Proceedings of the Steklov Institute of Mathematics, Vol. 218, 1997, pp. 352–371. Translated from Trudy Matematicheskogo Instituta imeni V. A. Steklova, Vol. 218, 1997, pp. 354–373. Original Russian Text Copyright \begin{matrix}c \\ ◯ \end{matrix} 1997 by Popov. English Translation Copyright \begin{matrix}c \\ ◯ \end{matrix} by ���� �����/Interperiodica Publishing (Russia).

Spherical Convergence of the Fourier Series and Integral of the Indicator of a Two-Dimensional Domain

D. A. Popov

Received February, 1997

1. INTRODUCTION

In [1, 2], attention was drawn to the problem of pointwise convergence of the method of summation over disks (balls) of Fourier series and integrals for indicators (characteristic functions) of domains in ℝ^{N}.

In [1] only spherically symmetric functions are considered, and it is shown that for dimension N ≥ 3, the method of summation over disks [3–5] diverges for the characteristic function of a ball at the point which is the center of this ball. In [6], in the two-dimensional case, the pointwise convergence of the method of summation of the Fourier series over disks for the characteristic function of a domain with a convex boundary is proved. The author is not familiar with other results concerning the problem under consideration.

The main results of this paper refer to the two-dimensional case, but when we state the problem and describe the method, we consider the general case of an arbitrary dimension. This is due to the fact that the proposed approach is suitable for N = 1, 2, 3 and, possibly, for N ≥ 4. In this paper, by examining the case N = 2, we demonstrate the difficulties that arise and the character of the results obtained. The case N = 3 is briefly discussed in Section 5.

We begin with the definition of the general spherical method of summation of Fourier integrals and series [5]. Everywhere in the sequel, we consider only bounded, absolutely integrable functions f on ℝ^{N} with the support in the cube [−π, π)^{N}. We denote by ⟨x, y⟩ the inner product in ℝ^{N},

by |x| = \sqrt{x_{1}^{2} + ⋯ + x_{N}^{2}} the norm of the element x = (x_{1}, …, x_{N}) ∈ ℝ^{N}, and by

f̂(ω) = ∫_{ℝ}^{N}e^{i⟨ω,x⟩}f(x) dx,  ω ∈ ℝ^{N},  (1)

the Fourier transform of the function f. When considering the Fourier series, we always assume that x ∈ [−π, π)^{N} and denote by f^{T} a periodic continuation of the function f from the cube [−π, π)^{N} to ℝ^{N}. Thus, f^{T} is a function on the torus T^{N} which is obtained by the identification of the corresponding sides of the cube [−π, π]^{N}. In this case, the Fourier coefficients a(n), n ∈ ℤ^{N}, of the function f^{T} have the form

a(n) = (2π)^{−N} f̂(−n),  n ∈ ℤ^{N}.  (2)

By definition, the general spherically symmetric method A of summation of Fourier integrals and series is defined by the even function A: ℝ → ℝ and associates the function f_{A} with the function f, i.e.,

f_{A}(x) = (2π)^{−N} ∫_{ℝ}^{N}e^{−i⟨ω,x⟩}A(|ω|)f̂(ω) dω,

and the function f_{A}^{T} with the function f^{T}, i.e.,

f_{A}^{T} (x) = ∑_{n∈ℤ^{N}} a(n)A(|n|)e^{i⟨n,x⟩}.  (3)

In any case, both functions f_{A} and f_{A}^{T} exist if A is a bounded piecewise continuous function with compact support, A(|ω|) = 0, |ω| > Ω′. Below we consider only the summation methods depending on one “large” parameter Ω for which

\begin{matrix}A(|ω|) = G(ξ) = G(−ξ),  ξ = \frac{|ω|}{Ω}, \\ G(ξ) → 1, ξ → 0,  G(ξ) = 0, ξ > ξ_{0}. \end{matrix}  (4)

In particular,

G(ξ) = \{ \begin{matrix}1, & |ξ| ≤ 1 \\ 0, & |ξ| > 1, \end{matrix}

corresponds to the summation over disks, and

G(ξ) = \{ \begin{matrix}(1 − ξ^{2})^{s}, & |ξ| ≤ 1 \\ 0, & |ξ| > 1. \end{matrix}

corresponds to the Riesz–Bochner summation [3, 4]. Below, for brevity, the method of summation over disks is called the Ω-method (Ω-summation), and, for this method, the functions f_{A} and f_{A}^{T} are denoted by f_{Ω} and f_{Ω}^{T}, respectively.

Let the function f be continuous in the neighborhood of the point x. The rate of convergence of the method A at the point x for the Fourier integral and series will be characterized by the quantities σ_{A}(f|x) and σ_{A}^{T}(f|x), respectively, which are defined as follows. Let

Δ_{A}f(x) = f_{A}(x) − f(x),  Δ_{A}f^{T} (x) = f_{A}^{T} (x) − f^{T} (x).  (5)

Then σ_{A}(f|x) and σ_{A}^{T}(f|x) are least upper bounds of the quantities α and α′ for which the following asymptotic estimates hold:

|Δ_{A}f(x)| ≤ O(Ω^{−α}), |Δ_{A}f^{T} (x)| ≤ O(Ω^{−α′}), Ω → ∞.

By σ_{A}(f) and σ_{A}^{T}(f) we denote the lower bounds of the quantities σ_{A}(f|x) and σ_{A}^{T}(f|x) on the set of points of continuity of the function f. If the function f belongs to a certain class {f}, then σ_{A}({f}) and σ_{A}^{T}({f}) are lower bounds of the quantities σ_{A}(f) and σ_{A}^{T}(f) on this class. For the method of summation over disks, the corresponding quantities σ_{A} and σ_{A}^{T} are denoted by σ and σ^{T}.

In this paper, for N = 2, we consider the problem of computing the quantities σ(f|x) and σ^{T} (f|x) for f ∈ F_{p}, where F_{p} is the class of characteristic functions of domains (f = χ_{D}, D ⊂ ℝ^{2}) with a piecewise-smooth boundary ∂D. In what follows, we assume that the domains D are simply connected; obviously, this does not lead to the loss of generality. The exact definition of the class F_{p} will be given in Section 2. It should be pointed out that all the obtained results are directly extended to a wider class of functions {F_{p}, I}, and f ∈ {F_{p}, I} if

f(x) = ∑_{i=1}^{I_{0}} g_{i}(x)f_{i}(x), f_{i} ∈ F_{p}, g_{i} ∈ C^{∞}, I_{0} ≤ I.

Moreover,

σ_{A}({F_{p}, I}) = σ_{A}(F_{p}), σ_{A}^{T}({F_{p}, I}) = σ_{A}^{T}(F_{p});

this follows immediately from the method of decomposition with respect to smoothness which is presented in [7].

It follows from the results of [6] that for a convex domain we have

σ^{T} (f|x) ≥ \frac{1}{3}, N = 2.

A method is proposed in [8], which allows one to solve completely the problem of determination of the quantities σ_{A}(f|x) for N = 2. In particular, it follows from the results of [8] that

σ(F_{p}) = \frac{1}{2},  N = 2.  (6)

Let us formulate the results of [8] for the case f = χ_{D}. Let ∂D be a general position smooth contour. This means that the curvature k of the curve ∂D as a function of the natural parameter s (s is the arc length) has only a finite number of isolated extrema (vertices) on ∂D, i.e., if k^{(1)}(s_{0}) = 0, then k^{(2)}(s_{0}) ≠ 0. Any contour can be transferred to the general position by a small perturbation. For the general position contour, the evolute Γ ≡ Γ(∂D), which is defined as the envelope of normals or as the locus of curvature centers, has only simple cusps, as for an ellipse; we denote by Γ_{0} the union of these cusps. Then, for f = χ_{D}, N = 2, we have

σ(f|x) = \{ \begin{matrix}1, & x & ∉ Γ \\ \frac{5}{6}, & x & ∈ Γ, x ∉ Γ_{0} \\ \frac{3}{4}, & x & ∈ Γ_{0}. \end{matrix}

For nongeneral position contours, σ(f|x) can assume a certain sequence of rational numbers from the interval [1/2, 1] as values, and the equality σ(f|x) = 1/2 is realized only for the characteristic function of a disk if x is the center of this disk.

