Exponentially localized polynomial frames on compact subset of the real line and the Euclidean sphere H. N. Mhaskar∗ Department of Mathematics, California State University Los Angeles, California, 90032, USA email:
[email protected] J. Prestin Institute of Mathematics, University of L¨ ubeck Wallstraße 40, 23560, L¨ ubeck, Germany email:
[email protected]
Abstract We describe an exponentially localized polynomial frame on compact subsets of the real line and on the unit sphere of a Euclidean space. The frame coefficients of a function f can be computed as finite linear combinations of the coefficients of f in an orthogonal polynomial expansion. In spite of the fact that the coefficients in the orthogonal expansion represent global information about the function, the frame coefficients are localized exponentially in the time domain. In particular, their behavior near a point determines whether f admits an extension as an analytic function in a complex neighborhood of the point. We describe a method to construct localized approximation kernels for a large class of orthogonal polynomials satisfying some mild conditions, such as the uniform boundedness of the Ces`aro means.
Keywords: Polynomial frames, analytic continuation, detection of analytic singularities. AMS classification: 41A25, 30B40, 42C15
1
Introduction
One of the most classical methods to approximate a continuous function f : [−1, 1] → R is to use an expansion of f in terms of a system of orthogonal polynomials, such as Legendre ∗
The research of this author was supported, in part, by grant DMS-0204704 from the National Science Foundation and grant W911NF-04-1-0339 from the U.S. Army Research Office.
1
polynomials. Although the partial sums of such an expansion cannot converge for every continuous function, one can often find a summability method with which the expansion can be made to converge to f uniformly on [−1, 1] in the sense of that method. For example, the arithmetic means of the partial sums of the expansion in Legendre polynomials converge for every continuous function on [−1, 1]. Let kf k∞ = maxx∈[−1,1] |f (x)|, Πn denote the class of all polynomials of degree at most n, and En,∞ (f ) := min kf − P k∞ . P ∈Πn
In the case of Legendre polynomial expansions, one can even construct linear operators {σn } such that for each integer n ≥ 1, σn (f ) ∈ Π2n−1 , and kf − σn (f )k∞ ≤ cEn,∞ (f ), where c is a positive constant independent of n and f . Such operators can be constructed both using the coefficients {fˆ(k)} in the Legendre expansions or samples of the function at judiciously chosen sites. We will elaborate on these constructions in Section 2 in a greater generality. One of the main advantages of polynomial approximation is that there is no saturation; given any nonincreasing sequence {δn } of positive numbers such that δn → 0 as n → ∞, one can construct a continuous function f such that En,∞ (f ) ≤ δn , n ≥ 1. On the other hand, the polynomial approximants in general are not expected to be localized. For example, if P is the best polynomial approximation from Πn for the function x 7→ |x|, then ||x| − P (x)| remains bounded from below by c/n at points away from the origin as well. Many mathematicians, including Gaier, Ivanov, Saff, and Totik ([4, 19], and references therein), have studied the construction of polynomials that provide a near best approximation to piecewise analytic functions on the whole interval [−1, 1], and an exponentially fast decaying approximation at points of analyticity of the function. A related question is to detect, without knowing the function itself, the intervals where the function is smooth so as to admit good approximation. We note that orthogonal polynomials, unlike classical wavelets, are not compactly supported. Hence, even though the sequence {fˆ(k)} contains enough information to determine all of the local features of f , the sequence itself does not directly reveal these local features. Therefore, it is an interesting question to determine the local smoothness properties of f , given {fˆ(k)}; for example, the locations of the points where a certain derivative of f has a jump discontinuity, or the Besov spaces to which the restrictions of f to the neighborhoods of different points belong. The problem (and the analogous problem for the simpler case of trigonometric Fourier series) appears in many different contexts; for example, the problem of hidden periodicities [7], direction finding in phased array antennas [6, 18, 15], and solution of partial differential equations [2, 5, 8]. A recent survey of the research arising from this problem is given in [16]. In both of the above problems, one seeks linear operators, applicable for every continuous function (universal operators) whose values are polynomials which adjust their behavior near each point according to the smoothness of the target function near that point. Naturally, a key component of the research in this direction is to construct kernels of the form Φ∗n (x, y) which are polynomials in x and y, and are localized in the sense that limn→∞ Φ∗n (x, y) = 0 if x 6= y. The rate at which the limit takes place determines which local Besov spaces can be characterised using these kernels. Unlike many of the classical, compactly supported wavelets, it is possible to choose the kernels so that there is no saturation on the order of the local smoothness of functions which can be characterised. 2
So far, all the constructions, which we are aware of, are based on the special function properties of the underlying orthogonal polynomial system. One purpose of this paper is to give a general purpose construction, which is based only on the assumption that the system admits a summability operator in the sense to be described precisely in Section 4. The summability conditions are satisfied, for example, by most of the Jacobi polynomials as explained in Section 2 and a large class of other orthogonal polynomials studied by Freud [3]. The second purpose of this paper is to illustrate the exponential localization of our kernels by characterizing local analyticity of functions. In the case of expansions in terms of polynomials orthogonal with respect to a measure µ on [−1, 1], there are well known Hadamard type formulas in terms of {fˆ(k)} to decide whether the function f allows an analytic continuation to the interior of an ellipse containing [−1, 1] ([14, Chapter VII, Section 3.1]) when µ satisfies certain mild conditions. In this paper, we are interested in determining, given the global information {fˆ(k)}, whether f admits an analytic continuation to a complex neighborhood of a point in [−1, 1], as opposed to a neighborhood of the whole interval [−1, 1]. Towards this goal, will assume certain very general P we ˆ conditions on the summability of the expansion f (k)pk , and define a frame transform τn (µ; f ) (See Section 4 for details) that can be calculated using the coefficients {fˆ(k)}, P such that f = τn (µ; f ) in the sense of uniform convergence on [−1, 1]. We will show that the function f admits an analytic continuation to a complex neighborhood of a point x ∈ [−1, 1] if and only if there is a nondegenerate interval I containing x, such that
lim sup max |τn (µ; f, y)| n→∞
y∈I
1/2n
< 1.
In Section 2, we present the theoretical results in the context of Jacobi polynomials. In Section 3, these are illustrated using some numerical examples. Our main theorems are applicable in a very general context, where one considers orthogonal polynomials with respect to general measures having a finite logarithmic energy, supported on arbitrary compact subsets of R, subject to certain technical conditions. In addition to the expansion coefficients and point evaluations, one may also consider other linear functionals. These generalizations are presented in Section 4. The constructions in Section 4 can be extended easily to the context of the unit sphere in a Euclidean space. This construction is explained in Section 5. The proofs of the new results are given in Section 6.
