Constructive Function Theory on Sets of the Complex Plane through Potential Theory and Geometric Function Theory

Theorem 2.1 ([12]) A set E ⊂R is uniformly perfect if and only if C \ KE, defined in Subsection 2.1, is a John domain. Since the behavior of a conformal mapping of a John domain onto the unit disk is well-studied (see, for example, [82]), the theorem above can be useful in the investigation of metric properties of the Green function for the complement of a uniformly perfect subset of R. In particular, Theorem 2.1 can be used to solve the inverse problem of approx- imation theory of functions that

Theorem 2.2 (Totik [104, (2.8) and (2.12)]) There are absolute positive con- stants C1, C2, D1 and D2 such that for 0 < r < 1, gΩ(−r) ≤C1 √r exp  D1 Z 1 r θ2 E(t) t3 dt  log 2

Theorem 2.3 ([13]) For 0 < r < 1 gΩ(−r) ≥c1 √r exp

Theorem 2.3 provides a lower bound for the Green function (cf. [104, (3.5)]). Since in (2.7) only the size of the components of E∗ r influences this bound, one cannot expect to find an upper bound of the same form. We believe that in a Totik-type theorem not only the size of the components (αj,r, βj,r) but also their mutual disposition must be important.

Theorem 2.4 ([13]) For 0 < r < 1, q > 1 and any finite q-covering of E∗ r the inequalities gΩ(−r) ≤c2 √r exp

Corollary 2.5 ([13]) The estimates (2.5) and (2.6) hold with C1 = 384, C2 = 80 and D1 = D2 = 120.

Corollary 2.6 ([13]) For the compact set ˜E := {0} ∪ ∞ [ n=1 n2 [ j=1 n2 + j −1 2n+1n2 , 2n2 + 2j −1 2n+2n2  . we have

Theorem 2.7 ([14]) The condition (2.10) implies lim r→0 cap(E ∩[0, r]) cap([0, r]) = 1. (2.11) The converse of Theorem 2.7 is slightly weaker.

Theorem 2.8 ([14]) If E satisfies (2.11), then lim r→0 gΩ(−r) r1/2−ε = 0, 0 < ε < 1 2 . (2.12)

Theorem 2.9 ([18]) The following two conditions are equivalent: (i) gC\C satisfies (2.10) with E = C; (ii) P j ε2 j < ∞. In the case K(j) = 1, j ∈N, this statement is equivalent to [104, Theorem 5.3], but the latter is stated for the equilibrium measure on C. Interestingly, (ii) does not depend on the sequence {K(j)}. 2.5 On sparse sets with Green function of the highest smoothness Let E ⊂R be a compact set with positive logarithmic capacity. For simplicity we assume that E ⊂[−1, 1] and ±1 ∈E. Le

Theorem 2.10 ([16]) There exists a regular set E0 ⊂R with the following prop- erties: (i) gC\E0 satisfies (2.16); (ii) dim(E0) = 1/2. Next, we describe the construction of E0 in Theorem 2.10. For −1 ≤a < b ≤1 we consider two sequences of real numbers · · · < x−2 < x−1 < x0 < x1 < x2 < · · · , xk −x0 = x0 −x−k and y0 > y±1 > y±2 > · · · , yk = y−k, such that x0 = a + b 2 ,

Theorem 2.11 ([16]) There exists a regular set E1 ⊂R with the following prop- erties: (i) gC\E1 satisfies (2.19); (ii) dim(E1) = 0. We describe the construction of E1 in Theorem 2.11. We begin with two sequences of real numbers 1 = x0 > x1 > x2 > · · · > −1 and 4 = y0 > y1 > y2 > · · · > 0 such that yk = (xk + 1)2, k ∈N, lim k→∞xk = −1, lim k→∞yk = 0, yk

Theorem 2.1 and Corollary 2.11 in [50] assert that for 0 < s < 2 the condition |Eν,c ∩I| ≥2 −s (3.24) implies ||Qν,c||I ≤ √ 2 + √s √ 2 −√s !ν(C) . (3.25)

Theorem 3.1 ([9]) Let 0 < δ < 1/2. Then the condition cap(Eν,c ∩I) ≥1 2 −δ (3.26) yields that ||Qν,c||I ≤

Theorem 2], [37, p. 230]), we have ||pn||T ≤e2sn, 0 < s ≤π 2 . (3.31) Our next objective is to discuss an analogue of (3.30)–(3.31) in which we use logarithmic capacity instead of linear length. As before, our main result deals not only with polynomials, but also with exponentials of potentials.