Thus, the behavior of the quantity σ(f|x) depends on the geometry of the boundary ∂D. It is interesting to clarify to what extent these results can be extended to Fourier series. One of the main results of this paper is the equality σ^{T} (F_{p}) = 1/2, N = 2.

The approach to the problem of computing the quantities σ(f|x) and σ_{A}^{T}(f|x) presented below is based on the reduction of this problem to the problem of estimation of oscillatory integrals (OI) of the form

J[Ω|u, Φ] = ∫_{∂D} e^{iΩΦ(s)}u(s) ds.  (7)

In the case of a Fourier series, the quantity Δ_{A}^{T}f(x) (4) can be represented (for N = 2) in the form of a double series of these integrals. To estimate the sum of such a series, asymptotic estimates of OI which enter into this series are no longer sufficient, and uniform bounds of these integrals with the indication of constants in estimates, estimates with constants, are necessary. The methods of obtaining these estimates, which date back to van der Corput and Vinogradov, were developed in the works by Arkhipov, Karatsuba, and Chubarikov; the results obtained in these works can be found in [9, 10]. In turn, on the basis of [9, 10], estimates were obtained in [8] which are used below.

Everywhere in this paper, we accept the convention concerning constants.

Convention concerning constants. All constants whose values are not indicated are denoted by the letter C; one letter can be used to denote different constants. In all case, we indicate on what quantities the constant C depends; if these quantities are not indicated, then C is an absolute constant. All constants C can be given in explicit form. If constants C which are not indicated enter into the conditions of validity of a certain assertion or its formulation, then this means that there exist and can be explicitly indicated the values of the constants such that these conditions are fulfilled together with the corresponding assertions.

2. THE METHOD

We shall show how and in what sense the problem of convergence of the summation method for a Fourier series can be reduced to the corresponding problem for a Fourier integral. This is done with the aid of the Poisson integral. This reduction is widely used in the theory of Fourier series, for example, when solving the problem of equiconvergence of Fourier integral and series [4].

Lemma 1. Assume that the following conditions hold:

(1) the method of summation is defined by a continuous even function A; moreover, A(|ω|) = 0 (|ω| > Ω′),

(2) the function f ∈ L^{1} has a support in the cube [−π, π)^{N}.

If x ∈ [−π, π)^{N}, then

Δ_{A}f^{T} (x) = ∑_{n∈ℤ^{N}} Δ_{A}f(x_{n}),  x_{n} = x − 2πn,  (8)

under the condition of absolute convergence of the series on the right-hand side of this relation.

Proof. The Poisson formula [11]

∑_{n∈ℤ^{N}} g(n) = ∑_{n∈ℤ^{N}} ĝ(2πn)  (9)

holds if the following conditions are fulfilled:

(1) the function g is continuous at the points of the lattice, is integrable on each finite interval, and the series ∑_{n} g(x + n) converges absolutely and uniformly in the cube |x_{i}| ≤ 0.5,

(2) the series on the right-hand side of the relation (9) converges.

We apply this formula to the series f_{A}^{T} (x) (3). In this case, g(ω) = (2π)^{−N} f̂(ω)A(|ω|)e^{i⟨ω,x⟩}, and the conditions of applicability of the Poisson formula hold by the virtue of assumptions that we have made. Thus,

f_{A}^{T} (x) = ∑_{n∈ℤ^{N}} f_{A}(x_{n})

and it remains to note that f(x_{n}) = 0 (n ≠ 0) for x ∈ [−π, π)^{N}.

For the method of summation over disks (Ω-summation), which is of interest to us, we have

f_{Ω}^{T} (x) = ∑_{|n|≤Ω} a(n)e^{i⟨n,x⟩}

and the conditions of Lemma 1 are not satisfied. In this case, we proceed as follows. Let A be a certain spherical summation method defined by a continuous even function A such that

A(|ω|) ≤ 1 ∀ω,  A(|ω|) = \{ \begin{matrix}1, & |ω| & ≤ Ω \\ 0, & |ω| & > Ω′, Ω′ = Ω + ΔΩ, ΔΩ > 0. \end{matrix}  (10)

We introduce the quantity

Δ_{AΩ}f^{T} (x) = f_{Ω}^{T} (x) − f_{A}^{T} (x) = − ∑_{Ω<|n|≤Ω′} a(n)A(|n|)e^{i⟨n,x⟩},  (11)

for which, by (2) and (10), we have the estimate

|Δ_{AΩ}f^{T} (x)| ≤ (2π)^{−N} ∑_{Ω<|n|≤Ω′} |f̂(n)|.  (12)

By the definition of Δ_{AΩ}f^{T},

Δ_{Ω}f^{T} (x) = Δ_{A}f^{T} (x) + Δ_{AΩ}f^{T} (x).

Since relation (8) holds for Δ_{A}f^{T} (x), we have

|Δ_{A}f^{T} (x)| ≤ ∑_{n∈ℤ^{N}} |Δ_{A}f(x_{n})|,  x ∈ [−π, π)^{N}.  (13)

The estimate

|Δ_{Ω}f^{T} (x)| ≤ |Δ_{A}f^{T} (x)| + |Δ_{AΩ}f^{T} (x)|,  (14)

together with estimates (12) and (13), is a starting point for the estimation of the quantity Δ_{Ω}f^{T} (x).

From this time on, we assume that N = 2 and use the vector notation a, r, r_{0}, ω for points from ℝ^{2}. Let us give the definition of the class F_{p} of functions being considered. We fix the integer p ≥ 0. The contour ∂D ⊂ ℝ^{2} is p-piecewise-smooth if it consists of a finite number of p-smooth arcs. An arc is p-smooth if it is a segment of a straight line or if it is defined by the function r(s) (s is a natural parameter) which has at least p + 3 continuous derivatives, and if only a finite number of planar points s_{i}^{0} (k(s_{i}^{0}) = 0) can exist on this arc; moreover,

k^{(n)}(s_{i}^{0}) = 0, n ≤ p_{i} − 1,  |k^{(p_{i})}(s_{i}^{0})| = K_{i} > 0, p_{i} ≤ p.  (15)

We say that f ∈ F_{p} if f = χ_{D} and ∂D is a p-piecewise-smooth contour.

The contour ∂D is said to be strictly quasiconvex if it is p-piecewise-smooth and its curvature k does not vanish anywhere, i.e.,

|k(s)| ≥ k_{0}.  (16)

In this case, we can assume that p = 0. The contour ∂D is said to be quasiconvex if it consists of strictly quasiconvex arcs and segments of lines. Note that a smooth quasiconvex contour is convex.

The use of estimates (12) and (13) will be based on the fact that the quantities Δ_{A}f(x) and f̂(ω) can be represented (with the required accuracy) in the form of OI over the boundary (7), and, for f ∈ F_{p}, these integrals admit an effective estimate [8].

Before considering these representations, note that everywhere below ν is an outer normal to the contour ∂D; this contour is assumed to be positively oriented. In this case, the curvature k can assume both signs, and k(s) > 0 if the vector ν is directed to the center of curvature. Moreover, the Serre–Frenet equations preserve their form

\frac{dτ}{ds} = k(s)ν(s), \frac{dν}{ds} = −k(s)τ (s),  |ν| = 1, |τ | = 1.  (17)

Here and in what follows, τ (s) is a tangent vector to ∂D at the point r(s) ∈ ∂D.