2
Jacobi polynomials
Let α, β > −1. The Jacobi measure is defined by dµ
(α,β)
(x) =
(1 − x)α (1 + x)β dx, 0, 3
if x ∈ (−1, 1), otherwise,
(α,β)
where α, β ≥ −1/2. There exists a unique system of polynomials {pk ∈ Π k }∞ k=0 , called orthonormalized Jacobi polynomials, with positive leading coefficients such that Z 1, if j = k, (α,β) (α,β) (α,β) pk pj dµ = 0, otherwise. If 1 ≤ p ≤ ∞, and A ⊆ [−1, 1], the space Lp (µ(α,β) ; A) consists of measurable functions f for which Z 1/p p (α,β) |f | dµ , if 1 ≤ p < ∞, kf kµ(α,β) ;A := A ess sup |f (t)|, if p = ∞ t∈A
is finite, with the usual convention that two functions are considered equal if they are equal almost everywhere. The space X p (µ(α,β) ; A) denotes Lp (µ(α,β) ; A) if 1 ≤ p < ∞ and the space of bounded uniformly continuous functions on A (equipped with the supremum norm) if p = ∞. The mention of the set A will be omitted if A = [−1, 1]. For real x ≥ 0, let Πx denote the class of all algebraic polynomials of degree at most x. We note that Πx = Πbxc , where bxc is the integer part of x. We prefer the simpler notation Πx regardless of whether x is an integer or not. For 1 ≤ p ≤ ∞ and f ∈ Lp (µ(α,β) ), we define the degree of approximation of f from Πx by Ex,p (α, β; f ) := min kf − P kµ(α,β) ;p . P ∈Πx
We adopt the following convention regarding constants: the symbols c, c1 , · · · will denote generic positive constants, dependent only on such fixed parameters in the discussion as p, α, β, etc. Their value may be different at different occurrences, even within the same formula. It is readily seen that the partial sum of order 2n of the Jacobi polynomial expansion R (α,β) of a function f ∈ L1 (µ(α,β) ) is given by f (y)Kn (◦, y)dµ(α,β) (y), where the Dirichlet– Darboux kernel Kn is defined by n
Kn(α,β) (x, y)
:=
2 X
(α,β)
pk
(α,β)
(x)pk
(y).
k=0
However, the sequence of these partial sums need not converge to f for every f ∈ L1 (µ(α,β) ). To get convergent sums, we need to use a summability method. It is known [21, Theorem 9.1.4] that for every continuous f , the Ces´aro means of order k > α + β + 1 of the Jacobi polynomial expansion of f converge uniformly to f . In order to get a near best approximation and exponential localization, we need to introduce an operator based on another related kernel. Let K ≥ α + β + 2 be an integer, h : [0, ∞) → R be a function which is a K times iterated integral of a function of bounded variation, h(x) = 1 for 0 ≤ x ≤ 1/2, and h(x) = 0 for x > 1. Then for x, y ∈ C, n = 0, 1, · · ·, we define the kernel Φn (µ(α,β) ; h, x, y) :=
2n X
(α,β)
h(k/(2n))pk
k=0
4
(α,β)
(x)pk
(y).
(2.1)
Using a summation by parts argument or directly as in [11], one can prove that Z sup |Φn (µ(α,β) ; h, x, y)|dµ(α,β) (y) < ∞. n≥0,x∈[−1,1]
R In addition, it is easy to verify that ΦnR(µ(α,β) ; h, x, y)P (y)dµ(α,β)(y) = P (x) for every P ∈ Πn/2 . Therefore, the polynomials f (y)Φn (µ(α,β) ; h, ◦, y)dµ(α,β) (y) converge uniformly to f for every continuous f , at a rate comparable to En/2,∞ (α, β; f ). In [11], we have shown that the smoother the h, the better localized the kernels Φn are; in particular, if h is infinitely often differentiable, then for every integer Q, |Φn (µ(α,β) ; h, x, y)| ≤ c(Q, x, y)/nQ. However, this rate is not enough to detect the possibility of analytic continuation of a function near a point. In order to obtain an exponential rate of decay, we use the following kernel instead. n 4 − (x − y)2 ∗ (α,β) Φn (µ ; h, x, y) := Φ3n (µ(α,β) ; h, x, y). (2.2) 4 The summability operators σnC and the frame operators τnC are defined for f∈L1 (µ(α,β) ), x ∈ C, n = 0, 1, · · ·, by Z C σn (α, β; h, f, x) := f (y)Φ∗n (µ(α,β) ; h, x, y)dµ(α,β) (y) C σ1 (α, β; h, f, x), if n = 0, C τn (α, β; h, f, x) := C C σ2n (α, β; h, f, x) − σ2n−1 (α, β; h, f, x), if n = 1, 2, · · ·. (2.3) We note that σnC (α, β; h, f ) ∈ Π8n , while τnC (α, β; h, f ) ∈ Π2n+3 . Since is a symmetric polynomial of degree 8n in each of its variables, one has the representation Φ∗n (µ(α,β) ; h, x, y)
Φ∗n (µ(α,β) ; h, x, y)
=
8n X 8n X
(α,β)
(α,β)
an;k,j (h)pk
(α,β)
(x)pj
(y),
k=0 j=0
(α,β)
where, for each integer n ≥ 0, (an;k,j (h)) is a symmetric matrix. Defining the Jacobi coefficients of f ∈ L1 (µ(α,β) ) by Z (α,β) ˆ j = 0, 1, · · · , f (α, β; j) = f pj dµ(α,β) , it follows that σnC (α, β; h, f ) =
8n 8n X X k=0
!
(α,β) (α,β) an;k,j (h)fˆ(α, β; j) pk .
j=0
Thus, the operators σnC and τnC can be computed using finitely many Jacobi coefficients of f . 5
From a computational point of view, we would like to define discrete versions of these operators, which are obtained using Gauss quadrature formulas. For m ≥ 1, let xk,m , (α,β) k = 1, · · · , m be the zeros of pm , and ! m 2 −1 X (α,β) λk,m := pj (xk,m ) j=1
be the corresponding Cotes numbers. We define the discretized versions of the operators by σnD (α, β; h, f, x)
:=
8n+1 X
λk,8n+1 f (xk,8n+1 )Φ∗n (µ(α,β) ; h, x, xk,8n+1 ),
k=1 σ1D (α, β; h, f, x), τnD (α, β; h, f, x) := σ2Dn (α, β; h, f, x) − σ2Dn−1 (α, β; h, f, x),
if n = 0, if n = 1, 2, · · ·.
(2.4)
The following theorem summarizes some of the properties of the operators introduced so far. Theorem 2.1 Let 1 ≤ p ≤ ∞, α, β ≥ −1/2, f ∈ Lp (µ(α,β) ). For integer n ≥ 0, let σn (f ) denote either σnC (α, β; h, f ) or σnD (α, β; h, f ), and similarly for τn (f ). (a) We have σ2n (P ) = P for P ∈ Π2n , kσ2n (f )kp ≤ ckf kp ,
(2.5)
and E2n+3 ,p (α, β; f ) ≤ kf − σ2n (f )kµ(α,β) ;p ≤ E2n ,p (α, β; f ). P p (α,β) ). (b) If f ∈ X p (µ(α,β) ) then f = ∞ n=0 τn (f ), with convergence in the sense of X (µ One has the Littlewood–Paley type decomposition n+3
f=
∞ 2X +1 X n=0
k=1
n o (α,β) (α,β) λk,2n+3 +1 τnC (α, β; f, xk,2n+3 +1 ) Kn+3 (◦, xk,2n+3 +1 ) − Kn−1 (◦, xk,2n+3 +1 ) .
Moreover, we have for every f ∈ L2 (µ(α,β) ), c1
Z
n+3
1 2
−1
α
β
f (t) (1 − t) (1 + t) dt ≤
∞ 2X +1 X n=0
≤ c2
Z
λk,2n+3 +1 τnC (α, β; f, xk,2n+3 +1 )2
k=1 1
−1
f (t)2 (1 − t)α (1 + t)β dt.
(c) Let x0 ∈ [−1, 1] and f ∈ X ∞ (µ(α,β) ). Then f has an extension as an analytic function in a complex neighborhood of x0 if and only if there exists a nondegenerate interval I ⊆ [−1, 1] containing x0 such that 1/2n
lim sup kτn (f )kµ(α,β) ;∞,I < 1. n→∞
6
(2.6)
3
Numerical examples
In this section, we illustrate the construction of our localized kernels and their approximation properties using some numerical examples. In our examples below, we use the Chebyshev polynomials; i.e., the polynomials Tn defined by Tn (cos θ) = cos nθ, θ ∈ [0, π], n = 0, 1, · · ·. The polynomials 1, if n = 0, T pn = √ 2Tn , if n = 1, 2, · · ·, are orthonormalized with respect to the measure dµT (x) =
dx dµ(−1/2,−1/2) (x) = , π π(1 − x2 )1/2
x ∈ (−1, 1).
