We propose a generalized Papoulis-Gerchberg algorithm for the recovery of gaps in general analytic functions. The continuous-time algorithm is based on signal expansions in terms of the Chebyshev polynomials of the first kind, and the discrete-time implementation is based on a suitable nonuniform sampling scheme and the discrete cosine transform.

1. Introduction

Gapped (or missing) data is encountered when it is hard to obtain contiguous measurements of a function for a long time. The recovery of the missing function values is possible, at least in theory, given a certain a priori knowledge about the function, for example, analyticity on a certain interval. A real-valued function 𝑓, defined on a closed interval 𝐼 on the real line, is called analytic on 𝐼, if there exists an analytic extension of 𝑓 onto some open set 𝐺 of the complex plane ℂ that contains 𝐼 [1]. That is, there is a unique single-valued analytic function, defined on 𝐺, that coincides with 𝑓 on 𝐼. Let 𝑓 be analytic on 𝐼. When the values of 𝑓 on a subinterval of 𝐼 are given, 𝑓 can be determined on 𝐼, by means of analytic continuation.

In many missing data problems, the functions are assumed to be bandlimited. Let 𝐿2(ℝ) denote the Hilbert space of all square integrable functions defined on the real line ℝ. For 𝜏>0, and a function 𝑓∈𝐿2(ℝ), we define the operators 𝑃𝜏 and 𝑄𝜏 as follows:𝑃𝜏𝑄𝑓=𝑓(𝑡),|𝑡|≤𝜏,0,|𝑡|>𝜏,𝜏𝑓=𝑓−𝑃𝜏𝑓.(1) Let ∫ℱ𝑓=𝑓(𝜔)=âˆžâˆ’âˆžğ‘“(𝑡)𝑒−𝑗𝜔𝑡𝑑𝑡, and, ℱ−1𝑓 be the Fourier transform and inverse Fourier transform of 𝑓, respectively. For ğœŽ>0, and a function 𝑓∈𝐿2(ℝ), we define the bandlimiting operator ğ‘ƒğœŽ as follows:ğ‘ƒğœŽğ‘“=ℱ−1ğ¿ğœŽî€œâ„±ğ‘“=âˆžâˆ’âˆžğ‘“(𝑡−𝜏)sinğœŽğœğœ‹ğœğ‘‘ğœ,(2) whereğ¿ğœŽî‚»(𝜔)=1,|𝜔|â‰¤ğœŽ,0,|𝜔|>ğœŽ.(3) The bandlimited signal extrapolation is defined as follows. Find 𝑓 having 𝑔=𝑃𝜏𝑓 and knowing that 𝑓=ğ‘ƒğœŽğ‘“. Various methods were proposed for solving the bandlimited extrapolation problem; [2] contains a detailed comparison of some of the methods. The iterative bandlimited signal extrapolation algorithm of Papoulis and Gerchberg [3, 4] is attractive due to its relative simplicity, requiring only Fourier transform and inverse Fourier transform in each iteration. Based on the identity𝑔+ğ‘„ğœğ‘ƒğœŽğ‘“=𝑃𝜏𝑓+𝑄𝜏𝑓=𝑓,(4) Papoulis [3] and Gerchberg [4] proposed the following algorithm:Initializationstep:𝑓0=𝑔,𝑚=1,2,…:𝑓𝑚=𝑔+ğ‘„ğœğ‘ƒğœŽğ‘“ğ‘šâˆ’1.(5) A proof for the convergence of the algorithm was given by Papoulis [3] using signal expansions in terms of the prolate spheroidal wave functions. Gerchberg [4] actually solved the dual problem of extrapolating the spectrum of a finite object beyond its diffraction limits, leading to superresolution and image enhancement.