The representation of the quantity f̂(ω) (for N = 2) in the form of an OI results directly from definition (1). Passing to polar coordinates ω = λη, λ = |ω|, η(cos φ, sin φ) and applying the Green formula, we obtain

f̂(λη) = −\frac{i}{λ} ∫_{∂D} e^{iλ⟨r(s),η⟩}⟨ν(s), η⟩ ds.  (18)

We are interested in the method of summation over disks; the consideration of other methods is of an auxiliary character and is connected only with the necessity of using estimate (14). Therefore, we shall assume below that the function G, (4), which defines the summation method, is piecewisedifferentiable and can only have discontinuities of the first kind at the points ξ_{p} ≠ 0. Moreover,

∑_{p} |Δ_{p}G| < ∞, Δ_{p}G ≡ G(ξ_{p} + 0) − G(ξ_{p} − 0),

the derivative G^{(1)} is piecewise-continuous, and G^{(1)} ∈ L^{1}. In this case, as is shown in [8], we have the representation

Δ_{A}f(a) = ∑_{ξ_{p}≠0} \frac{Δ_{p}G}{2π} ∫_{∂D} ds \frac{⟨l, ν⟩}{|l|^{2}}J_{0}(ξ_{p}Ω|l|)

+ \frac{1}{2π} ∫_{∂D} ds \frac{⟨l, ν⟩}{|l|^{2}} ∫_{0}^{∞}G^{(1)}(ξ)J_{0}(Ωξ|l|) dξ,  l ≡ r(s) − a,  (19)

where J_{0} is a Bessel function. The possibility of this representation is based on the Gauss formula [12] according to which

χ_{D}(a) = \frac{1}{2π} ∫_{∂D} \frac{⟨l, ν⟩}{|l|^{2}} ds,  N = 2, a ∉ ∂D.  (20)

Relations (19) and (20) yield the required representation of the quantity Δ_{Ω}f(a) in the form of an OI [8], i.e.,

Δ_{Ω}f(a) = Re \frac{1}{\sqrt{Ω × 2π^{3}}} ∫_{∂D} \frac{⟨l, ν⟩}{|p|^{5/2}} e^{iΩ|l|} ds + Δ_{Ω}^{1}f(a),  |Δ_{Ω}^{1}f(a)| ≤ \frac{CL}{Ω^{3/2}d^{5/2}} .  (21)

Here and below L is the length of ∂D, and it is assumed that

|l| ≥ d,  Ωd ≥ 1,  (22)

where d is the distance from the point a to ∂D.

For N = 2, 3, and, possibly, for all N, in order to consider the Ω-summation, it is sufficient to restrict ourselves by a special class of methods of summation (q-summation), which are given by the integer q ≥ 0. For the q-summation,

G(ξ) = \{ \begin{matrix}1, & 0 & ≤ ξ ≤ 1 \\ H(ξ), & 1 & ≤ ξ ≤ 1 + δ, δ = \frac{ΔΩ}{Ω} \\ 0, & ξ & ≥ 1 + δ, \end{matrix}  (23)

where H is a polynomial of degree 2q + 1 satisfying the conditions

H(1) = 1, H^{(1)}(1) = 0, …, H^{(q)}(1) = 0,
H(1 + δ) = 0, H^{(1)}(1 + δ) = 0, …, H^{(q)}(1 + δ) = 0.

A polynomial of this kind does exist. It is unique, monotonic, and has the form

H(ξ) = h(1 − \frac{ξ − 1}{δ}), h^{(1)}(y) = \frac{(2q + 1)!}{(q!)^{2}}y^{q}(1 − y)^{q},

h(y) = \frac{(2q + 1)!}{(q!)^{2}} ∑_{j=0}^{q}(−1)^{j}\binom{q}{j}\frac{y^{q+j+1}}{q + j + 1}.

In particular,

h(y) = \{ \begin{matrix}y, & q = 0 \\ 3y^{2} − 2y^{3}, & q = 1 \\ 10y^{3} − 15y^{3} + 6y^{5}, & q = 2. \end{matrix}

When considering the q-summation for the quantities Δ_{A}f and Δ_{A}f^{T}, we use the notation Δ_{q}f, Δ_{q}f^{T}.

It follows from the general results of [8] that

Δ_{q}f(a) = Im ∑_{ξ_{p}} ∑_{j=q}^{2q} \frac{C_{jq}}{(ΔΩ)^{j+1}} \frac{1}{\sqrt{ξ_{p}Ω}}J_{jp}(a) + Δ_{q}^{1}(a),  J_{jp}(a) = ∫_{∂D} ds \frac{⟨l, ν⟩}{|l|^{j+7/2}} e^{iξ_{p}Ω|l|}. (24)

In this formula, ξ_{p} are points, where the smoothness of the function G is violated, i.e., ξ_{p} = 1, 1 + δ, and the estimate

|Δ_{q}^{1}(a)| ≤ \frac{C_{q}L}{(Ωd)^{3/2}d} ∑_{j=q}^{2q} \frac{1}{(ΔΩ × d)^{j+1}}

holds for the quantity Δ_{q}^{1}(a). In particular, for q = 0 we have

Δ_{0}f(a) = Im \frac{A_{0}}{ΔΩΩ^{1/2}} ∫_{∂D} ds \frac{⟨l, ν⟩}{|l|^{7/2}}[\frac{e^{iΩ(1+δ)|l|}}{\sqrt{1 + δ}} − e^{iΩ|l|}] + Δ_{0}^{1}(a),

|Δ_{0}^{1}(a)| ≤ \frac{CL}{(Ωd)^{3/2}d^{2}ΔΩ}, A_{0} = \frac{1}{π^{3/2}} e^{i3π/4}.

(25)

Thus, the estimation of the quantity Δ_{q}f(a) reduces to the estimation of one-dimensional OI of the form (7) with phase |l|. It should be emphasized that the estimates with an explicit indication of constants in them (estimate with constants) are necessary since later on these estimates are substituted into series (13). These estimates will be considered in the next section. The required estimates for the quantities f̂(ω) (18) are already contained in [8] (see also [13]).

3. THE FOURIER INTEGRAL

Relations (21), (24), and (25) show that it is necessary to estimate one-dimensional OI of the form

J(a) = ∫_{∂D} ds e^{iΩΦ(s)}u(s),  u(s) = \frac{⟨l, ν⟩}{|l|^{t}},  (26)

with the phase Φ(s) = |l(s)| = |a − r(s)|, r(s) ∈ ∂D, which does not depend on the summation method. When carrying out the estimation, we shall follow [8].

The estimation of OI begins with the analysis of critical points. In the case which is of interest to us, the critical points s_{c} ≡ s_{c}(a) are determined from the conditions

Φ^{(1)}(s_{c}) = 0 ⟺ l(s_{c}) = ε_{c}|l(s_{c})|ν(s_{c}), ε_{c} = ±1,

and at these points,

Φ^{(2)}(s_{c}) = |l(s_{c})|^{−1} + ε_{c}k(s_{c}).  (27)

Let us consider the conditions of degeneration (Φ^{(2)}(s_{c}) = 0). The evolute Γ of the curve ∂D is defined by the equation

a_{Γ}(s) = r(s) + \frac{ν(s)}{k(s)}, a_{Γ}(s) ∈ Γ.