In our numerical computations, we will approximate integrals with respect to dµT by suitable quadrature formulas. We define the de la Vall´ee Poussin kernel by ΦTn (x, y)
:=
n X
pTk (x)pTk (y)
+
k=0
2n X
(2 − k/n)pTk (x)pTk (y).
k=n+1
We define the discrete de la Vall´ee Poussin operator by 8n+1 X (2k − 1)π 1 (2k − 1)π T Vn (f, x) = f cos Φ4n x, cos , 8n + 1 k=1 16n + 2 16n + 2 and note that Vn (f ) ∈ Π8n . With Φ∗T n (x, y)
=
4 − (x − y)2 4
n
ΦT3n (x, y),
we define the exponentially localized operator, denoted here for brevity by σn , by 8n+1 X 1 (2k − 1)π (2k − 1)π ∗T σn (f, x) = f cos Φn x, cos . 8n + 1 k=1 16n + 2 16n + 2 In particular, σn (f ) ∈ Π8n . To illustrate the approximation properties and localization of the operators, we consider two functions, the first of which is fa (x) := |x − 1/4|,
x ∈ [−1, 1].
To define the second function, we recall that for q ≥ 1, the cardinal B-spline of order q is the function defined by (cf. [1, p. 131]) 1, if 0 < x ≤ 1, M1 (x) := 0, otherwise, 1 (xMq−1 (x) + (q − x)Mq−1 (x − 1)), q ≥ 2. Mq (x) := q−1 7
We define fb (x) = M4 (2x + 2). Thus, fb is analytic on (−1, 1), except at ±1/2, 0, where it is twice continuously differentiable. In this section only, let C denote the set of 10, 000 equidistant points on [−1, 1], n (f, V ) := max |f (x) − Vn (f, x)|,
n (f, σ) := max |f (x) − σn (f, x)|,
x∈C
and δn (f, V ) := log2
x∈C
n (f, V ) , 2n (f, V )
δn (f, σ) := log2
n (f, V ) . 2n (f, V )
Table 1 shows the decay of errors n (f, V ) and n (f, σ) for different values of n. n 8 16 32 64
n (fa , σ) 1.8065 ∗ 10−2 9.8889 ∗ 10−3 4.8372 ∗ 10−3 2.3075 ∗ 10−3
n (fa , V ) n (fb , σ) −2 1.3609 ∗ 10 1.3838 ∗ 10−4 −3 8.0887 ∗ 10 1.684 ∗ 10−5 3.8924 ∗ 10−3 2.0823 ∗ 10−6 1.7856 ∗ 10−3 2.5918 ∗ 10−7
n (fb , V ) 4.7691 ∗ 10−5 6.0351 ∗ 10−6 7.5158 ∗ 10−7 9.3814 ∗ 10−8
Table 1: Maximum absolute errors. In light of the direct theorems of approximation theory, the quantities δn (fa , V ) and δn (fa , σ) should be close to 1, and the corresponding quantities for fb should be close to 3. Table 2 confirms this fact (cf. Theorem 2.1(a).) n δn (fa , σ) δn (fa , V ) δn (fb , σ) δn (fb , V ) 8 0.8693 0.7506 3.0387 2.9823 16 1.0316 1.0552 3.0156 3.0054 32 1.0678 1.1243 3.0062 3.0021 Table 2: The smoothness index as predicted by δn ’s. It is clear from Table 1 that the maximum error is less with the de la Vall´ee Poussin operators than the exponentially localized operators. However, Figure 1 shows with example of n = 64 that the later are more localized in the sense that on parts of the interval where the functions are analytic, the error with the exponentially localized operator is substantially smaller than that with the de la Vall´ee Poussin operators. Figure 1 shows the logplot of the errors in approximating fa and fb by the discrete de la Vall´ee Poussin operators and the exponentially localized operators. Thus, on the y axis in all the subfigures below, the value −k corresponds to the value 10−k for the errors plotted in the figures. We note that only 8n + 1 values of the function are used in the computation of the transforms Vn and σn . In particular, in the top left figure in Figure 1, an absolute error of less than 10−20 is obtained away from the singularity, using only 513 samples of the function fa . To illustrate this phenomenon further, we evaluated the errors at 2048 points chosen randomly according to the probability measure µT (again with polynomials of degree 512), and arranged them in an increasing order. Thus, Figure 2 shows the probability 8
-3
-3
-5
-5
-7
-7
-9
-9
-11
-11
-13
-13
-15
-15
-17
-17
-19
-19
-21
-21 -1
0
1
-7
-7
-8
-8
-9
-9
-10
-10
-11
-11
-12
-1
0
1
-1
0
1
-12 -1
0
1
Figure 1: Clockwise, the graphs represent log10 |fa −σ64 (fa )|, log10 |fa −V64 (fa )|, log10 |fb − V64 (fb )|, and log10 |fb − σ64 (fb )|. distribution of the errors, where the lowest 15 terms have been discarded as outliers. In the case of fa , the probability that the error does not exceed 10−5 times the minimum of the two maximum errors is 25.88% with the de la Vall´ee Poussin operators and 79.39% with the exponentially localized operators. In the case of the spline function fb , the corresponding probabilities are 3.22% and 18.46% respectively. When we compared polynomials of degree 1024 in the case of fb , the difference became more dramatic, with the probabilities being 5.96% for the de la Vall´ee Poussin operator, and 40.28% for the exponentially localized operator. Finally, we note that the plots in Figure 1 are produced with symbolic computations in Mathematica to illustrate an error much less than the usual double precision floating point accuracy would allow. On the other hand, we used Matlab to produce Figure 2, where we obtain the same results as with Mathematica, but up to error of 10−16 .
4
Polynomial frames on compact subsets of R
In this section, we state our main results in a very general form. Thus, instead of the Jacobi measure, we will consider an arbitrary measure, supported on an arbitrary compact subset of [−1, 1]. Instead of achieving the discretization of the summability and frame operators using Gauss quadrature formula, we will formulate our “discretization” using general functionals. Let ν be a (possibly signed) measure on R that is either positive and finite, or has a 9
100
100
90
90
80
80
70
70
60
60
50
50
40
40
30
30
20
20
10 0 −16
10
−14
−12
−10
−8
−6
−4
0 −14
−2
−13
−12
−11
−10
−9
−8
−7
−6
Figure 2: On the left, the continuous line represents on the x axis, the increasing rearrangement of the array log10 |fa − V64 (fa )| at 2048 randomly chosen points, and on the y axis, the percentage of samples where the error is below the corresponding value. The dashed line represents the corresponding graph for log10 |fa − σ64 (fa )|. The graphs on the right represent the same quantities for the function fb . bounded variation on R, |ν| denote ν if ν is a positive measure, and its total variation measure if it is a signed measure. We recall that the support of ν, denoted by supp (ν) is the set of all x ∈ R such that |ν|(I) > 0 for every interval I containing x. If A ⊆ R is |ν|-measurable, and f : A → R is |ν|-measurable, we write kf kν;p,A :=
Z
p
|f | d|ν|
1/p
,
|ν| − ess supt∈A |f (t)|,
Z
A
if 1 ≤ p < ∞, if p = ∞.
