n ∈ N. Given a triangular scheme (aN,k )n∈N0 ,k=0,...,N of complex numbers, define for x, y ∈ [−1, 1] N N X X A%N (x, y) = aN,k p%k (x) p%k (y) = aN,k Tk% (x) Tk% (y) h% (k) k=0
k=0
For f ∈ B define the k-th Fourier coefficient with respect to π% Z 1 fˆ(k; %) = f (x) p%k (x) dπ% (x) −1
and the weighted Fourier expansions A%N f
=
n X
aN,k fˆ(k; %) p%k .
k=0
It is well-known that the Banach-Steinhaus theorem yields the following equivalence: We have lim kA%N f − f kB = 0 for all f ∈ B if and only if the two conditions hold, N →∞
(i)
lim aN,k = 1 for all k ∈ N0
N →∞
(ii) there exists a constant C > 0 such that kA%N f kB ≤ C kf kB for all f ∈ B and N ∈ N0 . Denoting by kA%N kB the operator norm of the operator A%N we have, moreover, (see e.g. [5, Proposition 1]) (iii) kA%N kB ≤
sup −1≤x≤1
kA%N (x, ·)k1 for all N ∈ N 3
and (iv) in the case of B = L1 ([−1, 1], π% ) or B = C([−1, 1]) we have kA%N kB =
2
sup −1≤x≤1
kA%N (x, ·)k1
for all N ∈ N.
The kernels
We have to derive an essential inequality to obtain upper bounds for kernels in two variables x, y for Bernstein-Szeg¨o polynomials. Lemma 1 Let (aN,k ) be a triangular scheme of complex numbers with |aN,k | ≤ C1 . Given n, m ∈ N0 , m ≥ n, assume further that N −t X
|aN,k+t − aN,k | ≤ C2
for t ∈ {m, n}.
k=0
(C1 is independent of N and k, C2 is independent of N .) Then for x, y ∈ [−1, 1] and N ∈ N we have N X
aN,k (Tk−n (x) Tk−m (y) + Tk−m (x) Tk−n (y))
k=m
= (Tm−n (x) + Tm−n (y))
N X
aN,k Tk (x) Tk (y) + RN (x, y),
k=m−n
where RN (x, y) is continuous and |RN (x, y)| ≤ A, where A is a constant independent of N and x, y. Proof. We assume m ≥ n. Then for any N ≥ m − n and x, y ∈ [−1, 1] we have with the cosine addition formula N X
(Tm−n (x) + Tm−n (y))
aN,k Tk (x) Tk (y)
k=m−n N X aN,k (Tk+m−n (x) Tk (y) + Tk (x) Tk+m−n (y)) = 2 k=m−n
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N X aN,k + (Tk+n−m (x) Tk (y) + Tk (x) Tk+n−m (y)). 2 k=m−n
Putting simply j = k + m in the first sum and j = k + n in the second one, we see that N X aN,k Tk (x) Tk (y) (Tm−n (x) + Tm−n (y)) k=m−n
N +m X
1 aN,j−m (Tj−n (x) Tj−m (y) + Tj−m (x) Tj−n (y)) 2 j=2m−n
=
N +n X
+
j=m
=
1 aN,j−n (Tj−m (x) Tj−n (y) + Tj−n (x) Tj−m (y)) 2
N +n X
1 (aN,j−m + aN,j−n ) (Tj−n (x) Tj−m (y) + Tj−m (x) Tj−n (y)) 2 j=2m−n +
+
N +m X
1 aN,j−m (Tj−n (x) Tj−m (y) + Tj−m (x) Tj−n (y)) 2 j=N +n+1 2m−n−1 X j=m
1 aN,j−n (Tj−m (x) Tj−n (y) + Tj−n (x) Tj−m (y)). 2
The absolute values of the second and third sum are bounded by (m − n)C1 . To deal with the first sum we observe +n NX 1 aN,j − (aN,j−m + aN,j−n ) (Tj−n (x) Tj−m (y) + Tj−m (x) Tj−n (y)) 2 j=2m−n
≤
N +n X
N +n X
|aN,j − aN,j−m | +
j=2m−n
|aN,j − aN,j−n | ≤ 2C2 .