A bandlimited function 𝑓(𝑡), defined on the real-line ℝ, is an entire function of exponential type when 𝑡 is extended to ℂ [5]. Hence, bandlimited signals are a very special case of analytic functions, and other analytic signals may be encountered in practice. In this paper, we propose a method for the recovery of gaps in general analytic signals using a generalization of the Papoulis-Gerchberg algorithm, based on signal expansions in terms of the Chebyshev polynomials of the first kind. In Section 2, we briefly review the properties of Chebyshev polynomials. We generalize in Section 3 the Papoulis-Gerchberg algorithm from continuous-time bandlimited signals to continuous-time signals in polynomials spaces and general analytic signals. We then focus, in Section 4, on the discrete implementation of the continuous-time algorithm, based on a suitable nonuniform sampling scheme and the discrete cosine transform. The performance of the recovery algorithm is demonstrated, in Section 6, by a numerical example.

2. Chebyshev Polynomials

The Chebyshev polynomial 𝑇𝑛(𝑥) of the first kind is a polynomial in 𝑥 of degree 𝑛, defined by the relation𝑇𝑛(𝑥)=cos𝑛𝜃when𝑥=cos𝜃,(6) for 𝑥∈[−1,1] [6]. The range of the corresponding variable 𝜃 is [0,𝜋] (these ranges are traversed in opposite directions). Intervals [ğ‘Ž,𝑏] other than [−1,1] are easily handled by the change of variables𝑠=2𝑥−(ğ‘Ž+𝑏).ğ‘âˆ’ğ‘Ž(7) Without loss of generality, we focus hereafter on the interval [−1,1].

Let 𝐿2𝑤([−1,1]) be the Hilbert space of all real-valued square integrable functions on [−1,1] with respect to the nonnegative weight function1𝑤(𝑥)=√1−𝑥2.(8) This is the space of functions 𝑓 such that the norm‖𝑓‖=1−1||||𝑤(𝑥)𝑓(𝑥)2𝑑𝑥1/2(9) is finite. The associated inner product is⟨𝑓,𝑔⟩=1−1𝑤(𝑥)𝑓(𝑥)𝑔(𝑥)𝑑𝑥.(10) The set {𝑇𝑖(𝑥),𝑖=0,1,…} forms a complete orthogonal polynomial system in 𝐿2𝑤([−1,1]). The orthogonality relation is given by [6]⟨𝑇𝑚(𝑥),𝑇𝑛(𝑥)⟩=1−1𝑤(𝑥)𝑇𝑚(𝑥)𝑇𝑛=𝑑(𝑥)𝑑𝑥0,𝑚≠𝑛,𝑚𝜋2,𝑚=𝑛,(11) where𝑑𝑚=2,𝑚=0,1,𝑚≥1.(12) The 𝑁 zeroes of the Chebyshev polynomial 𝑇𝑁(𝑥) are𝑥𝑘=cos(2𝑘+1)𝜋2𝑁,𝑘=0,1,…,𝑁−1.(13) These zeroes are listed in decreasing order in 𝑥. The (𝑁−1)th degree polynomial 𝑝𝑁−1(𝑥), interpolating a function 𝑓(𝑥) in the zeroes of 𝑇𝑁, can be written as a sum of Chebyshev polynomials in the form𝑝𝑁−1(𝑥)=𝑁−1𝑛=0𝑐𝑛𝑇𝑛(𝑥),(14) where the coefficients 𝑐𝑛 in (14) are given by the explicit formula𝑐𝑛=𝛾𝑛𝑁−1𝑘=0𝑓𝑥𝑘𝑇𝑛𝑥𝑘,(15) where𝛾𝑛=⎧⎪⎨⎪⎩1𝑁2,𝑛=0,𝑁,0<𝑛≤𝑁−1.(16) It follows [7–9] that𝑐𝑛=𝛾𝑛𝑁−1𝑘=0𝑓𝑥𝑘cos𝜋(2𝑘+1)𝑛,2𝑁(17) which is the well-known discrete cosine transform (type II) [10] applied on the samples of the function 𝑓(𝑥) taken at a nonuniform sampling grid corresponding to the zeroes of 𝑇𝑁. Efficient algorithms exist for computing the discrete cosine transform (see, e.g., [11] and references therein).