Theorem 3.2 ([10]) Let 0 < δ < 1. Then the condition cap(Eν,c ∩T) ≥δ implies that ||Qν,c||T ≤ 1 + √ 1 −δ2 δ ν(C) . In order to examine the sharpness of Theorem 3.2 we consider the following example. Let 0 < α < π/2, and let L = Lα := {eiθ : 2α ≤θ ≤2π −2α}. (3.32)

Theorem 3.3 ([25]) Let G be a bounded domain and z ∈L := ∂G. Suppose that there are constants ε1, ε2 such that ε1r ≤|γz(r)| ≤(2π −ε1)r, 0 < r < ε2. (3.41) Then there exist constants c1, c2, c3 > c2 and ε3 <  ε2 2c3 2 depending only on ε1 and ε2 such that Rn(z, s) ≤exp  

Theorem 3.4 ([25]) Let L = ∂G consist of finitely many Dini-smooth arcs lj, which form exterior angles αjπ, 0 < αj < 2, at their junction points zj, j = 1, . . . , m. Let z ∈L be arbitrary and let zk be the nearest point to z among the zj, i.e., |zk −z| = min 1≤j≤m |z −zj|. Then, for 0 < s < m2(G) the inequality Rn(z, s) ≤exp  c6 n√s (√s + |zk −z|)1−1/αk log  c4 m2(G) −s 

Theorem 3.5 ([25]) Let G be a quasidisk. There exists a constant ε8 = ε8(ε6) such that for z ∈L and 0 < s ≤m2(G) −ε6 the inequality lim inf n→∞ log Rn(z, s) nδ(z, √s) ≥ε8 (3.50) holds. The inequality (3.50) demonstrates that (3.48) is asymptotically sharp (with respect to n) for all z ∈L and 0 < s ≤m2(G) −ε6. 3.3 Pointwise Remez-type inequalities in the unit disk Observe that (3.40) states a uniform bound. Our objective is to derive the point- wise extension of this bound. Note that the pointwis

Theorem 3.6 ([17]) The condition m2(D ∩Eν,c) ≥π −s, 0 < s < π, yields Qν,c(z) ≤exp  c1ν(C)√s exp  −c2 (1 −|z|)2 s  , z ∈D, where c1 and c2 are positive absolute constants.

Theorem 3.6 is sharp in the following sense. Given 0 < s < π/2 and 0 ≤x < 1. Let δ := 1 −x. Set for 0 < r < 1, ˜Er,δ := D \ ({ζ : |ζ −x| < r} ∪{ξ + iη : x < ξ ≤1, |η| < r}) . (3.51) Since m2( ˜Er,δ) > π −πr2 2 −2δr, taking r such that (πr2)/2 + 2δr = s, i.e., r = r(δ, s) := s δ + p δ2 + (π/2)s ,

Theorem 3.7 ([24]) Let L be an arbitrary Jordan arc or curve, and let σL(E∗ ν,c ∩L) diam(L) =: u < 1 2. Then ||Qν,c||L ≤

Theorem 3.7 extends [50, Theorem 2.1] from the case where L = [−1, 1] to the case where L is an arbitrary Jordan arc or curve.

Theorem 3.9 and the left-hand side of (3.61) below show that (3.56) is sharp (with respect to the degree 1/2 of u) even for the case of Jordan curves. However, if we take into consideration additional information about the geometry of L, the estimate (3.56) can be improved. Let L be a quasismooth (in the sense of Lavrentiev) curve which is defined by the following condition. For any z1, z2 ∈L, min {|L′|, |L′′|} ≤c1|z1 −z2|, c1 = c1(L) ≥1, (3.57) where L′ and L′′ are the connected components of L

Theorem 3.8 ([24]) Let L be a quasismooth curve and suppose that |E∗ ν,c ∩L| ≤s < 1 2 diam(L). Then ||Qν,c||L ≤exp(c2 δ(s, L) ν(C)) holds with c2 = c2(L). In order to discuss the sharpness of the bound of Theorem 3.8 we consider an important particular case of exponentials of potentials. Let V ⊂L consist of a finite number of open subarcs of L whose closures are disjoint, J := L \ V , and let c = c(V ) := −log cap(J). Since UµJ(z) = −gC\J(z) −log cap(J), z ∈C, we have QµJ,c(z) = exp(gC\J(z)),

Theorem 3.9 Let L be a quasismooth curve. Then λ(s, L) ≥ε2δ(s, L), 0 < s < diam(L) (3.59) holds with ε2 = ε2(L).

