Applied and Computational Harmonic Analysis 11, 439–454 (2001) doi:10.1006/acha.2001.0362, available online at http://www.idealibrary.com on Approximation of the Singularities of a Bounded Function by the Partial Sums of Its Differentiated Fourier Series George Kvernadze Department of Mathematics, Weber State University, Ogden, Utah 84408 E-mail: [email protected] Communicated by Leslie F. Greengard Received February 12, 2001; revised May 31, 2001 In our earlier work we developed an algorithm for approximating the locations of discontinuities and the magnitudes of jumps of a bounded function by means of its truncated Fourier series. The algorithm is based on some asymptotic expansion formulas. In the present paper we give proofs for those formulas. 2001 Academic Press Key Words: Fourier series; approximation of singularities. INTRODUCTION The problem of locating the discontinuities of a function by means of its truncated Fourier series, an interesting question in and of itself, arises naturally from an attempt to overcome the Gibbs phenomenon: the poor approximative properties of the Fourier partial sums of a discontinuous function. In [3], Cai et al. developed the idea already introduced in their previous papers and suggested a method for the reconstruction of a discontinuous function from the partial sums of its Fourier series. A key step of the method is the accurate approximation of the locations of singularities and the magnitudes of jumps of the function. Namely, let g be a2π-periodic function with a finite number, M, of jump discontinuities that is piecewise smooth on the period. In addition, let us assume that the first 2n + 1 Fourier coefficients of the function are known. If G(θ) = (π − θ)/2, θ ∈ (0, 2π),isthe2π-periodic sawtooth function, then the function g can be represented as − 1 M1 g(θ) = [g] G(θ − θ (g)) + g (θ), (1) π m m c m=0 where θm(g) and [g]m, m = 0, 1,...,M − 1, are the locations of discontinuities and the associated jumps of the function g,andgc is a 2π periodic continuous function, which is piecewise smooth on [−π,π]. 439 1063-5203/01 $35.00 Copyright 2001 by Academic Press All rights of reproduction in any form reserved. 440 GEORGE KVERNADZE Thus, the problem is to find a good approximation to the constants θm(g) and [g]m, m = 0, 1,...,M − 1, given the first 2n + 1 Fourier coefficients of the function g.Then gc could be recovered from the partial sums of its Fourier series using identity (1) and the undesirable Gibbs phenomenon could be avoided. Eckhoff [5, 6] utilized Prony’s method for the approximation of the constants θm(g).As a result, he developed an efficient method of approximating the locations of singularities and the jumps of a piecewise smooth function with multiple discontinuities. The approximations are found as the solution of a system of algebraic equations. Later, Bauer [2] introduced the idea of band-pass filters to find the discontinuity locations. He utilizes a global filter to find a number of discontinuities and their approximate locations. Then a local subcell filter is used to refine the accuracy of the singularity locations. Another approach to the recovery of a piecewise smooth function was suggested by Geer and Banerjee (see [1, 8, 9]). The authors introduced a family of periodic functions with “built-in” discontinuities to reconstruct a piecewise smooth function with exponential accuracy. The main assumption of the method is knowledge of the jumps and the locations of discontinuities of a given function. To find these, the authors suggested the following: use the well-known formula of symmetric difference of the partial sums of Fourier series that determines the jumps of a bounded function to obtain a first estimate for the location of discontinuities; then utilize a modified least-squares method to improve the accuracy of approximation. Recently Gelb and Tadmor [10] utilized the generalized conjugate partial sums of the Fourier series to detect the jump discontinuities of a piecewise smooth function. They introduced so called “concentration factors” in order to improve the convergence rate. It should be mentioned that one of the families of concentration factors they considered corresponds to a differentiated Fourier series. In our earlier work (see [15, 16]) we developed an algorithm for approximating the locations of discontinuities