3. Gap Recovery in General Analytic Signals Based on A Chebyshev Polynomial Series Expansion

Let ℙ𝑁−1 denote the subspace of all polynomials of degree <N (𝑁>1) in 𝐿2𝑤([−1,1]). Let 𝑃𝑁−1 denote the orthogonal projection operator onto ℙ𝑁−1𝑃𝑁−1𝑓=𝑁−1𝑘=0𝑓𝑘𝑇𝑘(𝑥),(18) where𝑓𝑘=2𝜋𝑑𝑘⟨𝑓,𝑇𝑘⟩,(19)𝑑𝑘 is defined in formula (12) and 𝑓∈𝐿2𝑤([−1,1]).

Let 𝐶𝑆 be the subspace of all the functions in 𝐿2𝑤([−1,1]) which vanish outside a collection of disjoint closed intervals in [−1,1] and let 𝐶+𝑆 be the orthogonal complement of 𝐶𝑆 in 𝐿2𝑤([−1,1]). We denote by 𝑃𝑆 and 𝑄𝑆 the orthogonal projection operators onto subspace 𝐶𝑆 and 𝐶+𝑆, respectively. It follows that 𝑄𝑆=𝐼𝑑−𝑃𝑆, where 𝐼𝑑 is the identity operator in 𝐿2𝑤([−1,1]). If we assume that 𝑓=𝑃𝑁−1𝑓, then we have:𝑓=𝑃𝑆𝑓+𝑄𝑆𝑓=𝑃𝑆𝑓+𝑄𝑆𝑃𝑁−1𝑓(20) The continuous-time recovery problem can be stated as follows. Find 𝑓 having 𝑔=𝑃𝑆𝑓 and knowing that 𝑓=𝑃𝑁−1𝑓. We propose the following Papoulis-Gerchberg type algorithm for recovering gaps in polynomial signals in 𝐿2𝑤([−1,1]) (PGP algorithm).

The PGP Algorithm
Initializationstep:𝑔=𝑓0=𝑃𝑆𝑓,(21)𝑚=1,2,…:𝑓𝑚=𝑔+𝑄𝑆𝑃𝑁−1𝑓𝑚−1.(22) The iteration process is stopped when ‖𝑓𝑚−𝑓𝑚−1‖≤𝜖, where 𝜖>0 is a user specified threshold. 𝑓𝑚 or 𝑃𝑁−1𝑓𝑚 is taken as the result of the recovery algorithm after 𝑚 iterations.
If 𝑓∈ℙ𝑁−1, then 𝑓=𝑃𝑁−1𝑓, and it follows that 𝑔=𝑓−𝑄𝑆𝑃𝑁−1𝑓.(23) We know from (22) and (23) that 𝑓𝑚−𝑓=𝑄𝑆𝑃𝑁−1𝑓𝑚−1.−𝑓(24) Therefore, ‖𝑓𝑚‖‖𝑄−𝑓‖≤𝑆‖‖‖‖𝑃𝑁−1‖‖‖‖𝑓𝑚−1‖‖≤‖‖𝑓−𝑓𝑚−1‖‖,−𝑓(25) which proves that the PGP algorithm has the property that it reduces the error energy during the iteration process.

Lemma 1. A signal 𝑓∈ℙ𝑁−1 can be uniquely determined from 𝑃𝑆𝑓 by using the PGP algorithm (21), and (22).

Proof. Iterating (22), we obtain 𝑓𝑚=𝑔+𝑄𝑆𝑃𝑁−1𝑓𝑚−1=𝑔+𝑄𝑆𝑃𝑁−1𝑔+𝑄𝑆𝑃𝑁−1𝑓𝑚−2=𝑔+𝑄𝑆𝑃𝑁−1𝑔+𝑄𝑆𝑃𝑁−1𝑄𝑆𝑃𝑁−1𝑔+𝑄𝑆𝑃𝑁−1𝑓𝑚−3=𝑚𝑛=0𝑄𝑆𝑃𝑁−1𝑛𝑔.(26) Substituting (23) in (26), we obtain 𝑓𝑚𝑄=𝑓−𝑆𝑃𝑁−1𝑚𝑓.(27) According to von Neumann's theorem on alternating projections [12], for every 𝑓∈ℙ𝑁−1, limğ‘šâ†’âˆžî€·ğ‘„ğ‘†ğ‘ƒğ‘âˆ’1𝑚𝑓=𝑓𝑐,(28) where 𝑓𝑐 is the projection of 𝑓 onto the closed subspace 𝐶+𝑆∩ℙ𝑁−1. Any 𝑓∈ℙ𝑁−1 is analytic on [−1,1] and can vanish in an interval if and only if it is the zero function. Hence, the subspace 𝐶+𝑆∩ℙ𝑁−1 contains only the zero function. We obtain that 𝑓𝑐=0 and that the PGP converges in norm to the required function 𝑓.