This implies that all critical points s_{c} are nondegenerate (Φ^{(2)}(s_{c}) ≠ 0) if a ∉ Γ, and there are degenerate critical points s_{c}^{0} among them if a ∈ Γ. Using the Serre–Frenet equations (17), we find that under the conditions

k^{(n)}(s_{c}^{0}) = 0, 1 ≤ n ≤ j − 1,  |k^{(j)}(s_{c}^{0})| = K_{j}^{0}, j ≥ 1,  (28)

we have

Φ^{(n)}(s_{c}^{0}) = 0, n ≤ j + 1,  Φ^{(j+2)}(s_{c}^{0}) = ε_{c}k^{(j)}(s_{c}^{0})  (29)

at this point. The case j = 1 corresponds to the fact that a ∈ Γ, and the point a is not a singular point of the evolute (a ∉ Γ_{0}), where Γ_{0} is a set of singular points. If j ≥ 2, then a ∈ Γ_{0} ⊂ Γ, j = ∞ if and only if ∂D is a circle and a is its center. The obtained results immediately imply

Theorem 1. Let f = χ_{D} ∈ F_{p}, and suppose that we can draw a normal from the point a to ∂D. Then

σ(f|a) = \{ \begin{matrix}1, & a & ∉ Γ \\ \frac{5}{6}, & a & ∈ Γ, a ∉ Γ_{0} \\ \frac{1}{2} + \frac{1}{j + 2}, & a & ∈ Γ_{0}, \end{matrix}  (30)

where j ≡ j(a) is the value of j, maximal for different points s_{c}^{0} ≡ s_{c}^{0}(a), such that conditions (28) are fulfilled.

If it is impossible to draw a normal from the point a to ∂D (∂D is a nonsmooth contour), then

σ(f|a) = \frac{3}{2}.  (31)

Proof. To determine the quantity σ(f|a), it suffices to know the asymptotics of the quantity Δ_{Ω}f(a) as Ω → ∞. By (21), this asymptotic behavior is defined by the contribution of the critical point s_{c} of maximal degeneration (provided that critical points exist) [14]. Taking now into account (27) and (29), we immediately obtain (30). If it is impossible to draw a normal from the point a to ∂D, then the contour ∂D is nonsmooth, and the asymptotic behavior of the integral J(a) is defined by the contribution of angle points. Integrating by parts, we obtain (31).

The knowledge of the asymptotics of the integral J(a) is no longer sufficient for analyzing the quantity Δ_{Ω}f^{T} (a), and the estimates with constants are necessary. Although the method of [8] allows us to obtain asymptotically exact estimates with constants for this integral at all a for an arbitrary p-piecewise-smooth contour ∂D, we shall prove only the simplest estimates.

Recall that we always assume that conditions (22) are fulfilled and ∂D ⊂ [−π, π)^{2}. By definition, for a strictly quasiconvex contour |k(s)| ≥ k_{0}, (16), we shall assume that

k_{0} ≤ 1,  |k(s)| ≤ \overline{k},  |k^{(1)}(s)| ≤ \overline{k^{(1)}}.

If |l(s)| ≤ \overline{l}, then it suffices to consider the estimates in the following two cases:

(1) \overline{l} ≤ Ck_{0}^{−1},
(2) d ≥ Ck_{0}^{−1},

since the condition \overline{l} ≥ Ck_{0}^{−1} implies that d ≥ Ck_{0}^{−1}.

Lemma 2. Let f = χ_{D} ∈ F_{p}, p ≥ 0, and let ∂D be a strictly quasiconvex contour; suppose that we consider the domain of points a such that |a − a_{Γ}| ≥ d_{Γ} ∀a_{Γ} ∈ Γ (Γ is the evolute of the curve ∂D).

Then for l ≤ Ck_{0}^{−1}, we have the following estimate for the integral J(a), (26):

|J(a)| ≤ \frac{CN}{Ω^{1/2}} (\frac{1}{d^{t−2}\sqrt{d_{Γ}}} + \frac{1}{d^{t−3/2}} ) + \frac{B_{1}(∂D)(N + 1)}{Ωd^{t+1}d̃_{Γ}^{4}},  d̃_{Γ} ≡ min\{1, d, d_{Γ}\};  (32)

PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS  Vol. 218  1997

and for d ≥ Ck_{0}^{−1} we have

|J(a)| ≤ \frac{CN}{Ω^{1/2}k_{0}^{1/2}d^{t−1}} + \frac{B_{2}(∂D)(N + 1)}{d^{t−1}Ω}.  (33)

Here N is the number of different normals from the point a to ∂D and the constants B_{i}(∂D) are indicated below (see (36) and (38)).

Proof. Since a ∉ Γ in the case being considered, all critical points are nondegenerate, and it follows from the results of [8] that

|J(a)| ≤ C_{1} ∑_{s_{c}} \frac{|u(s_{c})| + \overline{u_{c}^{(1)}}δ_{c}}{|ΩΦ^{(2)}(s_{c})|^{1/2}} + \frac{C_{2}(\overline{u} + ∥u^{(1)}∥_{L^{1}})}{ΩΦ_{1}}.  (34)

In this formula,

Φ_{1} = min_{s_{c},s_{ex}}\{|Φ^{(2)}(s_{c})δ_{c}|, \frac{(ψ^{(1)}(s_{c})Δ_{c})^{2}}{∥Φ^{(2)}∥_{L^{1}}} , \frac{|ψ(s_{ex})|^{2}}{∥Φ^{(2)}∥_{L^{1}}} \}

and ψ is twice piecewise-differentiable function such that |Φ^{(1)}(s)| ≥ |ψ^{(1)}(s)| and s_{ex} ≠ s_{c} are its extremal points (ψ^{(1)}(s_{ex}) = 0). We assume that

|u(s)| ≤ \overline{u}, |u^{(1)}(s)| ≤ \overline{u_{c}^{(1)}}, |s − s_{c}| ≤ δ_{c},
|Φ^{(3)}(s)| ≤ \overline{Φ_{c}^{(3)}}, |s − s_{c}| ≤ δ_{c}, |Φ^{(2)}∥_{L^{1}} ≡ ∫ |Φ^{(2)}(s)| ds.

Estimate (34) holds for any δ_{c} > 0 such that

δ_{c} ≤ Δ_{c} ≤ min_{s_{c}}\{\frac{|Φ^{(2)}(s_{c})|}{\overline{Φ_{c}^{(3)}}}, \frac{ψ^{(1)}(s_{c})}{\overline{ψ_{c}^{(2)}}}\}.

In the case which is of interest to us,

\begin{matrix}|u(s)| ≤ Cd^{−t+1},  |u^{(1)}∥_{L^{1}} ≤ Cd^{−t+1}(\overline{k} + d^{−1})L, \\ |Φ^{(2)}(s)| ≤ C(|l|^{−2} + \overline{k}|l|^{−1} + \overline{k} + \overline{k}^{(1)}),  ∥Φ^{(2)}∥_{L^{1}} ≤ CL(\overline{k} + d^{−1}). \end{matrix}  (35)

The quantity Φ^{(2)}(s_{c}) in (27) can be written in the form

Φ^{(2)}(s_{c}) = |l(s_{c})| × |k(s_{c})| × |a_{Γ}(s_{c}) − a|

and therefore

|Φ^{(2)}(s_{c})| ≥ \{ \begin{matrix}C|l(s_{c})|^{−2}d_{Γ}, & |k(s_{c})l(s_{c})| ≥ 0.5 \\ C|l(s_{c})|^{−1}, & |k(s_{c})l(s_{c})| ≤ 0.5. \end{matrix}

In the first case (for \overline{l} ≤ Ck_{0}^{−1}), the function ψ can be determined from the relation ψ(s) = \overline{l}^{−1}⟨l, τ ⟩. Then |Φ^{(1)}(s)| ≥ |ψ(s)|, and it is easy to find the extremal points of the function ψ (⟨a_{Γ}(s_{ex}) − a, ν(s_{c})⟩ = 0), and therefore

|ψ(s_{ex})| ≥ d_{Γ}\overline{l}^{−1}.