Theorem 3.8 and inequalities (3.58) and (3.59) show that the growth of the ex- ponentials of logarithmic potentials can be related to the function δ(s, L) which depends only on the geometry of L. Below we state some remarks concerning its properties. By the Ahlfors criterion (3.49), any quasismooth curve is quasiconformal. Therefore, Φ can be extended to a quasiconformal homeomorphism Φ : C → C. Taking into account the distortion properties of conformal mappings with a quasiconformal extension (

Theorem 3.10 ([24]) Let L be a quasismooth curve which is Dini-convex with respect to G. Then ε3s ≤δ(s, L) ≤c5s, 0 < s < diam(L). (3.60) Comparing Theorem 3.8 with the right-hand side of (3.60) we obtain an analogue of (3.31) for curves L instead of the unit circle T (see also [11]). This result is sharp because of Theorem 3.9 and the left-hand side of (3.60). If L consists of a finite number of Dini-smooth arcs which meet in the angles α1π, · · · , αmπ (with respect to Ω), 0 < αj < 2, α := max(1

Theorem 4.1 (Meinardus [70], Meinardus, Reddy, Taylor, Varga [71]) Let f satisfy (4.63). Then (i) the function f can be extended from R+ to an entire function of finite order;

Theorem 4.2 (Blatt, Kovacheva [36]) Assume that f is an entire function with (4.62) and, in addition to condition (4.64), the inequality ||f||[0,r] ≤µ(r)λ, (4.65) where µ(r) := minx≥r{f(x)}, holds for some number λ > 1 and for every r > r0. Then (4.63) is true. On the other hand, Henry and Roulier [59] have shown that the conditions (i) and (ii) of Theorem 4.1 are not sufficient for geometric convergence. For example, in [59] it was proved that f(x) = 1 + x + ex sin2 x (4.66) cannot be approximate

Theorem 4.3 ([21]) Let f satisfy (4.63). Then for every 1 < s < q there exist positive constants c = c(s, q), θ = θ(s, q) and r0 = r0(s, q) such that ||f||Er(f,s) ≤c rθ, r ≥r0 . (4.67) Next, we are going to discuss the geometrical meaning of condition (4.67). For ∞> H > h > minx∈R+ f(x) > 0, we introduce the strip domain S(h, H) := { (x, y) : −∞< x < ∞, h < y < H } as well as the intersection of this strip with the graph of f, i.e., Y (f, h, H) := S(h, H) ∩{ (x, y) : x ≥0, y = f(x) } , and define

Theorem 4.4 ([21]) Let f be an entire function satisfying (4.62). If, in addition, for some s > 1 and θ > 1, the function f satisfies (4.67), then, for each M > θ, lim sup h→∞ N(f, h, hM) log h < ∞. (4.68) Note that the result of Theorem 4.4 is sharp in the following sense: For each M > 1 there exists an entire function f = fM which satisfies (4.67) with some s > 1 and 1 < θ < M and lim sup h→∞ N(f, h, hM) log h

Theorem 4.3, f satisfies (4.67) in which we can take s so close to 1 that θ < M. The relation (4.69) immediately follows, if we set h = ek, k ∈N, and let k →∞. The new sufficient condition for geometrical convergence of best approximants can be stated in the following form.

Theorem 4.5 ([21]) Let f be an entire function satisfying (4.62) and (4.67) with some s > 1 and θ > 1. In addition, assume that there exists a constant M = M(f) > 1 such that lim sup h→∞ N(f, h, hM) < ∞. Then f satisfies (4.63).