The class of measurable functions f for which kf kν;p,A < ∞ is denoted by Lp (ν; A), with the standard convention that two functions are considered equal if they are equal |ν|almost everywhere on A. The class of all uniformly continuous, bounded functions on A (equipped with the norm of L∞ (ν)) will be denoted by C(ν; A). The class X p (ν; A) will denote Lp (ν; A) if 1 ≤ p < ∞ and C(ν; A) if p = ∞. In the sequel, we will assume that µ is a fixed, finite, positive, Borel measure with supp (µ) being an infinite subset of [−1, 1]. The mention of the set A will be omitted if A = supp (µ). Thus, for example, we will write X p (µ) = X p (µ; supp (µ)) and C(µ) = C(µ; supp (µ)). It is well known [3, Chapter 1] that there exists a unique system of polynomials {pk (x) = γk xk + · · · , γk > 0}∞ k=0 such that for k, j = 0, 1, · · ·, pk pj dµ =
1, 0,
if k = j, otherwise.
Moreover (cf. [3, Chapter 2]), any function f ∈ L1 (µ) is uniquely determined by the sequence of its coefficients fˆ(k) :=
Z
f pk dµ, 10
k = 0, 1, · · · .
(4.1)
Next, we define the kernels. For x, y ∈ C, we will write n
Kn (x, y) :=
2 X
pk (x)pk (y),
k=0
n = 0, 1, 2, · · · .
(4.2)
and K−1 (x, y) = 0. For an integer n ≥ 0, a function Φn : C × C → R will be called a reproducing summability kernel (of order n), if each of the following four conditions is satisfied. For each x, y ∈ C, Φn (x, y) = Φn (y, x), Φn (x, ◦) ∈ Π2n , Z Φn (x, y)P (y)dµ(y) = P (x), P ∈ Πn , (4.3) and sup
x∈supp (µ)
Z
|Φn (x, y)|dµ(y) ≤ c,
(4.4)
where c is a constant independent of n, depending at most on µ and the whole sequence {Φn }. We assume in the sequel that there exists a sequence {Φn } of reproducing summability kernels. In [11], we have proved that the kernels Φn (µ(α,β) , h, x, y) defined in (2.1) are reproducing summability kernels if α, β ≥ −1/2. In [3, Section IV.3], Freud has shown the strong (C, 1) summability of a very general class of orthogonal polynomials. If h is an integral of a function of bounded variation, h(x) = 1 for 0 ≤ x ≤ 1/2, and h(x) = 0 for x > 1, it can be shown using a summation by parts argument that a kernel similar to the one defined in (2.1) with these orthogonal polynomials satisfies all the properties mentioned above. In the case of the Jacobi measure, we were able to use the special function properties of (α,β) Jacobi polynomials to obtain localization estimates on the kernels Φn (cf. [11]). These techniques cannot be used to obtain localization estimates on the kernels in general, for example, for the orthogonal polynomial systems discussed by Freud. Nevertheless, our simple construction below allows one to construct exponentially localized kernels based only on the summability estimates. In turn, the localization allows one to use the ideas in [11] to obtain a characterization of local Besov spaces on the interval also in the case of these more general systems of orthogonal polynomials. For x, y ∈ C, and n = 0, 1, · · ·, let n 4 − (x − y)2 ∗ Φn (x, y) = Φ3n (x, y). 4 N N We observe PNthat if N ≥ 1 is an integer, {xk }k=1 , {wk }k=1 are Rreal numbers, a sum of the form k=1 wk f (xk ) can be expressed as a Stieltjes integral f dν, where ν is the measure that associates the mass P wk with each point xk . (The total variation measure in this case is given by |ν|(B) = xk ∈B |wk |, B ⊂ R.) We prefer to use the integral notation rather than the more explicit sum notation for a number of reasons. First, the precise locations of the points xk , the values of wk , and sometimes, even the value of N do not play a significant role in our theory. The use of the integral notation avoids the need to prescribe these quantities explicitly, and develop additional notation for these. Second, we wish our theory to be applicable to all Lp spaces. If p < ∞, point evaluations are
11
not well defined for every f ∈ Lp (µ), and we have to use some other local measurements, for example, averages over small subintervals around certain points. Again, the details of exactly what these points and the corresponding subintervals are, and even the nature of the local measurements do not play any significant role in our theory. The integral notation allows us to treat both the case of continuous functions and elements of Lp in a unified manner. If ν is a Borel, finite, positive or signed measure (with bounded variation), and f ∈ 1 L (ν), we define the operators Z σn (ν; f, x) := Φ∗n (x, y)f (y)dν(y), x ∈ C, n = 0, 1, · · · . With ν = µ(α,β) and Φn (µ(α,β) , h, x, y) in place of Φn (x, y), σn (ν; f ) reduces to σnC (α, β; h, f ). We observe that Φ∗n being a symmetric polynomial in x and y, has an expansion of the form 8n X 8n X ∗ Φn (x, y) = an;j,k pj (x)pk (y), j=0 k=0
where (an;j,k ) is a symmetric matrix. Therefore, taking ν to be the measure µ, we see that ! 8n 8n X X an;j,k fˆ(k) pj σn (µ; f ) = j=0
k=0
is a polynomial with coefficients given as a finite linear combination of the coefficients {fˆ(k)}8n k=0 . The more general definition allows us to compute these operators using, for example, values of f . If {νn } is a sequence of finite positive or signed Borel measures having a bounded variation on [−1, 1], we define for z ∈ C σ1 (ν0 ; f, z), if n = 0, τn (νn ; f, z) = (4.5) σ2n (νn ; f, z) − σ2n−1 (νn−1 ; f, z), if n ≥ 1. Clearly, the operator τn depends upon two measures: νn and νn−1 . Although we have to mention the measure to distinguish between the general case and the continuous case, when each νn = µ, we prefer to keep the notation simpler rather than using the more cumbersome notation τn (νn , νn−1 ; f, x). In the Jacobi case, we choose Φn (µ(α,β) ; h, x, y) in place of Φn (x, y). Choosing each νn to be µ(α,β) we obtain τn (µ(α,β) ; f, x) = τnC (α, β; h, f, x). We obtain τnD (α, β; h, f, x) by choosing νn to be the measure that associates the mass λk,2n+3 +1 with each xk,2n+3 +1 , k = 1, · · · , 2n+3 + 1. We now formulate certain assumptions on our measures. Definition 4.1 A sequence {νn } will be called an M–Z (Marcinkiewicz–Zygmund) sequence if each of the following conditions is satisfied. 1. Each νn is a Borel, finite, positive or signed measure having bounded variation on [−1, 1]. 12
2. kT kνn ;p ≤ ckT kµ;p , 3.
Z
T1 T2 dνn =
Z
T ∈ Π2n+4 , p = 1, ∞.
(4.6)
T1 , T2 ∈ Π2n+3 .