𝑃𝑁−1𝑃𝑆𝑃𝑁−1 is a bounded positive self-adjoint compact operator. Hence, there exists an orthonormal basis of ℙ𝑁−1 consisting of eigenfunctions 𝜙𝑛, 𝑛=0,1,…,𝑁−1, of the operator 𝑃𝑁−1𝑃𝑆𝑃𝑁−1, with corresponding real positive eigenvalues 𝜆𝑛, such that 1>𝜆0>𝜆1>⋯>𝜆𝑁−1>0 [13].

Lemma 2. Let 𝑓=𝜙𝑘 be an eigenfunction of 𝑃𝑁−1𝑃𝑆𝑃𝑁−1 with corresponding eigenvalue 𝜆𝑘, and let 𝑔=𝑃𝑆𝑓 be the given values of 𝑓. Then, 𝑃𝑁−1𝑓𝑚=𝐴𝑚𝜙𝑘where𝐴𝑚=1−1−𝜆𝑘𝑚+1.(29)

Proof. We will prove (29) by induction. It is true for 𝑚=0 that 𝑃𝑁−1𝑓0=𝑃𝑁−1𝑔=𝑃𝑁−1𝑃𝑆𝜙𝑘=𝑃𝑁−1𝑃𝑆𝑃𝑁−1𝜙𝑘=𝜆𝑘𝜙𝑘=𝐴0𝜙𝑘,(30) where we used the identity 𝑃𝑁−1𝜙𝑘=𝜙𝑘 (since 𝜙𝑘∈ℙ𝑁−1).
Suppose that it is true for some 𝑚≥1. By using (22), we obtain 𝑃𝑁−1𝑓𝑚+1=𝑃𝑁−1𝑔+𝑃𝑁−1𝑄𝑆𝑃𝑁−1𝑓𝑚=𝑃𝑁−1𝑃𝑆𝑃𝑁−1𝜙𝑘+𝑃𝑁−1𝐼𝐿2𝑤([−1,1])−𝑃𝑆𝑃𝑁−1𝑓𝑚=𝑃𝑁−1𝑃𝑆𝑃𝑁−1𝜙𝑘+𝑃𝑁−1𝑓𝑚−𝑃𝑁−1𝑃𝑆𝑃𝑁−1𝑓𝑚.(31) Substituting 𝑃𝑁−1𝑃𝑆𝑃𝑁−1𝜙𝑘=𝜆𝑘𝜙𝑘 and using the induction assumption, we obtain 𝑃𝑁−1𝑓𝑚+1=𝜆𝑘𝜙𝑘+𝐴𝑚𝜙𝑘−𝜆𝑘𝐴𝑚𝜙𝑘=𝐴𝑚+1−𝐴𝑚𝜆𝑘𝜙𝑘.(32) In (31) and (32), we used the idempotent property of the orthogonal projection operator 𝑃𝑁−1 (i.e., 𝑃𝑁−1=𝑃𝑁−1𝑃𝑁−1). It is easy to see that 𝐴𝑚+1−𝐴𝑚𝜆𝑘=1−1−𝜆𝑘𝑚+1+1−𝜆𝑘𝑚+1𝜆𝑘=1−1−𝜆𝑘𝑚+2.(33) Hence, 𝑃𝑁−1𝑓𝑚+1=𝐴𝑚+1𝜙𝑘,(34) which completes the proof.