j=2m−n
Therefore we have N +n X
aN,k (Tk−n (x) Tk−m (y) + Tk−m (x) Tk−n (y))
k=2m−n
= (Tm−n (x) + Tm−n (y))
N X
k=m−n
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˜ N (x, y), aN,k Tk (x) Tk (y) + R
˜ N (x, y)| ≤ 2C1 (m − n) + 2C2 . Hence where |R N X
aN,k (Tk−n (x) Tk−m (y) + Tk−m (x) Tk−n (y))
k=m
= (Tm−n (x) + Tm+n (y))
N X
aN,k Tk (x) Tk (y) + RN (x, y),
k=m−n
with |RN (x, y)| ≤ A, where the constant A is independent of N and x, y ∈ [−1, 1]. Remark: We note that both sums in Lemma 1 are symmetric in x and y. This special symmetric form is essential as the following example shows. Consider N X k κN (x) = 1− Tk (x) Tk−1 (0) for x ∈ [−1, 1] N +1 k=0 Then Z
1
|κN (x)| dπ1 (x) ≥
−1
Z
1
|κN (x)| dπ1 (x)
0
Z π/2 N X k π 1− ≥ cos kt cos(k − 1) dt N +1 2 0 k=0
=
N X k=1
Thus
N P
k=0
1−
k N +1
k 1− N +1
sin2 (k π2 ) −→ ∞ k
with N → ∞.
Tk (x) Tk−1 (0) cannot be uniformly bounded on [−1, 1] whereas
Lemma 1 yields that N X k=0
k 1− N +1
(Tk (x) Tk−1 (0) + Tk−1 (x) Tk (0))
is uniformly bounded on [−1, 1]. Next we deal with the kernel functions A%n (x, y). For that we prove the following identity. Lemma 2 Assume degree(%) = r, and let (aN,k ) be a triangular scheme. Then for N >r N r−1 X X aN,k Tk% (x) Tk% (y) aN,k Tk% (x) Tk% (y) = k=0
k=0
6
r X
+
2 αm
m=0
+
X
αm αn
0≤n 1 for k = 0, ..., N − 1. Hence −k % (An )n∈N0 is an approximate identity in all B for all %.
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(4) Fej´ er-Korovkin kernel. Put k+1 k−1 aN,k = cN (N − k + 3) sin( π) − (N − k + 1) sin( π) N +2 N +2 k 1 k+1 = cN (N − k + 1) 2 cos( π) sin( π) + 2 sin( π) , N +2 N +2 N +2 1 > 0, see [3, p.80]. where cN = 2(N + 2) sin( Nπ+2 ) With standard trigonometric identities we obtain for k = 0, ..., N − 1 π k k+1 aN,k − aN,k+1 = cN (N − k) 2 sin( ) cos( π) − cos( π) N +2 N +2 N +2 π k+1 k+2 k π) sin( ) + 2 sin( π) − 2 sin( π) + 2 cos( N +2 N +2 N +2 N +2 π k + 1/2 1/2 = 2cN (N − k) sin( ) 2 sin( π) sin( π) N +2 N +2 N +2 k π k+1 π) sin( ) + sin( π) + cos( N +2 N +2 N +2 k+1 π k+1 π − sin( π) cos( ) − cos( π) sin( ) N +2 N +2 N +2 N +2 π k + 1/2 1/2 ) 2 sin( π) sin( π) = 2cN (N − k) sin( N +2 N +2 N +2 k k+1 π + cos( π) − cos( π) sin( ) N +2 N +2 N +2 k+1 π + sin( π) 1 − cos( ) ≥ 0 N +2 N +2 Hence Lemma 3 can be applied, and we get that (A%n )n∈N0 is an approximate identity in all B for all %. (5) Generalized Dirichlet kernel. Fix λ > 1, and put aN,k
(λ)N +k (λ)N −k (N !)2 = (N + k)! (N − k)! ((λ)N )2
for k = 0, ..., N.
For these non-classical kernels see [6] or [7, p.115].We have aN,k (N + k + 1)(λ + N − k − 1) = . aN,k+1 (N − k)(λ + N + k) x 1+N +k N −k Now (λ−1)+x is an increasing function in x > 0, and so (λ−1)+1+N ≥ (λ−1)+N . +k −k aN,k % Therefore aN,k+1 ≥ 1, and Theorem 1 of [6] yields that (An )n∈N0 is an approximate identity in all B for all %.
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