Theorem 4.6 ([21]) There exists an entire function f satisfying the assumptions of Theorem 4.5, but not possessing property (4.65). The proof of Theorem 4.5 is based on an analogue of the classical result due to Bernstein, concerning polynomial approximation of functions analytic in the neighborhood of a subinterval of the real axis, for the case of several intervals. Let E = Sk j=1 Ij be the union of k disjoint intervals Ij = [αj, βj] of the real axis R and let Ω:= C\E. The set Es := {z ∈Ω: gΩ(

Theorem 4.7 ([21]) For each f ∈C(E) satisfying the following two conditions: for some s > 1, f can be extended analytically into int(Es) , (4.70) f has at least one zero on each Ij , (4.71) there exist constants q > 1 and c > 0 depending only on s and k such that inf pn∈Πn ||f −pn||E =: En(f, E) ≤c ||f||Es q−n, n ∈N . (4.72) Note that (4.72) can be interpreted as a result concerning geometric convergence of the polynomials of best approximation to the function f, independent of the geometry of E

Theorem 4.8 (Nikol’skii [76], Timan [100], Dzjadyk [44]) Let f ∈C([−1, 1]) and let ω be a function of modulus of continuity type satisfying the inequality δ 1 Z δ ω(t) t2 dt ≤c2 ω(δ), 0 < δ < 1, (4.73) with some constant c2 > 0. Then the following assertions are equivalent: (i) f ∈Cω([−1, 1]);

Theorem 4.9 ([15]) Let E = ∪m j=1Ej consist of m ∈N, m ≥2, disjoint con- tinua Ej, f ∈A(E), ||f||E ≤1, and let z1, · · · , zN ∈E be distinct points. Let for any n > n0 ∈N and j = 1, · · ·, m there be a polynomial pn,j ∈Pn such that |fj(z) −pn,j(z)| ≤εj 1 n, z  , z ∈∂Ej, pn,j(zl) = fj(zl), zl ∈Ej, where fj := f|Ej is the restriction of f to Ej, and the function εj(δ, z), 0 < δ ≤ 1, z ∈∂Ej, satisfies, for any j = 1, · · ·, m and z ∈∂Ej, the properties: (i) εj(δ, z) is monotonically increasing in

Theorem 4.10 ([8]) There exist a regular compact set E0 ⊂R and for any 0 < α ≤1 a function fα ∈Cα(E0) such that the following assertion is false: for any n ∈N there is a polynomial pn ∈Πn with the property: |fα(x) −pn(x)| ≤c ρα 1/n(x), x ∈E0, (4.75) where the constant c > 0 is independent of n and x.

Theorem 4.11 ([8]) Let the regular set E ⊂R consist of a finite number of disjoint compact sets, each of which belongs to the class E(α, c) with some α, c > 0. Suppose that f ∈C(E) and that the function ω of the modulus of continuity type satisfies (4.73). Then the following conditions are equivalent: (i) f ∈Cω(E); (ii) for any n ∈N there exists a polynomial pn ∈Πn such that |f(x) −pn(x)| ≤c8 ω(ρ1/n(x)), x ∈E, where the constant c8 > 0 does not depend on x and n. The simplest example of E satisfyi

Theorem 4.12 ([22]) Let E ∈H∗, f ∈A(E), k ∈N, and let z1, · · ·, zN ∈E be distinct points. Then for any n ∈N, n ≥N + k, there exists a polynomial pn ∈Pn such that |f(z) −pn(z)| ≤c1 ωf,k,E(ρ1/n(z)), z ∈L, (4.83) pn(zj) = f(zj), j = 1, · · ·, N (4.84) with c1 > 0 independent of n. Moreover, if E0 ̸= ∅and for 0 < δ < 1, δ Z 0 ωf,k,E(t) dt

Theorem 4.13 ([22]) Let E ∈H∗, f ∈Ar(E), r ∈N, k ∈N, and let z1, · · · , zN ∈ ∂E be distinct points. Then for any n ∈N, n ≥Nr+k, there exists a polynomial pn ∈Pn such that for l = 0, · · ·, r, |f (l)(z) −p(l) n (z)| ≤c ρr−l 1/n(z) ωf(r),k,E(ρ1/n(z)), z ∈L, p(l) n (zj) = f (l)(zj), j = 1, · · ·, N with c independent of n. Our next goal is to allow the number of interpolation nodes N in Theorem 4.12 to grow infinitely with the degree of the approximating polynomial n. To this end, we specify the ch

Theorem 4.14 ([22]) Let E be a closed Jordan domain bounded by a quasicon- formal curve L. Let f, r, k be as in Theorem 4.12 and let z1, · · ·, zN ∈E be the points of an N-th Fekete point set of E. Then for any ε > 0 there exists a polynomial pn ∈Pn, n ≤(1 + ε)N, satisfying conditions (4.83) and (4.84). Moreover, if (4.85) holds then in addition to (4.83) and (4.84) we have (4.86), and the constants c1, c3, c4 and α are independent of N. 4.4 Open problems We begin with a question that would be a