(4.7)
T1 T2 dµ,
If 1 ≤ p ≤ ∞, a sequence {νn } will be called p-compatible if each µ–measurable function is also νn –measurable for each n, and kf kνn ;p ≤ ckf kµ;p for every f ∈ X p (µ). In the case of Jacobi polynomials, it is proved by Lubinsky, M´at´e, and Nevai [9, Theorem 5] that the measures νn that associate the mass λk,2n+3 +1 with each xk,2n+3 +1 , k = 1, · · · , 2n+3 + 1, form an ∞–compatible M–Z sequence. In general, it is natural to construct measures to satisfy (4.7) using Gauss quadrature formulas based on the zeros of a sufficiently high degree orthogonal polynomial pN . However, if supp (µ) is not an interval, then the zeros of the corresponding orthogonal polynomials might not be all in supp (µ), in which case, such a measure would not be ∞–compatible. We will prove the following proposition to demonstrate the existence of ∞–compatible, M–Z sequences of measures supported on finite subsets of supp (µ). Proposition 4.1 Let µ({x}) = 0 for every x ∈ [−1, 1]. Then there exists an ∞– compatible M–Z sequence {νn } of measures such that each of the sets supp (νn ) is a finite subset of supp (µ). Our first theorem, generalizing part (b) of Theorem 2.1, shows a representation of any function in X p , 1 ≤ p ≤ ∞, in terms of the operators and kernels introduced so far. The theorem uses two sequences of measures. The sequence {νn } is determined by the kind of information we have regarding the target function f . Thus, if one starts with the coefficients {fˆ(k)}, then each of the measures νn = µ. On the other hand, if a set of values of the form {f (xk,Nn )} are available at a system of points, we should choose νn to be an M–Z measure supported at the points {xk,Nn : k = 1, · · · , Nn }, if such a measure can be found. The choice of the sequence {µn } is required only to satisfy (4.7), and may be used judiciously to obtain a parsimoneous representation, or a representation with other desirable properties depending upon the application. Theorem 4.1 Let 1 ≤ p ≤ ∞, {νn } be a p-compatible M–Z sequence, {µn } be an sequence of measures satisfying (4.7), and f ∈ X p (µ). We have f=
∞ X
τn (νn ; f ),
(4.8)
n=0
where the convergence of the series is in the norm of X p (µ). In the case when each νn = µ, we have further f=
∞ Z X n=0
τn (µ; f, y) {Kn+3 (◦, y) − Kn−1 (◦, y)} dµn (y). 13
(4.9)
Moreover, for f ∈ L2 (µ), we have the frame property: c1 kf k2µ;2 ≤
∞ X n=0
kτn (µ; f )k2µ;2 =
∞ X n=0
kτn (µ; f )k2µn ;2 ≤ c2 kf k2µ;2 .
(4.10)
Next, we describe our main theorem of this paper, generalizing part (c) of Theorem 2.1. The proof of this theorem relies upon certain potential theoretical notions. We review these briefly, based on the discussion in [19, Chapter 2.4]. The logarithmic energy of a positive measure ν on C is defined by Z Z 1 E(ν) := log dν(x)dν(y), |x − y| whenever the integral is well defined. If A ⊆ C is a compact set, the capacity of A, cap (A) is defined by log(1/cap (A)) = inf E(ν),
where the infimum is taken over all unit, positive, Borel measures ν, with supp (ν) ⊆ A. The set A is called regular if there exists a function GA , called the Green’s function for C \ A with pole at ∞, with the following properties: (i) GA is continuous and nonnegative on C and harmonic on C \ A, (ii) lim (GA (z) − log |z|) = log(1/cap (A)),
|z|→∞
and (iii) lim GA (z) = 0,
z→x
x ∈ A.
For example, if A = [a, b], one has cap ([a, b]) = (b − a)/4, and p |2z − a − b + 2 (z − a)(z − b)| G[a,b] (z) = log , z ∈ C \ [a, b]. b−a Theorem 4.2 Let E(µ) < ∞, supp (µ) be a regular set, f ∈ C(µ), and {νn } be an ∞-compatible M–Z sequence of measures. If x0 ∈ supp (µ), then f has an analytic continuation to a complex neighborhood of x0 if and only if there exists a non-degenerate interval I with x0 ∈ I such that 1/2n
lim sup kτn (νn ; f )kµ;∞,I < 1.
(4.11)
n→∞
We note that E(µ) ≥ log(1/2)(µ([−1, 1]))2 . The condition E(µ) < ∞ implies, in particular, that µ({x}) = 0 for each x ∈ [−1, 1]. Thus, in view of Proposition 4.1, the measures {νn } as required in Theorem 4.2 always exist. We observe again that the operators τn (νn ; f ) are defined using global information about f ; the coefficients {fˆ(k)} in the case when each νn is equal to µ. Nevertheless, the exponential localization of these operators enables us to obtain the characterization of local analyticity of the function. Similar characterizations of local Besov spaces can also be obtained, using the ideas in [11]. 14
We end this section by describing a connection of our results with those of Gaier in [4]. If 1 ≤ p ≤ ∞, x ≥ 0, and f ∈ Lp (µ), we define Ex,p (f ) := Ex,p (µ; f ) := min kf − P kµ;p . P ∈Πx
Gaier constructed a sequence of linear operators Gn such that for each continuous f : [−1, 1] → R, and integer n ≥ 1, Gn (f ) ∈ Πn , and satisfies the following conditions: max |f (x) − Gn (f, x)| ≤ M (f )e−αn + c1 En/6,∞ (0, 0; f ),
x∈[−1,1]
(4.12)
and if f is analytic in the complex neighborhood |z − x0 | ≤ d of a point x0 ∈ [−1, 1], then |f (x0 ) − Gn (f, x0 )| ≤ M (f )d−4 exp(−cd2 n), where M (f ) is a positive constant depending only on f , and c1 , c, α are absolute positive constants. Gaier’s construction is based on the Fourier-Chebyshev coefficients of f . As a corollary of Theorem 4.2, we are able to drop the extra term M (f )e−αn in (4.12) at the expense of a higher estimate c1 En/8,∞ (0, 0; f ) in place of c1 En/6,∞ (0, 0; f ). Our construction can be based either on the coefficients in more general orthogonal polynomial expansions, or based on values of the function. Finally, we think that our proofs are simpler than those given by Gaier in [4]. Corollary 4.1 Let E(µ) < ∞, supp (µ) be a regular set, f ∈ C(µ), {νn } be an ∞compatible M–Z sequence of measures, x0 ∈ supp (µ), and f have an analytic continuation to a complex neighborhood of x0 . Then kf − σ2n (νn ; f )kµ;∞ ≤ cE2n ,∞ (f ), and there exists a nondegenerate interval J and ρ ∈ (0, 1) (depending upon x 0 and f ), such that x0 ∈ J and n kf − σ2n (νn ; f )kµ;∞,J ≤ c(f, x0 )ρ2 .
5
Frames on the sphere
The constructions described for the unit interval can be adapted easily for the unit sphere of a Euclidean space. We now sketch these adaptations, and indicate the differences. Let q ≥ 1 be an integer, S := {(x1 , · · · , xq+1 ) ∈ R q
q+1
:
q+1 X
x2j = 1}.
j=1
A spherical cap, centered at x0 ∈ Sq and radius α is defined by Sqα (x0 ) := {x ∈ Sq : x · x0 ≥ cos α}. 15
We note that for any x0 ∈ Sq , Sqπ (x0 ) = Sq . The surface area (volume element) measure on Sq will be denoted by µ∗q , and we write µ∗q (Sq ) =: ωq . For any Borel measure ν which is either positive and finite, or signed (and hence, of bounded variation), and a Borel measurable subset A ⊂ Sq , the spaces Lp (ν; A) := Lp (|ν|; A) are defined as usual. We will omit the mention of the measure ν if it is µ∗q . The spaces X p (Sq ) on the sphere are also defined analogously to the case of the interval. A spherical polynomial of degree m is the restriction to Sq of a polynomial in q + 1 real variables with total degree m. For x ≥ 0, the class of all spherical polynomials of degree at most x will be denoted by Πqx . For integer ` ≥ 0, the class of all homogeneous, harmonic, spherical polynomials of degree ` will be denoted by Hq` , and its dimension by dq` . For each integer ` ≥ 0, let {Y`,k : k = 1, · · · , dq` } be a µ∗q -orthonormalized basis for Hq` . It is known (cf. [20, 17]) that for any integer n ≥ 0, {Y`,k : ` = 0, · · · , n, k = 1, · · · , dq` } is an orthonormal basis for Πqn . The connection with the theory of orthogonal polynomials on [−1, 1] is the following addition formula (cf. [17], where the notation is different): q
d` X
(q/2−1,q/2−1)
−1 Y`,k (x)Y`,k (y) = ωq−1 p`
(q/2−1,q/2−1)
(1)p`
k=1
(x · y),
` = 0, 1, · · · .