Any 𝑓∈ℙ𝑁−1 can be represented as𝑓=𝑁−1𝑘=0𝛼𝑘𝜙𝑘,(35) where𝛼𝑘=1−1𝑤(𝑥)𝑓(𝑥)𝜙𝑘(𝑥)𝑑𝑥.(36) Using (29) and (35), we obtain that after 𝑚 iterations of the PGP algorithm𝑃𝑁−1𝑓𝑚=𝑁−1𝑘=0𝛼𝑘1−1−𝜆𝑘𝑚+1𝜙𝑘.(37) The energy of error is given by‖‖𝑓−𝑃𝑁−1𝑓𝑚‖‖2=𝑁−1𝑘=0𝛼2𝑘1−𝜆𝑘2(𝑚+1).(38) It follows that the rate of convergence of the PGP algorithm is dependent on the spectral representation of the signal in terms of the eigenfunctions of the operator 𝑃𝑁−1𝑃𝑆𝑃𝑁−1. If the signal contains only significant components which correspond to relatively high eigenvalues, the convergence is fast. If this is not the case, the PGP will be slowly convergent.

The PGP can also be viewed as an example of signal restoration by the method of projections onto convex sets (POCS) [14, 15]. The two convex sets, or constraints, are the affine set 𝑉1, which contains all the functions 𝑓∈𝐿2𝑤([−1,1]) with 𝑔=𝑃𝑆𝑓 and the subspace ℙ𝑁−1. If 𝑓∈𝑉1 but 𝑓∉ℙ𝑁−1, we have two inconsistent convex constraints. The convergence of a POCS algorithm with two inconsistent convex constraints is a well-studied problem [16, 17]. In this case, the PGP algorithm iterates 𝑃𝑁−1𝑓𝑚 converge to the signal ğ‘žâˆˆâ„™ğ‘âˆ’1 closest to 𝑉1, that is, with the minimum distance 𝑑, where 𝑑 is given byğ‘‘î€·ğ‘ž,𝑉1=inf𝑦∈𝑉1â€–ğ‘žâˆ’ğ‘¦â€–.(39) Thus, the PGP algorithm is exact for all 𝑓∈P𝑁−1, and a certain “best polynomial approximation” in the sense of (39) is obtained when 𝑓∉ℙ𝑁−1.

The expansion of analytic function in terms of the Chebyshev polynomials converges rapidly (exponentially). It can be proved that if the function 𝑓(𝑥) can be extended to a function 𝑓(𝑧) analytic on the ellipse √{𝑧∶|𝑧+𝑧2−1|=𝑟}, where 𝑟>1, then |𝑓−𝑃𝑁−1𝑓|=𝑂(𝑟−(𝑁−1)) for all 𝑥 in [−1,1] [6]. This property of rapid convergence combined with the approximation property (39) makes the proposed algorithm a practical gap recovery tool for general analytic functions.

4. Discrete Implementation of the Continuous-Time-Recovery Algorithm

A Chebyshev polynomial-based series expansion allows a convenient and accurate discrete implementation by a nonuniform sampling scheme, where the samples are taken at time locations that are Chebyshev polynomial zeroes and a discrete cosine transform (DCT) applied on the nonuniform samples [8, 9]. Equation (15) is a Gauss-Chebyshev quadrature for the original integral (19), which is known to be a very accurate numerical method [6]. In addition, if the maximum absolute error is used as the optimality criterion, then the interpolating polynomial given by (14) provides optimal approximation among polynomials of degree = 𝑁−1 to the function 𝑓 [18].

Let ℝ𝑁 denote the real 𝑁-dimensional space. The 𝑙2 inner product is given by⟨𝑦,ℎ⟩=𝑁−1𝑚=0𝑦[𝑚]ℎ[𝑚],(40) and the induced norm is given by‖𝑥‖2=⟨𝑥,𝑥⟩.(41) Let 𝐶 be the 𝑁×𝑁 unitary DCT matrix defined as𝐶𝑚,𝑛=⎧⎪⎪⎨⎪⎪⎩1√𝑁,𝑚=0,0≤𝑛≤𝑁−1,2𝑁cos𝜋(2𝑛+1)𝑚2𝑁,1<𝑚≤𝑁−1,0≤𝑛≤𝑁−1.(42) The DCT of the vector 𝑦∈ℝ𝑁 (arranged as a column vector) is given by ̂𝑦=𝐶𝑦. The inverse DCT of the vector ̂𝑦 is given by 𝑦=𝐶𝑇̂𝑦, where 𝐶𝑇 is the transpose of 𝐶. A discrete signal 𝑦 is bandlimited if its DCT ̂𝑦 vanishes on some fixed set, that is, if[𝑚]̂𝑦=0,𝑚∈𝑆,(43) where 𝑆 is a fixed nonempty proper subset of {0,1,…,𝑁−1}. The set of signals bandlimited to a specific set 𝑆 is a linear subspace of ℝ𝑁, of dimension equal to the cardinal of the complement of 𝑆.

