Pn(z;µ, q) a degree polynomial having minimal Lq(µ)-norm pn(z;µ, q) the polynomialPn(z;µ, q) divided by its Lq(µ)-norm Pch(µ) convex hull polynomial of mass support μ. We will be interested in describing the asymptotic behavior of the polynomials {pn(z;µ)}n≥0 and{Pn(z;µ)}n≥0 as it tends to infinity.
Potential Theory
In the example above, we relied heavily on the fact that the equilibrium measure of a compact set is always supported on the boundary of the set. Several conditions closely related to the solvability of the Dirichlet problem are discussed in Section III.6 of [14] and Section I.4 of [50].
Conformal Maps
Let K be a compact set whose limit is a Jordan curve and let φ be a conformal mapping as described above. Similarly, we can take the measure νon ∂D and push it onto ∂Ω using ψ and get the measure ψ∗ν.
Chebyshev and Faber Polynomials
As in section 1.1, φ will denote the conformal mapping from the region Ω to the complement of the closed unit disk in the extended plane. Therefore – as in the case of the unit disc – Faber polynomials and Chebyshev polynomials are the same for the interval [−2,2].
The Szeg˝ o Function
For anyq∈(0,∞), we say that a function f is in the Hardy spaceHq(D) if it is analytic in the unit disk and. The Szeg˝o function can also be displayed in the context of targets supported on curves other than the unit circle.
The Keldysh Lemma
Since this reasoning can be applied to any subset of the functions{fn}n∈N, we conclude fn→1 in Hq(D) as claimed. Since the same reasoning can be applied to any subset of the functions{fn}n∈N, we get the desired conclusion.
Regularity
Our use of the expression 'locally uniform' is the same as in [65], i.e. for every z∈Cand series{zn}n∈N that converges toz, we have that. Consistent with the above observations, all these conditions will imply some degree of density near the outer limit of the measure's support. One of the weakest such conditions is known as condition Λ∗ and is satisfied exactly when.
As mentioned in section 1.1, it is a necessary and sufficient condition for root asymptotics of the orthonormal polynomials.
Szeg˝ o’s Theorem
It also has important consequences for the location of the zeros of orthonormal polynomials. Therefore (2.3.2) implies that this integral can be bounded from below by the value of the integrand at infinity. Returning to the proof of the upper bound, we recall that since Γ is an analytic Jordan curve, we know Fn(z)−φ(z)n → 0 uniformly for z in the closure of the unbounded component C\Γ.
Note that in deriving the lower bound in Theorem 2.3.1 we used only the absolutely continuous part of the measure µ (with respect to the arc length measure).
Bergman Polynomials
The leading order behavior of the polynomials is S(z)φ(z)n for an explicitly computable function S that is independent of the singular component of the measure µ. The simplest and most natural such setting is when the target is an area target of an area of the aircraft. Another key step is an analysis of the generalized Faber polynomial Bn(z, qm), which is the polynomial of g(z, qm)φ(z)n.
For each fixr, the angular integral is the integral of the absolute value of a function H2 around it.
Christoffel Functions
We will generalize it in the sense that we will consider a more general class of measures, but we will pay the price of replacing the error term in the expression for pn(z;µ) simply by o(1). Conversely, if we know something about the behavior of the Christoffel function, then we know something about the behavior of orthonormal polynomials. Therefore, the study of the properties of the Christoffl function is a natural problem related to the general study of orthogonal polynomials and will be studied in more detail in Chapter 3.
The main step in the proof will be to derive a uniformity quantity in the harmonic measures for a sequence of interconnected domains whose union carries the measure µ.
The Bergman Kernel Method
Note that the Bieberbach polynomials are not specific to the regionG, but are determined by the region and the fixed point z0∈G. Bieberbach polynomials provide an explicit application of Bergman polynomials to an important problem in approximation theory, namely the numerical approximation of the conformal mapping of an arbitrary simply connected Jordan domain onto the unit disk. Let Γ be the limit of Gand. Assume that Γ is time differentiable and the p-th derivative is a Lipschitz volume of order α∈ (0,1).
We conclude that if the boundary of the region in question is sufficiently smooth, then we have the convergence of the polynomials πn to the conformal map ϕ0 in the closed region G and a reasonable estimate of the rate of convergence.
Ratio Asymptotics
When µ is supported on the unit circle or real line, there are detailed results regarding ratio asymptotic behavior. However, we use essentially the same argument as in the second proof of Theorem 2.1 in [55]. Using the argument in the proof of Theorem 2.9.4 (see below), Theorem 2.7.1 implies that if µ has compact support inR, is regular and has essential support equal to [−2,2], then we have relation asymptotics along a sequence of asymptotic density 1.
Generalizations of Theorem 2.7.1 to the setting when the measureµ is supported on a closed unit disk will be discussed in Section 3.3.2.
Relative Asymptotics
First, it shows that a certain perturbation—in this case, a singular component—does not affect the leading-order behavior of orthonormal polynomials. These are the characteristics of the relative asymptotic results, which we will also try to cover in section 3.3. We will use techniques similar to Saff's later in this text, and much of our work is based on the conjecture of [48], so we mention some of his results here.
This is indeed a special case of a more general result that we will prove in section 3.3.4.
Weak Asymptotic Measures
Gaussian Quadrature
In fact, this reasoning holds for any measure with at least N+ 1 points in the support, since for such measures it still makes sense to speak of orthonormal polynomials up to degree N. Since orthonormal polynomials are determined by the recursion coefficients, it follows that this spectral measure has the same first 2N moments as mass μ. Since this coefficient is only used to determine pN+1, we see that the spectral mass of Jb(N) and e0. call itρN,b) has the same first N+ 1 orthonormal polynomials as µ and hence the same first 2N moments as µ. 3.1.3) The measure ρN,0 is often called the quadratic measure of degree N + 1 for the measure μ.