It is proved in [13] that if n ≥ 1 is an integer, C is a finite set of points on Sq such that maxq min dist (x, y) ≤ c2−n , x∈S
y∈C
for a judiciously chosen constant c > 0, then there exist nonnegative weights wx , x ∈ C such that X P ∈ Πq2n+4 , wx |P (x)| ≤ c1 kP kµ∗q ;1,Sq , x∈C
and
X
wx P1 (x)P2 (x) =
x∈C
Z
Sq
P1 P2 dµ∗q ,
P1 , P2 ∈ Πq2n+3 .
Thus, we may define an M–Z sequence of measures νn on the sphere by taking a sequence of suitably dense sets Cn of points, with each νn associating the mass wx with x ∈ Cn as described above. Let h : [0, ∞) → R be a q times iterated integral of a function of bounded variation, h(x) = 1 if x ∈ [0, 1/2], and h(x) = 0 if x ∈ [1, ∞), then it is proved in [12] that the kernels defined by ΦSn (q; h, x, y)
:=
−1 ωq−1
n X
(q/2−1,q/2−1)
p`
`=0
(q/2−1,q/2−1)
(1)p`
(x · y)
are reproducing summability kernels of order n on the sphere, defined analogously to such kernels on the unit interval. For n = 0, 1, · · ·, x, y ∈ Sq , let n 1+x·y ∗ ΦS3n (q; x, y). Φn (x, y) = 2 16
The operators τn are defined using Φ∗n and measures and functions on the sphere, exactly as in the case of the interval. The analogue of the kernels Kn in (4.2) is n
Kn (t) :=
−1 ωq−1
2 X
(q/2−1,q/2−1)
p`
(q/2−1,q/2−1)
(1)p`
(t).
`=0
The following theorem is the analogue of part (b) of Theorem 2.1, and can be proved in exactly the same way. Theorem 5.1 Let 1 ≤ p ≤ ∞, {νn } be a p-compatible M–Z sequence on Sq , {µn } be an M–Z sequence on Sq , and f ∈ X p (Sq ). We have ∞ ∞ Z X X f= τn (νn ; f ) = τn (µ∗q ; f, y) {Kn+3 (◦ · y) − Kn−1 (◦ · y)) dµn (y). (5.1) n=0
If f ∈ L2 (Sq ), then
c1 kf k2µ∗q ;2 ≤
n=0
∞ X n=0
Sq
kτn (µ∗q ; f )k2µ∗q ;2 =
∞ X n=0
kτn (µ∗q ; f )k2µn ;2 ≤ c2 kf k2µ∗q ;2 .
Since Sq is not a set of uniqueness of functions analytic on Cq+1 , we do not see how to define the analogue of analytic continuation of a function on Sq . Nevertheless, one can define an analogue from the approximation theory point of view as follows. Let x0 ∈ Sq , K be a spherical cap centered at x0 and f : K → C. We write En (K, f ) := inf q kf − P kµ∗q ;∞,K . P ∈Πn
The class Aq (x0 ) is defined to be the class of all functions f ∈ C(Sq ) such that for some spherical cap K centered at x0 , lim sup En (K, f )1/n < 1. n→∞
The analogue of part (c) of Theorem 2.1 is the following. Theorem 5.2 Let f ∈ X ∞ (Sq ), and {νn } be an M–Z sequence of measures on Sq . If x0 ∈ Sq , then f ∈ Aq (x0 ) if and only if there exists a non-degenerate spherical cap K ⊆ Sq with center at x0 such that 1/2n
lim sup kτn (νn ; f )kµ∗q ;∞,K < 1.
(5.2)
n→∞
In particular, if f ∈ Aq (x0 ), then kf − σ2n (νn ; f )kµ∗q ;∞,Sq ≤ cE2n (Sq , f ), and there exists a nondegenerate spherical cap K ⊆ Sq with center at x0 and ρ ∈ (0, 1) (depending upon x0 and f ) such that n
kf − σ2n (νn ; f )kµ∗q ;∞,K ≤ c1 (f, x0 )ρ2 .
17
6
Proofs
Proof of Proposition 4.1. This proof follows the ideas in [13]. Without loss of generality, we assume that µ([−1, 1]) = 1. Let n ≥ 1 be an integer. Since Πn is a finite dimensional space, there exists a constant, to be denoted in this proof only by Bn such that Z 1 Z 0 |P (t)|dt ≤ Bn |P (t)|dµ(t), P ∈ Πn . (6.1) −1
Our assumption that µ({x}) = 0 for each x ∈ [−1, 1] implies that the function x 7→ µ([−1, x)) is a continuous, nondecreasing function on [−1, 1], with the range of this function being [0, 1]. Therefore, there exist intervals Ik with mutually disjoint interiors such that [−1, 1] = ∪Ik , and µ(Ik ) ≤ 1/(4Bn ) for each Ik . In this proof only, let I be the set of integers k such that Ik ∩ supp (µ) is not empty, and we choose a point xk ∈ Ik ∩ supp (µ) for each k ∈ I. In view of (6.1), we have for any P ∈ Πn , Z X Z X µ(Ik )|P (xk )| = |P (xk )|dµ(t) |P (t)|dµ(t) − kP kµ;1 − Ik Ik k∈I k∈I Z X ≤ |P (t) − P (xk )|dµ(t) Ik
k∈I XZ Z
≤
Therefore, (3/4)kP kµ;1 ≤
Ik
k∈I
Z
1
Ik
|P 0 (u)|dudµ(t)
≤
1 4Bn
X
µ(Ik )|P (xk )| ≤ (5/4)kP kµ;1 .
k∈I
−1
|P 0 (u)|du ≤ (1/4)kP kµ;1 . (6.2)
In this proof only, let N be the number of elements in I, T denote the linear operator defined by T (P ) = (P (xk ))k∈I , and V be the range of T . The estimates (6.2) imply that T is invertible on RV . In this proof only, let x∗ denote the linear functional defined on P V by x∗ (T (P )) = P dµ. We equip RN by the norm k|(rk )k∈I |k = k∈I µ(Ik )|rk |. The estimates (6.2) imply that the dual norm of x∗ with respect to this norm is bounded from above by 4/3. The Hahn–Banach theorem, together with the characterization of the N dual existence of wk ∈ R, k ∈ I, such that the functional (rk )k∈I 7→ P of R , implies the ∗ k∈I wk rk extends x , and has the dual norm bounded from above by 4/3. In this proof only, let µn be the measure that associates the mass wk with xk , k ∈ I. It is easy to check that the total variation measure of µn associates the mass |wk | with each xk , k ∈ I. The statement about the extension means that Z Z P dµn = P dµ, P ∈ Πn . (6.3) The statement about the dual norm means that |wk | ≤ (4/3)µ(Ik ) for k ∈ I. Therefore, the estimates (6.2) imply that X X kP kµn ;1 = |wk ||P (xk )| ≤ (4/3) µ(Ik )|P (xk )| ≤ (5/3)kP kµ;1 , P ∈ Πn . k∈I
k∈I
18
We note that each µn is supported on a finite subset of supp (µ). Therefore, each µn is trivially ∞–compatible. We have thus shown that the sequence {νn = µ2n+4 } is an ∞-compatible M–Z sequence of measures, with each νn supported on a finite subset of supp (µ). 2 Lemma 6.1 Let 1 ≤ p ≤ ∞ and {νn } be a p-compatible M–Z sequence of measures. Then σ2n (νn ; P ) = P for all P ∈ Π2n . We have f ∈ Lp (νn ), 1 ≤ p ≤ ∞.
kσ2n (νn , f )kµ;p ≤ ckf kνn ;p ,
(6.4)
Further, for each f ∈ X p (µ), E2n+3 ,p (f ) ≤ kf − σ2n (νn ; f )kµ;p ≤ cE2n ,p (f ). Proof. Let P ∈ Π2n , x ∈ [−1, 1], and Q be defined by 2n 4 − (x − y)2 , Q(y) = P (y) 4
(6.5)
y ∈ R.