Let 𝑦[𝑘]=𝑓(𝑥𝑘), 𝑘=0,…,𝑁−1 be a vector in ℝ𝑁, whose elements are the samples of the analytic function 𝑓(𝑥) taken in the zeroes of 𝑇𝑁 (13). Since only 𝑃𝑆𝑓 is available, only certain 𝑄<𝑁 samples are known. Assuming that 𝑓∈ℙ𝑀−1 and 𝑀≤𝑄, the discrete recovery problem is to determine the DCT coefficients ̂𝑦[𝑚] for 𝑚=0,1,…,𝑀−1 from the 𝑄 known samples. Using the DCT coefficients (with an appropriate simple scaling) in (14), we obtain the (𝑀−1)th degree polynomial that approximates 𝑓(𝑥).

Let 𝑆1 be a subset of size 𝑄 in [0,1,2,…,𝑁−1] containing the indices of the known samples, let 𝑆2 denote the subset of indices [0,1,…,𝑀−1], and denote 𝐷=𝐶−1=𝐶𝑇. The required DCT coefficients can be determined by solving the least squares problem𝑖∈𝑆2𝐷𝑘𝑖[𝑖][𝑘]̂𝑦≃𝑦,𝑘∈𝑆1.(44) It can be proved [19] that any square submatrix (𝑀=𝑄) of the matrix 𝐷 in (44) is the product of a Vandermonde and upper triangular matrix, and therefore, it is invertible. It follows that the submatrix 𝐷𝑘𝑖, 𝑖∈𝑆2, 𝑘∈𝑆1, is a full column rank matrix when 𝑀≤𝑄. The resulting least squares problem (44) can be solved by many methods (e.g., singular value decomposition, QR factorization, conjugate-gradient method for solving the normal equations, etc.) [20, 21]. A simple popular option (not necessarily the best in terms of computational complexity) is a version of the discrete Papoulis-Gerchberg algorithm to be described below.

Let the operator 𝑃𝐴 map a signal 𝑦∈ℝ𝑁 into another, 𝑃𝐴𝑦=𝐴𝑦, by multiplying 𝑦 with a diagonal matrix 𝐴 containing only zeroes or ones. 𝐴𝑚𝑚=1, if 𝑚∈𝑆1, and 𝐴𝑚𝑚=0, otherwise. It is easy to see that this operation is idempotent and self-adjoint. Hence, 𝑃𝐴 is the orthogonal projector onto the subspace of ℝ𝑁 signals with zero-valued elements at the indices not belonging to 𝑆1.

Let the operator 𝑃𝐵 map a signal 𝑦 into another 𝑃𝐵𝑦=𝐶𝑇𝐵𝐶𝑦, where 𝐵 a diagonal matrix containing only zeroes or ones. 𝐵𝑚𝑚=1, if 𝑚∈𝑆2, and 𝐵𝑚𝑚=0, otherwise. Using the identities 𝐵2=𝐵 and 𝐵𝑇=𝐵, it follows that 𝑃𝐵 is idempotent𝑃𝐵𝑃𝐵𝑦=𝐶𝑇𝐵𝐶𝐶𝑇𝐵𝐶𝑦=𝐶𝑇𝐵𝐶𝑦=𝑃𝐵𝑦,(45) and self-adjoint ⟨𝑃𝐵𝐶𝑦,ℎ⟩=𝑇=𝐵𝐶𝑦,ℎ=⟨𝐵𝐶𝑦,ğ¶â„ŽâŸ©=⟨𝐶𝑦,ğµğ¶â„ŽâŸ©ğ‘¦,ğ¶ğ‘‡î¬ğµğ¶â„Ž=⟨𝑦,ğ‘ƒğµâ„ŽâŸ©,(46) for every 𝑦,â„Žâˆˆâ„ğ‘. Hence, 𝑃𝐵 is the orthogonal projector onto the subspace of bandlimited (in terms of the DCT and 𝑆2) signals in ℝ𝑁.