Since |d|l|/ds| ≤ 1, it follows from the obtained estimates that we can take

δ_{c} = Δ_{c} = \frac{Cd̃_{Γ}}{g_{0}} , g_{0} = 1 + \frac{\overline{k}}{k_{0}} + \frac{\overline{k}^{(1)}}{k_{0}^{2}} + \frac{\overline{k}^{2}}{k_{0}^{2}} ,

and then we obtain from (34) the required result (32) in which

B_{c}(∂D) = (L\overline{k}g_{0})^{2}.  (36)

In the second case (for d ≥ Ck_{0}^{−1}), we use the estimate

|Φ^{(2)}(s_{c})| ≥ Ck_{0},  (37)

which follows from (27). Now we can take ψ = Φ^{(1)} since the quantities Φ^{(1)}(s_{ex}) admit an effective lower bound. The equation Φ^{(2)}(s_{ex}) = 0 for extremal points can be written as

(1 + k⟨l, ν⟩)(s_{ex}) = \frac{(⟨l, τ ⟩(s_{ex}))^{2}}{|l(s_{ex})|^{2}} = (Φ^{(1)}(s_{ex}))^{2}.

This implies that |Φ^{(1)}(s_{ex})| ≥ 0.5 if |k(s_{ex})⟨l, ν⟩(s_{ex})| ≤ 3/4, or |k(s_{ex})⟨l, ν⟩(s_{ex})| ≥ 5/4.

It remains to consider the case where |k(s_{ex})⟨l, ν⟩(s_{ex})| ≃ 1, and therefore |⟨l, ν⟩(s_{ex})| ≤ Ck_{0}^{−1}. Since ⟨l, τ ⟩^{2}+⟨l, ν⟩^{2} = |l|^{2} and |l| ≥ Ck_{0}^{−1}, we have |⟨l, τ ⟩(s_{ex})| ≥ C|l(s_{ex})| in this case, and therefore |Φ^{(1)}(s_{ex})| ≥ C. Using estimates (35) and (37), we find that one can take δ_{c} = Δ_{c} = Ck_{0}^{−1}g^{−1},

g = 1 + \overline{k} + \overline{k}^{2} + \overline{k}^{(1)}. Now estimate (33) follows from (34); moreover,

B_{2}(∂D) = \frac{(1 + L\overline{k})L\overline{k}g^{2}}{k_{0}^{4}} .  (38)

The obtained results can be used to estimate the quantity Δ_{Ω}f(a) (21). In particular, these results imply that under the conditions of Lemma 2, there exist quantities α_{i} > 0 such that if

d ≥ Ω^{−α_{1}},  d_{Γ} ≥ Ω^{−α_{2}},  k_{0} ≥ Ω^{−α_{3}},
L ≤ Ω^{α_{4}},  \overline{k} ≤ Ω^{α_{5}},  \overline{k}^{(1)} ≤ Ω^{α_{6}},  N ≤ Ω^{α_{7}},

(39)

then the following estimate holds:

|Δ_{Ω}f(a)| ≤ \frac{CN}{Ωd} (\frac{1}{d} + \frac{1}{\sqrt{dd_{Γ}}} ).  (40)

It is shown in [8] that for a certain analog of conditions (39), estimate (40) is valid for an arbitrary f ∈ F_{p}. With the aid of the results of [8], we can obtain similar estimates as d_{Γ} → 0, but we shall not do this here.

Let us see how estimates (32) and (33) change for the case of a quasiconvex contour which has one or several plane segments of the boundary. Let a plane segment (a segment of a line) S correspond to the interval s_{1} ≤ s ≤ s_{2}, and let the other part of the contour ∂D \ S be strictly quasiconvex. Then, for the integral over ∂D \ S, estimates (32) and (33) hold true, since when deriving them, we never use the fact that ∂D is a closed curve. Thus, it remains to consider the contribution of the plane segment S, i.e., the integral

J_{S} = ∫_{s_{1}}^{s_{2}}u(s)e^{iΩ|l|} ds.

When estimating the integral J_{S}, we can proceed precisely in the same way as in the proof of Lemma 2. Now the critical points s_{c} exist only if we can draw a normal from a to S. These critical points are nondegenerate, and there are no extremal points s_{ex} for ψ = Φ^{(1)}. Setting δ_{c} = C \frac{d}{1+d}, we find from (34) that

|J_{S}| ≤ \frac{N_{S}}{Ω^{1/2}d^{t−3/2}} + \frac{N_{S} + 1}{Ω}(\frac{1}{d^{t−1}} + \frac{L_{S}}{d^{t}} )(1 + d + \frac{L_{S}}{d} ),  (41)

where N_{S} is the number of normals from a to S and L_{S} is the length of the plane segment S. This estimate is also valid for an arbitrary finite number of plane segments if we understand S as the union of these segments. Thus, for the quasiconvex contour we have

|J(a)| ≤ |J_{∂D\S}| + |J_{S}|,  (42)

and estimates (33) and (34) are valid for J_{∂D\S}, and estimate (41) holds for J_{S}. If a contour consists only of plane segments, i.e., is a polygon, then we must set J_{∂D\S} ≡ 0 in (42).

4. A FOURIER SERIES

We begin the analysis of convergence of the method of summation over disks for a Fourier series with a rough estimation of the quantity. We obtain from (25) that

|Δ_{0}f(a)| ≤ \frac{C}{ΔΩΩ^{1/2}} \frac{L}{d^{5/2}} ,  |l| ≥ d.  (43)

Let r_{0} ∈ [−π, π)^{2}; we introduce the notation

a_{n} = r_{0} + 2πn, l_{n} = r(s) − a_{n}, n ∈ ℤ^{2},
|l_{n}| ≥ d_{n} ≡ d(a_{n}).

Then we can take

d_{n} = \{ \begin{matrix}d_{0} = d, & n = 0, 1 \\ π|n|, & |n| ≥ 2. \end{matrix}  (44)

Recall that d_{0} ≡ d is the distance from r_{0} to ∂D, and we assume that Ωd > 1.

It follows from (43) and (44) that the series

∑_{n∈ℤ^{2}} Δ_{0}f(a_{n})

absolutely converges. Thus, for N = 2, it suffices to consider the o-summation (23). We assume below that on each nonplane arc belonging to ∂D the following estimates hold:

|k^{(p)}(s)| ≤ C_{p}\overline{k}b^{p},  b ≥ \overline{k},  |k(s)| ≥ k̃ (|s − s_{i}^{0}| ≥ C_{i}K_{i}(\overline{k}b^{p_{i}+1})^{−1}),

and in the notation C(∂D) we have C(∂D) ≡ C(p_{i}, K_{i}, K_{j}^{0}, \overline{k}, k̃, k_{0}, L). In the case being considered, inequality (14) can be written as

|Δ_{Ω}f^{T} (r_{0})| ≤ |Δ_{0}f(r_{0})| + |Δ_{0Ω}f^{T} (r_{0})|,  (45)

and it remains to estimate the quantity Δ_{AΩ}f(r_{0}), (11). To this end, we use estimate (12), which does not depend on the summation method, for N = 2 and Ω′ = Ω + ΔΩ.

Lemma 3. Let f = χ_{D} ∈ F_{p}. Then the following assertions hold for Ω ≥ C(∂D):

(1) if C_{1} ≤ ΔΩ ≤ C_{2}Ω (say, \frac{1}{2} ≤ ΔΩ ≤ \frac{1}{2}Ω), then

∑_{Ω<|n|≤Ω+ΔΩ} |f̂(n)| ≤ \frac{C(∂D)ΔΩ}{Ω^{1/2}}  (46)

and, in particular,

|Δ_{0Ω}f^{T} (r_{0})| ≤ \frac{C(∂D)ΔΩ}{Ω^{1/2}};  (47)
(2) if C_{1}Ω^{−4/11} ≤ ΔΩ ≤ C_{2}Ω and the constants γ_{0} and γ_{i} are arbitrary but such that

0 ≤ γ_{0} ≤ 1, 0 ≤ γ_{i} ≤ \frac{p_{i} + 1}{p_{i} + 2},

then the following estimate holds:

∑_{Ω<|n|≤Ω+ΔΩ} |f̂(n)| ≤ C(∂D)\{\frac{ΔΩ}{Ω^{1/2}} + ∑_{i}[\frac{ΔΩ}{Ω^{1/2−γ_{i}p_{i}/(2(p_{i}+1))}} + \frac{1}{Ω^{γ_{i}+1/(p_{i}+2)}} ]

+ L_{S}(Ω^{−1+γ_{0}}ΔΩ + Ω^{−γ_{0}})},  (48)

where L_{S} is the total length of plane segments of the boundary ∂D and the summation over i is the summation over planar points s_{i}^{0} of order p_{i}, (15).