and the magnitudes of jumps of a bounded function. The algorithm is based on special asymptotic expansion formulas. In the present paper we give proofs for those formulas. PRELIMINARIES Throughout this paper we use the following general notations: N, Z+, Z,andR are the sets of positive integers, nonnegative integers, integers, and real numbers, respectively. L[a,b] is the space of integrable functions on [a,b].ByCq [a,b], q ∈ Z+, we denote the space of q-times continuously differentiable functions on [a,b],whereC0[a,b]≡C[a,b] −1 is the space of continuous functions with uniform norm ·[a,b].ByC [a,b] we denote the space of functions on [a,b] that may have only jump discontinuities and are normalized by the condition g(θ) = (g(θ+) + g(θ−))/2, θ ∈ (a, b). (Here, and elsewhere, g(θ+) and g(θ−) mean the right- and left-hand-side limits of a function g at a point θ, respectively.) g(−r), r ∈ N, is defined as follows: for any g ∈ L[−π,π], g(−r)(τ) ≡ g(−r+1)(τ) dτ, SINGULARITIES OF A BOUNDED FUNCTION 441 where g(0) ≡ g, and the constants of integration are successively determined by the condition π − g( r)(τ) dτ = 0,r∈ N. −π By K we denote constants, possibly depending on some fixed parameters and in general distinct in different formulas. Sometimes the important arguments of K will be written explicitly in the expressions for it. For positive quantities An and Bn, possibly depending on some other variables as well, we write An = o(Bn), An = O(Bn),orAn Bn,if = ∞ ∈ limn→∞ An/Bn 0, supn∈N An/Bn < ,orK1 <An/Bn <K2, n N, respectively, where K1 > 0andK2 > 0 are some absolute constants. All functions below are assumed to be 2π-periodic with the obvious exceptions. If g ∈ L[−π,π],theng has a Fourier series with respect to the trigonometric system { }∞ 1, cosnθ, sin nθ n=1, and we denote the nth partial sum of the Fourier series of g by Sn(g;·);i.e., n π a0(g) 1 Sn(g; θ)= + (ak(g) coskθ + bk(g) sin kθ)= g(τ)Dn(τ − θ)dτ, 2 π − k=1 π where 1 π 1 π ak(g) = g(τ)coskτdτ and bk(g) = g(τ)sin kτdτ π −π π −π are the kth Fourier coefficients of the function g,and + 1 sin(n )θ n 2 for θ/∈ 2πZ, = 1 + = 2sin θ Dn(θ) coskθ 2 (2) 2 = 1 k 1 n + for θ ∈ 2πZ 2 is the Dirichlet kernel. ˜ By Sn(g;·) we denote the nth partial sum of the series conjugate to the Fourier series; i.e., n π ˜ 1 ˜ Sn(g; θ)= (ak(g) sin kθ − bk(g) coskθ)=− g(τ)Dn(τ − θ)dτ, π − k=1 π where n ˜ Dn(θ) = sin kθ k=1 is the kernel conjugate to the Dirichlet kernel. Correspondingly, by g˜ we denote the conjugate function; i.e., π + − − ˜ = − 1 g(θ τ) g(θ τ) g(θ) lim τ dτ , h→ 0 π h 2tan 2 which exists and is finite almost everywhere for any g ∈ L[−π,π] (cf. [13, Theorem, p. 79]). 442 GEORGE KVERNADZE = ∞ DEFINITION [21]. Let # (λk)k=1 be a nondecreasing sequence of positive numbers ∞ =∞ [ ] such that k=1 1/λk . A function g is said to have #-bounded variation on a,b , i.e., g ∈ #BV [a,b],if n |g(x ) − g(x − )| sup 2k 2k 1 < ∞, λk & k=1 where & is an arbitrary system of disjoint intervals (x2k−1,x2k) ⊂[a,b]. If λk = 1, k ∈ N,then#BV [a,b] coincides with the Jordan class V [a,b] of functions of bounded variation. We say that a function g is of harmonic bounded variation on [a,b], i.e., g ∈ HBV[a,b],ifλk = k, k ∈ N. If there is no ambiguity, we will usually suppress the dependence on the domain and simply write C, #BV ,etc. In what follows we need the following additional notations. If γ ∈ R is fixed, then for the sawtooth function G we set G(γ ; θ)≡ G(θ − γ), θ ∈ R. It is trivial to check that + − − S(r 1)(G( q)(γ ;·); θ)= Dˆ (r q)(θ − γ) n n n − (r − q)π = kr q cos k(θ − γ)+ (3) 2 k=1 and ˜(r+1) (−q) ;· ; = ˜ (r−q) − Sn (G (γ ) θ) Dn (θ γ) n − (r − q)π = kr q sin k(θ − γ)+ (4) 2 k=1 ∈ + ∈ ˆ ≡ − 1 for r, q Z and γ R,whereDn(τ) Dn(τ) 2 . By M ≡ M(g) we denote the number of discontinuities of the function g ∈ C−1.By θm ≡ θm(g) and [g]m ≡ g(θm+) − g(θm−), m = 0, 1,...,M− 1, we denote the points of discontinuity and the associated jumps of a function g ∈ C−1. For a fixed r ∈ N and g ∈ C−1,weset − rπ (−1)(r 1)/2S(r)(g;·), if r is odd, DS (r; g;·) ≡ n (5) n r − (r/2)−1 ˜(r) ;· n ( 1) Sn (g ), if r is even. For a fixed r ∈ N and M ∈ N, the points θm(r; g; n), m = 0, 1,...,M − 1, are defined via the condition |DSn(r; g; θm(r; g; n))|=max{|DSn(r; g; θ)| : θ ∈[θm − *m(g), θm + *m(g)]}, (6) = 1 {| − | = − = } where *m(g) 3 min θm K mod2π : k 0, 1,...,M 1andk m .