A discrete version of the PGP algorithm is obtained by working in the finite-dimensional space ℝ𝑁 and using the operators 𝑃𝐴 instead of 𝑃𝑆 and 𝑃𝐵 instead of 𝑃𝑁−1. The discrete version of the PGP is also a projection onto convex sets algorithm, and it follows that the iterates 𝑃𝐵𝑦𝑚 converge to the required least squares solution (see the discussion before (39)).

5. Discussion

Our model is based on two assumptions. First, the original analytic signal can be accurately approximated by a linear combination of a few Chebyshev polynomials (exponential convergence), and, secondly, that the Gauss-Chebyshev quadrature (15) provides a good approximation to the original integral (19). When these assumptions are not (approximately) satisfied (e.g., due to noise), the algorithm may converge (it always converges) to a crude approximation of the original signal. In such cases, a higher-order polynomial approximation and a denser sampling grid would provide an improvement. If the model assumptions are (approximately) satisfied, the resulting approximation is very accurate and minimizes a certain objective function in the sense of (39).

A Chebyshev polynomial series expansion is a Fourier cosine series with a change of variable 𝑥=cos𝜃 for 𝑓(cos𝜃) [6]. While 𝑓(𝑥) itself is not periodic in 𝑥, the function 𝑓(cos𝜃) is periodic in 𝜃. The smoother the function is, the more rapidly its Fourier coefficients decrease. Since 𝑓(cos𝜃) is periodic and analytic, its Fourier series must have exponential convergence. For a Fourier cosine series expansion for the nonperiodic function 𝑓(𝑥), the first derivative of the function is discontinuous at the interval's borders and for a Fourier series expansion, the function itself is discontinuous at the interval’s borders. It follows that a Chebyshev polynomial series expansion of an analytic function provides a better approximation than a Fourier series expansion (using a discrete Fourier transform in discrete implementations [22]) with the same number of terms, which makes it attractive to recovery problems. The improved approximation comes, however, at the price of a nonuniform sampling grid. The sampling points are clustered near the boundaries of the interval [−1,1]. It follows that the proposed algorithm is more suited to the recovery of gaps that occur in the middle of the signal or nearby. In such cases, we benefit from both the excellent approximation properties of the Chebyshev polynomials and the fact that less missing samples have to be recovered.

6. Numerical Example

We demonstrate the performance of the discrete implementation of the recovery algorithm by a gap recovery example. The function 𝑓(𝑥) defined on the interval [−1,1]1𝑓(𝑥)=1+25𝑥2.(47) It is analytic on [−1,1] but has a pole at 𝑥=±0.2𝑗, where √𝑗=−1. We assume that the piece [−0.4,0.4] is missing and that signal is well approximated by 64 Chebyshev polynomials. The samples are taken at the 128 zeroes of the polynomial 𝑇128, and it follows that 34 samples are missing. The proposed algorithm is used to recover the missing samples: the 𝑙2 norm of the error is 0.16, while the norm of the missing samples is 3.5244. We see that the recovery algorithm succeeds in approximating the missing data. Finally, the function and the resulting polynomial approximation are given at a uniform grid of 128 points in the interval [−1,1] in Figure 1.

7. Summary and Conclusions

The recovery of gaps in general analytic functions was obtained by using signal expansions in terms of the Chebyshev polynomials of the first kind. The recovery algorithm is exact for polynomial signals, and certain best polynomial approximations are obtained for general analytic signals. For the continuous-time case, we have extended the well-known Papoulis-Gerchberg algorithm for bandlimited signals to signals in polynomial subspaces. The discrete implementation is based on a specific nonuniform sampling grid (at the zeroes of a Chebyshev polynomial) and the discrete cosine transform and results in solving a linear least squares problem.