Proof. We begin with the proof of assertion (1). We denote by □_{n} a closed square with side 1 and the center at the point n ∈ ℤ^{2}. Let Q ⊂ ℝ^{2} be a closed finite domain, and let

Q_{□} = ⋃_{n∈Q} □_{n}.

We denote by |r, M| and |M_{1}, M_{2}| the Euclidean distance from the point r to the set M and the distance between two closed subsets M_{1} and M_{2}, respectively. Let the set Q_{1} ⊃ Q consist of points lying on the distance ≤ 1 from the set Q; thus, |r, Q| ≤ 1 for r ∈ Q_{1} \ Q. Let g_{f}(ω) be any function such that

g_{f}(ω) ≥ |f̂(n)|  ∀ω ∈ □_{n}, n ∈ Q_{1}.  (49)

Then

∑_{n∈Q} |f̂(n)| ≤ ∫_{Q_{1}} g_{f}(ω)| dω.  (50)

Indeed,

∑_{n∈Q} |f̂(n)| = ∑_{n∈Q} ∫_{□_{n}} |f̂(n)| dω ≤ ∑_{n∈Q} ∫_{□_{n}} g_{f}(ω) dω = ∫_{Q_{□}} g_{f}(ω) dω ≤ ∫_{Q_{1}} g_{f}(ω) dω,

and the last inequality in this expression follows from the fact that Q_{□} ⊂ Q_{1} since |ω, ∂Q| ≤ 2^{−1/2} if ω ∈ Q_{□}, ω ∈ Q. Thus,

∑_{Ω<|n|≤Ω+ΔΩ} |f̂(n)| ≤  ∫_{||ω|−Ω|≤C(1+ΔΩ)} g_{f}(ω) dω,  (51)

and we have to find a function g_{f} satisfying (49). Actually, the question is a sufficiently good upper

bound for the Fourier transform f̂(ω). All necessary estimates of the quantity f̂(ω) are contained in [8]; they are based on representation (18). In particular, it follows from the results of [8] that for a strictly quasiconvex contour we can take

g_{f}(ω) = \frac{C}{Ω^{3/2}k_{0}^{1/2}}  if |ω| ≥ 4(\overline{k}^{2} + \frac{\overline{k}^{(1)}}{k_{0}^{3}} ).  (52)

Assume that there are plane points s_{i}^{0} on the contour ∂D and planar segments of the boundary. Below by the angle of a line we mean the angle φ in its equation ⟨r, η⟩ = q, η = (cos φ, sin φ). Let φ_{0α} be an angle of the line constituting a plane segment of the boundary for which s_{1α} ≤ s ≤ s_{2α}, s_{2α} − s_{1α} = L_{α}. The contribution of this segment to integral (18) can be easily estimated:

|−\frac{i}{λ} ∫_{s_{1α}}^{s_{2α}}e^{iλ⟨r,η⟩}⟨ν, η⟩ ds| ≤ \{ \begin{matrix}\frac{CL_{α}}{λ} , & |φ − φ_{0α}| & ≤ \frac{C}{λL_{α}} \\ \frac{C}{λ^{2}|φ − φ_{0α}|}, & |φ − φ_{0α}| & ≥ \frac{C}{λL_{α}} . \end{matrix}  (53)

Let φ_{i}^{0} be angles of tangents at planar points. It follows from (53) and the results of [8, 13] that for λ ≥ C(∂D) the function g_{f} can be defined by the relation

g_{f}(ω) = \frac{C(∂D)}{λ^{3/2}} ∑_{i} \frac{1}{|φ − φ_{i}^{0}|^{p_{i}/(2(p_{i}+1))}} + \{ \begin{matrix}∑_{α} \frac{C}{λ^{2}|φ − φ_{0α}|}, & |φ − φ_{0α}| & ≥ \frac{C}{λL_{α}} \\ ∑_{α} \frac{CL_{α}}{λ} , & |φ − φ_{0α}| & ≤ \frac{C}{λL_{α}} . \end{matrix}  (54)

The required result (46) now follows directly from (50).

We pass now to the proof of assertion (2). We introduce the notation

Δ_{i}^{0} = C_{i}(∂D)λ^{−(p_{i}+1)/(p_{i}+2)}, Δ_{α}^{0} = C(λL_{α})^{−1}.

It follows from the results of [8] and estimate (53) that for λ ≥ C(∂D) and f = χ_{D} ∈ F_{p}, the

following inequalities hold for the Fourier transform f̂(ω), (1):

(1) if |φ − φ_{i}^{0}| ≥ Δ_{i}^{0}, |φ − φ_{0α}| ≥ Δ_{α}^{0} ∀α, i, then
|f̂(λη)| ≤ \frac{C(∂D)}{|λ|^{3/2}}[∑_{i} \frac{1}{|φ − φ_{i}^{0}|^{p_{i}/(2(p_{i}+1))}} + ∑_{α} \frac{1}{λ^{1/2}|φ − φ_{0α}|}];  (55)

PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS  Vol. 218  1997

(2) if |φ − φ_{j}^{0}| ≤ Δ_{j}^{0}, then

|f̂(λη)| ≤ \frac{C(∂D)}{λ^{1+1/(p_{j}+2)}} ;  (56)

(3) if |φ − φ_{0α}| ≤ Δ_{α}^{0}, then

|f̂(λη)| ≤ CL_{α}λ^{−1}.  (57)

We denote by M the domain Ω ≤ |ω| ≤ ω + ΔΩ (ω ∈ ℝ^{2}) which is of interest to us. We isolate the sectors |φ − φ_{i}^{0}| ≤ Δ_{i}, |φ − φ_{0α}| ≤ Δ_{0α} and assume that Δ_{i} ≥ Δ_{i}^{0}, Δ_{0α} ≥ Δ_{α}^{0}. We denote the intersection of these sectors with M by M_{i} and M_{0α}, respectively,

M = M′ ∪ (⋃_{i} M_{i}) ∪ (⋃_{α} M_{0α}), M′ = {ω ∈ M | |φ − φ_{i}^{0}| > Δ_{i}, |φ − φ_{0α}| > Δ_{0α}}.

The following estimate will be used as an assumption estimate:

∑_{n∈M} |f̂(n)| ≤ max_{n∈M} |f̂(n)| × N(M′) + ∑_{i} max_{n∈M_{i}} |f̂(n)| × N(M_{i}) + ∑_{α} max_{n∈M_{α}} |f̂(n)| × N(M_{0α}), (58)

where N(Q) is the number of integer points in the domain Q. Let S(Q) be the area of Q, and let L(Q) be the length of the boundary ∂Q. We shall prove that

N(Q) ≤ S(Q) + C(L(Q) + 1).  (59)

We introduce the set

Q′ = ⋃_{n∈Q} □_{n} ⊂ Q,

where the union is taken over all integer points n∈Q such that □_{n} ⊂Q. Then N(Q′)=S(Q′)≤S(Q), and it remains to consider the number of integral points in the domain Q \ Q′. If r ∈ Q \ Q′, then |r, ∂Q| ≤ 2^{−1/2}, since otherwise r ∈ Q′. We choose a certain point r_{0} ∈ ∂Q; traversing the contour ∂Q in a certain direction, we put the points r_{1}, r_{2}, …, r_{i}, … on it in such a way that the distance from r_{i} to r_{i+1} along ∂Q be equal to 1. Then the number of these points ≤ L(Q)+1. If Π_{i} is a disk of radius 1 centered at r_{i}, then the part ∂Q_{i} of the boundary lying between the points r_{i} and r_{i+1} lies entirely in the intersection Π_{i} ∩ Π_{i+1}. Hence, if we take the disks Π̃_{i} of radius 1 + 2^{−1/2} instead of Π_{i}, then n ∈ Q \ Q′, whence it follows that n ∈ ⋃_{i} Π̃_{i}. There are not more than nine integral

points in each disk Π̃_{i}, and therefore N(Q \ Q′) ≤ 9(L(Q) + 1). Thus, estimate (59) is proved with constant C = 9.