Now let's integrate both sides of (3.1.3) along R with respect to the measure π(1+bdb 2), which is a probability measure.
The Unit Circle Case
This example illustrates the fact that, in general, the measurements of µn and νn do not have to be similar to each other as the measurements on D, so it is really important to consider balayage.
The Real Line Case
Note that xkKn−k(x, x;µ) is a polynomial of degree 2n−k, while the denominator of the weight defining the measure ρn is a polynomial of degree 2n+ 2. We are done if we can show, that the second term in ( 3.1.8) tends to zero asO(n−1) asn→ ∞and for this it is sufficient to put a uniform bound on. Therefore, by the justification of the above proof, we can rewrite the right-hand side of (3.1.11) as.
The expressionxkKn−k(x, x;µ) is a polynomial of degree 2n−k, and since ρn,0 has the same first 2n moments as µ, we can rewrite this again as.
L p -Extremal Polynomials on Analytic Regions
Push-Forward of Product Measures on the Disk
In this section, we will derive the norm asymptotics for the extremal polynomials corresponding to measures of the form considered in Theorem 3.2.2. As mentioned in the proof of Lemma 3.2.4, all weak bounds of the measures considered there rest on ∂G, so it suffices to consider functions that are continuous in a neighborhood of ∂G. The proof of the exponential draw now proceeds exactly as in the last part of the proof of Theorem 8.1.11 in [56] and we provide it here for completeness.
By the inequality of the measure and the extremal property we have kPn(˜µ)kqµ˜≥ kPn(˜µ)kqµ≥ kPn(µ)kqµ,.
Szeg˝ o Asymptotics for Extremal Polynomials
Note that theorem 3.2.11 tells us that the asymptotic Szeg˝o behavior of the leading order polynomialsPn(µ, q) is independent of τ. Corollary 3.2.8 and the remark after it imply that every weak limit of the measures{|pn|qdµ}n∈N must be a measure over ∂Gand so that we, without loss of generality, assume that σ1=σ2= 0,`= 0 , andν is completely absolutely continuous with respect to the Lebesgue measure. We conclude that if γ is a weak limit point of the measures {|pn(µ)|qdµ}n∈N, then for every k∈N we have
Theorem 3.2.12 gives the following corollary, which can be interpreted in terms of the Christoffel functions discussed in section 2.5 (see (2.5.5)).
Christoffel Functions
Indeed, the proof of the lemma will show that when r is close enough to 1, ϕr,x is defined on all G, so the statement of the lemma makes sense. We thus obtain a lower bound for the leftmost side of (3.2.32) which is independent of the degree of Q. Now that we have some understanding of λ∞(x;µ, q) for all x ∈ G when µ has the form discussed in theorem 3.2.2, we want to try to calculate it exactly.
Due to the injectivity of ϕonG (here we used Carath´eodory's theorem; see Section 1.2), any measure µonG can be pushed forward over ϕ to obtain the measure ϕ∗µonDas in Section 1.2.
New Results on Ratio and Relative Asymptotics
The Key Fact
The key to our calculations will be to write the numerator on the right-hand side of (3.3.2) as a perturbation of the denominator and – under suitable hypotheses – show that the perturbation tends to zero as n → ∞ while the denominator has no. Our first hypothesis on Qn implies that the sum of the first two terms tends to 2 asn→. It is possible for the limiting function(z) to vanish at a point inside the convex hull of the mass support.
In a general setting, assume that σ is a weak limit point of the measures{|pn(z;µ)|2dµ(z)}n≥0 with corresponding subsequenceNσ.
Application: Measures Supported on the Unit Disk
- Ratio Asymptotics on the Disk
- Weak Asymptotic Measures on the Disk
Let µ be a probability measure supported on the unit circle and let {αn(µ)}n≥0 be the corresponding sequence of Verblunsky coefficients. Let µ be a probability measure supported on the unit circle and let Qn be as in Theorem 3.3.2. Let µ be a probability measure supported on the unit circle and assume µ is normal in the sense defined in [32].
To see this, we let N be the subsequence as in Corollary3.3.5 and let M be the subsequence of N constructed by Lemma2.9.3.
Application: Measures Supported on Regions
Since k∈N was arbitrary, this implies that σ is a normalized arc length measure at∂That is desired. For example, if G satisfies one of the three conditions in [17]), we say Gis of class K and write G∈ K. Although Theorem 3.3.9 is an analogue of Theorem 3.3.4 for more general supports, this proves a analogous to Corollary 3.3.5 or Theorem 2.7.1 is more challenging. The following example shows that we can strengthen the conclusion of Theorem 3.3.9 to be more similar to that of Theorem 2.7.1 if some power of the conformal map φ is a monic polynomial.
The same calculation applies in any situation where some power of the conformal mapφ is a monic polynomial.
Application: Stability Under Perturbation
- The Uvarov Transform
- The Christoffel Transform
L´opez Lagomasino, Ratio and relative asymptotics of orthogonal polynomials on an arc of the unit circle, J. Kaliaguine, A note on the asymptotics of orthogonal polynomials on a complex arc: The case of a mass with discrete parts, J. Simon, Fine structure of zeros of orthogonal polynomials IV: a priori bounds and clock behavior, Kom.
Stoiciu, The statistical distribution of the zeros of random paraorthogonal polynomials on the unit circle, J.