Then Q ∈ Π3(2n ) . Consequently, (4.7) and (4.3) imply that 2n Z 4 − (x − y)2 σ2n (νn ; P, x) = P (y) Φ3(2n ) (x, y)dνn (y) 4 Z Z = Q(y)Φ3(2n ) (x, y)dνn (y) = Q(y)Φ3(2n ) (x, y)dµ(y) = Q(x) = P (x).
Since |4 − (x − y)2 | ≤ 4 for all x, y ∈ [−1, 1], the conditions (4.6) and (4.4) imply that Z Z ∗ sup sup |Φ2n (x, y)|d|νn|(y) ≤ sup sup |Φ∗2n (x, y)|dµ(y) ≤ c. (6.6) n≥0 x∈supp (µ) n≥0 x∈supp (µ) Therefore, for any f ∈ L∞ (νn ) and x ∈ supp (µ), we have Z |σ2n (νn ; f, x)| ≤ |Φ∗2n (x, y)||f (y)|d|νn|(y) ≤ ckf kνn ;∞ .
(6.7)
Thus, (6.4) is satisfied if p = ∞. If f ∈ L1 (νn ) and g ∈ L∞ (µ), we verify using Fubini’s theorem that Z Z σ2n (νn ; f, x)g(x)dµ(x) = f (y)σ2n (µ; g, y)dνn(y).
Since σ2n (µ; g) ∈ Π2n+3 , the condition (4.6) implies that kσ2n (µ; g)kνn ;∞ ≤ ckσ2n (µ; g)kµ;∞. Therefore, using (6.7) with µ in place of νn , we obtain that for every f ∈ L1 (νn ) and g ∈ L∞ (µ), Z Z σ2n (νn ; f, x)g(x)dµ(x) ≤ |f (y)σ2n (µ; g, y)|d|νn|(y) ≤ kσ2n (µ; g)kνn ;∞ kf kνn ;1 ≤ ckσ2n (µ; g)kµ;∞ kf kνn ;1 ≤ ckgkµ;∞ kf kνn ;1 . 19
Therefore, the Hahn-Banach theorem implies that kσn (νn , f )kµ;1 ≤ ckf kνn ;1 for every f ∈ L1 (νn ). Thus, we have proved (6.4) for p = 1, ∞. An application of the Riesz–Thorin interpolation theorem now yields (6.4) for 1 < p < ∞. Consequently, for any P ∈ Π2n , E2n+3 ,p (f ) ≤ kf − σ2n (νn ; f )kµ;p = kf − P − σ2n (νn ; f − P )kµ;p ≤ ckf − P kµ;p . Since P is arbitrary, this implies (6.5).
2
Proof of Theorem 4.1. We note that f ∈ X p implies that En,p (f ) → 0 as n → ∞. The equation (4.8) follows from (4.5) and (6.5). Next, let each νn = µ. Since τn (µ;f ) ∈ Π2n+3 , we verify easily that Z τn (µ; f, x) = τn (µ; f, y)Kn+3 (x, y)dµ(y). (6.8) Further, since Kn−1 (x, y), as a function of y, is in Π2n−1 , τn (µ; Kn−1 (x, ◦)) = 0. Therefore, Z Z τn (µ; f, y)Kn−1 (x, y)dµ(y) = f (z)τn (µ; Kn−1 (x, ◦), z)dµ(z) = 0, and (6.8) may be rewritten in the form Z τn (µ; f, x) = τn (µ; f, y) (Kn+3 (x, y) − Kn−1 (x, y)) dµ(y). Since µn satisfies the quadrature formula (4.7), this implies (4.9). In the remainder of this proof only, let Z Pm (f, x) := f (y) (Km (x, y) − Km−1 (x, y)) dµ(y), x ∈ R, m = 0, 1, · · · . We note that the Parseval identity implies that f=
∞ X
m=0
Pm (f ),
kf k2µ;2
=
∞ X
m=0
kPm (f )k2µ;2 ,
where the convergence of the first series is in the sense of L2 (µ). Next, we observe that Pm (P ) = 0 if P ∈ Π2m−1 . If n ≥ 0 is an integer, and n + 4 ≤ m, then τn (µ; f ) ∈ Π2n+3 ⊆ Π2m−1 , and Pm (τn (µ; f )) = 0. Similarly, if n − 1 ≥ m, then for each x ∈ R, Km (x, ◦) − Km−1 (x, ◦) ∈ Π2m ⊆ Π2n−1 , and hence, for each x, t ∈ R, Z (Φ∗2n (y, t) − Φ∗2n−1 (y, t)) (Km (x, y) − Km−1 (x, y)) dµ(y) = 0. Therefore, if n − 1 ≥ m, then Pm (τn (µ; f )) Z Z ∗ ∗ = f (t) (Φ2n (y, t) − Φ2n−1 (y, t)) dµ(t) (Km (x, y) − Km−1 (x, y)) dµ(y) Z Z = f (t) (Φ∗2n (y, t) − Φ∗2n−1 (y, t)) (Km (x, y) − Km−1 (x, y)) dµ(y)dµ(t) = 0. 20
Hence, (4.8) implies that for any integer m ≥ 0, kPm (f )k2µ,2
2
2
∞ m
X
X
= Pm (τn (µ; f )) = Pm (τn (µ; f ))
n=0
n=max(0,m−3)
µ;2 µ;2 2 m X ≤ kPm (τn (µ; f ))kµ;2 n=max(0,m−3)
≤ 4
≤ 4
m X
n=max(0,m−3) m X
n=max(0,m−3)
kPm (τn (µ; f ))k2µ;2 kτn (µ; f )k2µ;2 .
This implies kf k2µ;2
=
∞ X
m=0
kPm (f )k2µ,2
∞ X
≤4
m X
m=0 n=max(0,m−3)
kτn (µ; f )k2µ;2
≤ 16
∞ X n=0
kτn (µ; f )k2µ;2 .
The proof of the second inequality in (4.10) is similar. Thus, arguing as before, we see that τn (µ; Pm (f )) = 0 except when n ≤ m ≤ n + 3. So, for any integer n ≥ 0,
2
2 n+3 ∞
X
X
2 τn (µ; Pm (f )) kτn (µ; f )kµ;2 = τn (µ; Pm (f )) =
m=n m=0 µ;2 µ;2 ! 2 n+3 n+3 X X ≤ kτn (µ; Pm (f ))kµ;2 ≤ 4 kτn (µ; Pm (f ))k2µ;2 ≤ c Consequently,
∞ X n=0
m=n n+3 X
m=n
m=n
kPm (f )k2µ;2 .
kτn (µ; f )k2µ;2 ≤ c
∞ X
m=0
kPm (f )k2µ;2 = ckf k2µ;2 . 2
The proof of Theorem 4.2 depends upon the following well known Bernstein–Walsh inequality (cf. [19, Estimate (2.4), p. 153]). Lemma 6.2 Let A ⊆ [−1, 1] be a regular set, m ≥ 0 be an integer, P ∈ Πm . Then for any z ∈ C, |P (z)| ≤ exp(mGA (z)) max |P (x)|. (6.9) x∈A
We need also the following lemma summarizing some of the technical details required in the proof. 21
Lemma 6.3 Let f ∈ C(µ), x0 ∈ supp (µ), ` > 0, I = [x0 − `, x0 + `] ∩ supp (µ), J = [x0 − 2`, x0 + 2`] ∩ [−1, 1], and {νn } be an M–Z sequence of measures. Then for every integer n ≥ 0 and x ∈ I, |σ2n (νn ; f, x)| ≤ ckf kνn ;∞,J + c1
4 − `2 4
2n
kf kνn ;∞ .