In (58), we use the estimates

\begin{matrix}N(M′) ≤ N(M) = N(Ω + ΔΩ) − N(Ω), \\ N(M_{i}) ≤ C(S(M_{i}) + L(M_{i}) + 1),  N(M_{0α}) ≤ C(S(M_{0α}) + L(M_{0α}) + 1); \end{matrix}  (60)

in the first of which N(Ω) is the number of integral points in the disk of radius Ω. The following estimate is obtained in [15]:

N(Ω) = πQ^{2} + ΔN(Ω), |ΔN(Ω)| ≤ CΩ^{−7/11},

from which it follows immediately that

N(Ω + ΔΩ) − N(Ω) ≤ CΩΔΩ  (61)

for CΩ^{−4/11} ≤ ΔΩ ≤ CΩ, and thus, for CΩ^{−4/11} ≤ ΔΩ ≤ CΩ, we find from (58), (60), and (61) that

∑_{Ω<|n|≤Ω+ΔΩ} |f̂(n)| ≤ C_{1} max_{n∈M′} |f̂(n)|ΩΔΩ + C_{2} ∑_{i} max_{n∈M_{i}} |f̂(n)|(S(M_{i}) + L(M_{i}) + 1)

+ C_{3} ∑_{α} max_{n∈M_{0α}} |f̂(n)|(S(M_{0α}) + L(M_{0α}) + 1).  (62)

We take Δ_{i} = C_{i}Ω^{−γ_{i}}, Δ_{0α} = CΩ^{−γ_{0}} and assume that |φ_{i}^{0} − φ_{j}^{0}| ≥ Δ_{i}, |φ_{i}^{0} − φ_{0α}| ≥ Δ_{i}, |φ_{0α} − φ_{0β}| ≥ Δ_{0α}. The estimates

max_{n∈M′} |f̂(n)| ≤ C(∂D)[Ω^{−3/2} ∑_{i} Ω^{γ_{i}p_{i}/(2(p_{i}+1))} + Ω^{−2+γ_{0}}],

max_{n∈M_{i}} |f̂(n)| ≤ \frac{C(∂D)}{Ω^{1+1/(p_{i}+2)}} ,  max_{n∈M_{0α}} |f̂(n)| ≤ \frac{CL_{α}}{Ω}

(63)

follow from (55)–(57). Note that

S(M_{i}) ≤ CΔ_{i}ΩΔΩ ≤ CΩ^{1−γ_{i}}ΔΩ,  L(M_{i}) ≤ C(Δ_{i}Ω + ΔΩ) ≤ C(Ω^{1−γ_{i}} + ΔΩ)  (64)

and similar estimates for S(M_{0α}) and L(M_{0α}) are obtained by the substitution of γ_{0} for γ_{i}. Substituting (63) and (64) into (62), we obtain (48).

Let us formulate and prove the fundamental theorem.

Theorem 2. We have σ^{T} (F_{p}) = 1/2.

Proof. It follows from (45), with due account of (43) and (47), that for f = χ_{D} ∈ F_{p} and ΔΩ ∼ 1,

|Δ_{Ω}f^{T} (r_{0})| ≤ \frac{C(∂D)}{Ω^{1/2}d^{5/2}} ,  Ω ≥ C(∂D)  (65)

and therefore,

σ^{T} (f|r_{0}) ≥ \frac{1}{2},  f ∈ F_{p}.  (66)

It remains to prove that there exist a contour ∂D and a point r_{0} such that the sign of equality is realized in (66). We can take a circle of radius R < π (say, R = 1) centered at the origin as such a contour and set r_{0} = 0. To prove this, we note that in the case of a circle of radius R, it follows from (52) that

|Δ_{0Ω}f^{T} (r_{0})| ≤ \frac{C(R)ΔΩ}{Ω^{1/2}},  CΩ^{−4/11} ≤ ΔΩ ≤ CΩ.  (67)

Let f = χ_{D}, D be a disk centered at the origin, and let r_{0} = 0. Relation (8) implies the equality

Δ_{0}f(0) = ∑_{n∈ℤ^{2}} Δ_{0}f(a_{n}), a_{n} = 2πn.

Since we can assume that d_{n} ≥ Ck_{0}^{−1} (k_{0} = R^{−1}) for |n| ≥ 1, we obtain from (43) and (44) the estimate

|Δ_{0}f(a_{n})| ≤ \frac{C}{|n|^{5/2}ΩΔΩ}, |n| ≠ 0.

On the other hand (see (25)),

Δ_{0}f^{T} (0) = Im \frac{A_{0}}{ΔΩΩ^{1/2}} × \frac{e^{iΩR}}{R^{5/2}} (\frac{e^{iRΔΩ}}{\sqrt{1 + ΔΩ/Ω}} − 1) + Δ_{0}^{1}(0),

A_{0} = \frac{1}{\sqrt{2}π^{3/2}} (−1 + i), |Δ_{0}^{1}(0)| ≤ \frac{C}{Ω^{3/2}R^{7/2}ΔΩ}.

This implies that for any fixed R, there exists a sequence of values Ω_{j}, Ω_{j} → ∞, j → ∞, such that

|Δ_{0}f^{T} (0)| ≥ \frac{A}{Ω_{j}^{1/2}ΔΩ}

for Ω = Ω_{j}. Taking into account estimate (67) and choosing ΔΩ, for example, from the condition (ΔΩ)^{2} ≤ 10^{−1}AC(R), we find that |Δ_{Ω}f^{T} (0)| ≥ CΩ^{−1/2} (Ω = Ω_{j}).

Estimate (65) can be improved with the use of various assumptions concerning the shape of the contour ∂D and the position of the point r_{0}. The roughness of estimates of the quantity Δ_{AΩ}f given by Lemma 3 is the main obstacle for obtaining of exact results.

Let us formulate one of the possible results. To this end, we define the set Γ̃ ∈ [−π, π]^{2}. Let Γ = Γ(∂D) be the evolute of the curve ∂D. The evolute Γ defines the curve Γ′ on the torus T^{2}, which is regarded as the square [−π, π]^{2} with identified sides. Unrolling the torus T^{2} into a square, we obtain the set Γ̃ from Γ′. For the quasiconvex curve ∂D this set consists of a finite number of arcs. This is due to the fact that the diameter of the evolute Γ ≤ Ck_{0}^{−1}. If there exist planar

points, the set Γ̃ can everywhere densely cover the square [−π, π]^{2}.

Lemma 4. Let f = χ_{D} ∈ F_{p}, p ≥ 0, ∂D be a strictly quasiconvex contour, and let |r_{0}−aΓ̃| ≥ dΓ̃

∀aΓ̃ ∈ Γ̃. Then there exist numbers α_{1} > 0 and α_{2} > 0 such that for d ≥ Ω^{−α_{1}}, dΓ̃ ≥ Ω^{−α_{2}}, the following estimate holds:

|Δ_{Ω}f^{T} (r_{0})| ≤ \frac{C(∂D)}{Ω^{3/4}d}(\frac{1}{\sqrt{ddΓ̃}} + \frac{1}{d}), d ≡ |r_{0}, ∂D|.

Proof. We shall again proceed from estimate (45). Applying Lemma 2 and taking into account (25) and (44), we find from (13) that under our assumptions

|Δ_{0}f^{T} (r_{0})| ≤ \frac{C(∂D)}{ΔΩΩd}(\frac{1}{\sqrt{ddΓ̃}} + \frac{1}{d}).  (68)

To estimate the quantity Δ_{0Ω}f^{T} (r_{0}), it remains to apply the analog of estimate (67) and set ΔΩ = Ω^{−1/4}.

For the quasiconvex contour ∂D with plane segments, the estimate of J_{S} (41) implies that it is necessary to consider the q-summation with q ≥ 1 in order to use Lemma 1, and it is sufficient to consider the case q = 1. Then, instead of (68), we obtain

|Δ_{1}f^{T} (r_{0})| ≤ C(∂D, d, dΓ̃)(ΔΩ)^{−3}Ω^{−1}, r_{0} ∉ ∂D ∪ Γ̃.

In this case, estimate (48) yields

|Δ_{1Ω}f^{T} (r_{0})| ≤ C(∂D)[\frac{ΔΩ}{Ω^{1/2}} + L_{S}(Ω^{−1+γ_{0}}ΔΩ + Ω^{−γ_{0}})].

Choosing ΔΩ = Ω^{−1/8}, γ_{0} = 9/16, we find

|Δ_{Ω}f^{T} (r_{0})| ≤ \frac{C(∂D, d, dΓ̃)}{Ω^{9/16}}, r_{0} ∉ ∂D ∪ Γ̃,

for a quasiconvex contour. Of course, in this estimate, C(∂D, d, dΓ̃) → ∞ as d → 0 or dΓ̃ → 0.

As one more example, we consider the case where ∂D is a polygon (not necessarily convex). In this case, we can reject the first term on the right-hand side of expression (54) for g_{f}(ω), and then, for ΔΩ ≥ C, (51) implies that

∑_{Ω<|n|≤Ω+ΔΩ} |f̂(n)| ≤ CΩ^{−1}.

Using estimate (41) for J_{S} and choosing q = 1, we obtain

|Δ_{1}f^{T} (r_{0})| ≤ \frac{C(∂D)}{Ωd^{5}}, Ω ≥ C(∂D),

and, thus, if f is a characteristic function of the polygon, then

|Δ_{Ω}f^{T} (r_{0})| ≤ \frac{C}{Ωd^{5}} , Ω ≥ C(∂D).

We can also slightly improve estimate (65) for general position contours. In this case, p_{i} =1, (15), and j ≤ 2, (28). Let f = χ_{D}, and let the boundary ∂D consist of plane segments and arcs of the general position. In this case, the estimation of the quantity Δ_{Ω}f^{T} is performed in the same way as above. The required estimate of integral (26), when the point a is near the evolute, is obtained with the aid of the results of [8], and for d ≥ Ω^{−α} we obtain

|Δ_{Ω}f^{T} (r_{0})| ≤ C(∂D)Ω^{−29/54};

if d ≥ Ω^{−α_{1}}, |r_{0}, Γ̃_{0}| ≥ Ω^{−α_{2}}, then

|Δ_{Ω}f^{T} (r_{0})| ≤ C(∂D)Ω^{−31/57}d^{−9/2},

where Γ̃_{0} is the image, under the mapping Γ → Γ̃, of the set Γ_{0} of the singular points of the evolute Γ.

5. CONCLUDING REMARKS

The results obtained above refer to the two-dimensional case. Note that in a one-dimensional case, the proposed approach is also applicable and yields the estimates

|Δ_{Ω}f(x)| ≤ \frac{C}{Ωd}, |Δ_{Ω}f^{T} (x)| ≤ \frac{C}{Ωd^{2}} .

Naturally, this approach is also applicable for analyzing the general spherically symmetric summation methods. In particular, it follows immediately from the results of [8] concerning the estimation of the quantity Δ_{A}f(r_{0}) (f = χ_{D} ∈ F_{p}) that for any M there exists a summation method such that |Δ_{A}f^{T} (r_{0})| ≤ C(d, ∂D)Ω^{−M}. The greatest difficulty for summation is precisely the summation over disks (Ω-summation). As was already mentioned, the main obstacle to the obtaining of exact results concerning the rate of convergence of the Ω-summation for the Fourier series is the absence of good estimates for the quantity Δ_{AΩ}f^{T} (11). If |Δ_{qΩ}f| ≤ CΩ^{−1−ε} for ΔΩ ∼ 1, then (for N = 2) we can obtain an analog of Theorem 1 for the Fourier series.

The obtained results can be regarded in the framework of the following general problem: with what accuracy can one reconstruct the characteristic function χ_{D} (D ∈ ℝ^{2}) for d > Ω^{−α} if ∼ Ω^{2} functionals of this function are known? It was shown above that with the use of ∼ Ω^{2} Fourier coefficients even using the Ω-summation, one can reconstruct χ_{D} (D is a general position domain) with an accuracy of Ω^{−29/54}. On the other hand, the maximal accuracy of reconstruction of this domain with the use of ∼ Ω^{2} values of the Radon transform does not exceed Ω^{−1/3} or Ω^{−1/2} at the point of the general position [7].

Consider the following question: to what extent the obtained results can be extended to the three-dimensional case (N = 3)? For N = 3, we can also consider the class F_{p} of characteristic functions of domains with a piecewise-smooth boundary. By the results of [1], σ(F_{p}) = σ^{T} (F_{p}) ≤ 0 (the Fourier series diverges at the center of the ball), and it is necessary to consider the quantities of the type σ(f|r_{0}), σ^{T} (f|r_{0}), r_{0} ∈ ℝ^{3}, f = χ_{D} ∈ F_{p}. For N = 3, we have again the Gauss formula

χ_{D}(a) = \frac{1}{4π} ∫_{∂D} dS \frac{⟨l, ν⟩}{|l|^{3}}

and analogs of formulas (18) and (19),

f̂(λη) = −\frac{i}{λ} ∫_{∂D} dS ⟨l, ν⟩e^{iλ⟨r,η⟩}, r ∈ ∂D, ω = λη, λ = |ω|,

Δ_{A}f(a) = \frac{1}{2π^{2}} ∫_{∂D} dS \frac{⟨l, ν⟩}{|l|^{3}} ∑_{p} Δ_{p}G sin Ωξ_{p}|l| + \frac{1}{2π^{2}} ∫_{∂D} dS \frac{⟨l, ν⟩}{|l|^{3}} ∫_{0}^{∞} \frac{G(ξ) + ξG^{(1)} − 1}{ξ} sin Ωξ|l| dξ,

(69)

(l = r − a, r ∈ ∂D, dS is an element of the surface area of ∂D).

With the aid of these formulas, the problem of estimation of the quantities σ(f|r_{0}) and σ^{T} (f|r_{0}) reduces to the problem of estimation of two-dimensional OI over the boundary ∂D with phases ⟨r, η⟩ and |l|. Using the results from the singularity theory, [16], we can prove the analog of Theorem 1 (at least for general position surfaces). The enveloping surface of the family of normals to ∂D — Γ(∂D) plays the role of the evolute in this case.

As to the Fourier series, in order to apply formula (8) for N = 3, it suffices to consider the q-summation (23) for q = 1. A preliminary analysis on the basis of formulas (69) shows us that at

least for convex domains D ⊂ ℝ^{3}, the method of summation over disks converges for r_{0} ∉ Γ̃, and the set Γ̃ ⊂ [−π, π]^{3} can be constructed from Γ precisely in the same way as in the two-dimensional case.

ACKNOWLEDGMENTS

The author expresses his gratitude to M. I. D’yachenko and P. L. Ul’yanov for useful discussions, and M. I. D’yachenko for the presented opportunity to read paper [6] before its publication.

This work was partly supported by the Russian Foundation for Fundamental Research (project 96-01-00767).

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Translated by S. Vakhromeev