(6.10)
Proof. Let x ∈ I. If y ∈ [−1, 1] \ J, then 4 − `2 4 − (x − y)2 ≤ , 4 4 and hence, |Φ∗2n (x, y)|
≤
4 − `2 4
2n
|Φ3(2n ) (x, y)|.
Therefore, (4.6) and (4.4) imply that Z
[−1,1]\J
Φ∗2n (x, y)f (y)dνn(y)
2n Z 4 − `2 kf kνn ;∞ |Φ3(2n ) (x, y)|d|νn|(y) ≤ 4 2n 4 − `2 ≤ c1 kf kνn ;∞ . 4
The estimate (6.6) implies that Z Z ∗ Φ n (x, y)f (y)dνn(y) ≤ kf kνn ;∞,J |Φ∗n (x, y)|d|νn|(y) ≤ ckf kνn ;∞,J . 2 2 J
Together with the definition of σ2n (νn ; f, x), we are thus led to (6.10).
2
Proof of Theorem 4.2. Let 2 > ` > 0, (4.11) be satisfied for J = [x0 − 2`, x0 + 2`] ∩ [−1, 1] in place of I, and 0 < ρ1 < 1 and integer N be chosen so that for all integer n ≥ N , 1/2n
kτn (νn ; f )kµ;∞,J < ρ1
.
Since τn (νn ; f ) ∈ Π2n+3 and kτn (νn ; f )kµ;∞ ≤ ckf kµ,∞ , we see from Lemma 6.3 applied with µ in place of νn that for every x ∈ I := [x0 − `, x0 + `] ∩ supp (µ), n
n
|τn (νn ; f, x)| = |σ2n+3 (µ; τn (νn ; f ), x)| ≤ cρ21 + c1 (1 − `2 /4)2 kf kµ,∞ .
(6.11)
In the remainder of the proof, constants denoted by c, c1 , · · · may depend upon x0 , `, and f . Since x0 ∈ supp (µ), µ(I) > 0, and hence, it is easy to see that the restriction of µ to I has a finite logarithmic energy. Therefore, cap (I) > 0. In view of a result of Wiener (cf. [19, Theorem 1.1, Appendix A]), I is a regular set. Letting ρ := max(ρ1 , (1 − `2 /4)), we obtain from (6.9) and (6.11) that for every z ∈ C, and integer n ≥ N , n
|τn (νn ; f, z)| ≤ c (ρ exp(GI (z)))2 . 22
P We observe that 0 < ρ < 1. Therefore, the series ∞ n=0 τn (νn ; f, z) converges uniformly and absolutely on compact subsets of the region {z ∈ C : |GI (z)| < log(1/ρ)}. Since I is regular, this is an open neighborhood of I. In view of (4.8), the sum of this series is an analytic function that coincides with f on I. Conversely, let ` > 0, and f be analytic on the complex disc of radius 4`, centered at x0 . Without loss of generality, we may assume that ` ≤ 1. Let J = [x0 − 2`, x0 + 2`] ∩ [−1, 1]. In view of Bernstein’s theorem (cf. [22, Theorem 5, p. 75]), there exists a sequence of polynomials {Pm ∈ Πm } such that |f (y) − Pm (y)| ≤ c1 e−cm ,
y ∈ J.
(6.12)
We note that supp (µ) does not have any isolated points. Therefore, the continuity of f implies that for any µ-measurable set A ⊆ supp (µ) with µ(A) > 0, sup y∈A |f (y)| = kf kµ;∞,A . The Bernstein–Walsh inequality (6.9), applied with J in place of A, implies that there exists a constant a, depending upon f , x0 , `, such that |f (y) − Pm (y)| ≤ kf kµ;∞ + |Pm (y)| ≤ c1 eam ,
y ∈ supp (µ).
In particular, since νn is ∞-compatible, kf − Pm kνn ;∞ ≤ c1 eam .
(6.13)
Let b := log(4/(4 − `2 )), and for integer n ≥ 1, mn the integer part of min(1/2, (b/4a))2n. Let I = [x0 − `, x0 + `] ∩ supp (µ) and x ∈ I. We note that µ(I) > 0. Using (6.10), we conclude from (6.12) and (6.13) that |σ2n (νn ; f − Pmn , x)| ≤ c1 (e−cmn + exp(−b2n + amn )) ≤ c1 exp(−c2n ). Similarly, |σ2n−1 (νn−1 ; f − Pmn , x)| ≤ c1 (e−cmn + exp(−b2n−1 + amn )) ≤ c1 exp(−c2n ). Therefore, recalling that Pmn ∈ Π2n−1 , we conclude that |τn (νn ; f, x)| = |τn (νn ; f − Pmn , x)| = |σ2n (νn ; f − Pmn , x) − σ2n−1 (νn−1 ; f − Pmn , x)| ≤ c1 exp(−c2n ). Since x ∈ I was arbitrary, this implies (4.11).
2
Proof of Corollary 4.1. This corollary follows immediately from Theorem 4.2 and (4.8). 2 Proof of Theorem 2.1. It is proved by Lubinsky, M´at´e, and Nevai [9, Theorem 5] that the measures νn that associate the mass λk,2n+3 +1 with each xk,2n+3 +1 , k = 1, · · · , 2n+3 + 1, form an ∞–compatible M–Z sequence. Further, it is proved in [11] that the kernel Φn (µ(α,β) ; h, x, y) is a reproducing kernel of order n. Therefore, part (a) follows from 23
Lemma 6.1, and part (b) follows from Theorem 4.1. We recall (cf. [19, Chapter I, Example 3.5]) that Z Z 2 log dµ(−1/2,−1/2) (x)dµ(−1/2,−1/2) (t) = π 2 log 4. |x − t| It follows that if α, β ≥ −1/2, then Z Z 2 log dµ(α,β) (x)dµ(α,β) (t) |x − t| α+β+1 2 (α + 1/2)α+1/2 (β + 1/2)β+1/2 ≤ α+β+1 Z Z 2 × log dµ(−1/2,−1/2) (x)dµ(−1/2,−1/2) (t) |x − t| α+β+1 2 2 = π (log 4) (α + 1/2)α+1/2 (β + 1/2)β+1/2 . α+β+1 Thus, µ(α,β) has finite logarithmic energy. Since [−1, 1] is a regular set, part (c) follows from Theorem 4.2. 2 We now turn our attention to the proofs of the results in Section 5. Theorem 5.1 is proved exactly as Theorem 4.1. There are no new ideas involved, and we omit the proof. In order to prove Theorem 5.2, we recall an analogue of the Bernstein–Walsh inequality from [10, Estimate (22)]: For m = 0, 1, · · ·, 0 < α < β ≤ π, maxq |P (x)| ≤ x∈Sβ
π(2β − α) 2m α
maxq |P (x)|, x∈Sα
P ∈ Πqm .
(6.14)
In [10], it was assumed that β < π. However, the same estimate holds clearly for β = π because of continuity. Proof of Theorem 5.2. The fact that (5.2) implies that f ∈ Aq (x0 ) is clear from the first equation in (5.1) and the definition of Aq (x0 ). The converse is proved exactly as in the proof of the corresponding statement in Theorem 4.2, using (6.14) in place of the Bernstein–Walsh inequality. The proof of the remainder of this theorem is exactly the same as that of Corollary 4.1, except that we use the first part of this theorem in place of Theorem 4.2 and (5.1) in place of (4.8). 2
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