Continuous window functions for NFFT

In this paper, we study the error behavior of the nonequispaced fast Fourier transform (NFFT). This approximate algorithm is mainly based on the convenient choice of a compactly supported window function. Here we consider the continuous Kaiser--Bessel, continuous $\exp$-type, $\sinh$-type, and continuous $\cosh$-type window functions with the same support and same shape parameter. We present novel explicit error estimates for NFFT with such a window function and derive rules for the optimal choice of the parameters involved in NFFT. The error constant of a window function depends mainly on the oversampling factor and the truncation parameter. For the considered continuous window functions, the error constants have an exponential decay with respect to the truncation parameter.


Introduction
The nonequispaced fast Fourier transform (NFFT), see [7,6,21,17] and [16,Chapter 7] is an important generalization of the fast Fourier transform (FFT). The window-based approximation leads to the most efficient algorithms under different approaches [8,19]. Recently a new class of window functions were suggested in [3] and asymptotic error estimates are given in [4]. After [21], the similarities of the window-based algorithms for NFFT became clear. Recently we have analyzed the window-based NFFT used so far and presented the related error estimates in [18]. Now we continue this investigation and present new error estimates for the some other window functions. More precisely, we consider the continuous Kaiser-Bessel window function and two close relatives of the sinh-type window function, namely the continuous exp-type and cosh-type window functions. All these window functions have the same support and the same shape parameter. We show that these window functions are very useful for NFFT, since they produce very small errors. In this paper, we present novel explicit error estimates (2.3) with so-called error constants (2.1). The error constants of NFFT are defined by values of the Fourier transform of the window function. We show that an upper bound of (2.1) depends only on the oversampling factor σ > 1 and the truncation parameter m ≥ 2 and decreases with exponential rate with respect to m. In numerous applications of NFFT, one uses quite often an oversampling factor σ ∈ 5 4 , 2 and a truncation parameter m ∈ {2, 3, . . . , 6}. Therefore we will assume that σ ≥ 5 4 . The outline of the paper is as follows. In Section 2 we introduce the set Φ m,N 1 of continuous, even window functions with support − m N 1 , m N 1 , where m ∈ N \ {1} and N 1 = σN ∈ 2N (with N ∈ 2N and 2m ≪ N 1 ) are fixed. We emphasize that a continuous window function ϕ ∈ Φ m,N 1 tends to zero at the endpoints ± m N 1 of its compact support − m N 1 , m N 1 ⊂ − 1 2 , 1 2 . In Section 3 we show that the simple rectangular window function (3.1) is not convenient for NFFT. The main results of this paper are contained in Sections 4 -7. For the first time, we present explicit estimates of the error constants (2.1) for fixed truncation parameter m and oversampling factor σ. In Section 4, we derive explicit error estimates for the continuous Kaiser-Bessel window function (4.1). In comparison, we show that the popular standard Kaiser-Bessel window function (4.13) has a similar error behavior as (4.1). A very useful continuous window function is the sinh-type window function (5.1) which is handled in Section 5. The main drawback for the numerical analysis of the exp-type and cosh-type window function is the fact that an explicit Fourier transform of this window function is unknown. In Sections 6 and 7, we develop a new technique. We split the continuous exp-type/coshtype window function into a sum ψ + ρ, where the Fourier transform of the compactly supported function ψ is explicitly known and where the compactly supported function ρ has small magnitude. Here we use the fact that both window functions (6.1) and (7.1) are close relatives of the sinh-type window function (5.1) which was introduced by the authors in [18]. The Fourier transform of ρ is explicitly estimated for small as well as large frequencies, where σ and m are fixed. We present many numerical results so that the error constants of the different window functions can be easily compared. After this investigation, we favor the use of a continuous window function with small error constant, which can be very fast computed, such as the sinh-type, standard/continuous exp-type, or continuous cosh-type window function.

Continuous window functions for NFFT
Let σ > 1 be an oversampling factor. Assume that N ∈ 2 N and N 1 := σN ∈ 2 N are given. For fixed truncation parameter m ∈ N \ {1} with 2m ≪ N 1 , we introduce the open interval I := − m N 1 , m N 1 and the set Φ m,N 1 of all continuous window functions ϕ : R → [0, 1] with following properties: • Each window function ϕ is even, has the supportĪ, and is continuous on R.
• For each window function ϕ, the Fourier transform Examples of continuous window functions of Φ m,N 1 are the (modified) B-spline window functions, algebraic window functions, Bessel window functions, and sinh-type window functions (see [18] and [16,Chapter 7]). More examples are presented in Sections 4 -7.
In the following, we denote the torus R/Z by T and the Banach space of continuous, 1-periodic functions by C(T). Let I N := {−N/2, . . . , N/2 − 1} be the index set for N ∈ 2N. We say that a continuous window function fulfills the condition e σ (ϕ) ≪ 1 for conveniently chosen oversampling factor σ > 1. Now we show that the error of the nonequispaced fast Fourier transform (NFFT) with a window function ϕ ∈ Φ m,N 1 can be estimated by the error constant (2.1). The NFFT (with nonequispaced spatial data and equispaced frequencies) is an approximate, fast algorithm which computes approximately the values p(x j ), j = 1, . . . , M , of any 1periodic trigonometric polynomial at finitely many nonequispaced nodes x j ∈ − 1 2 , 1 2 , j = 1, . . . , M , where c k ∈ C, k ∈ I N , are given coefficients. By the properties of the window function ϕ ∈ Φ m,N 1 , the 1-periodic functionφ is continuous on T and of bounded variation over − We approximate the trigonometric polynomial p by the 1-periodic function with the coefficients The computation of the values s(x j ), j = 1, . . . , M , is very easy, since ϕ is compactly supported. The computational cost of the algorithm is O(N 1 log N 1 + (2m + 1)M ), see [16,Algorithm 7.1] and [11] for details. We interpret s − p as the error function of the NFFT which we measure in the norm of C(T).
Then the error function of the NFFT can be estimated by 3) The proof of Theorem 2.1 is based on the equality For details of the proof see [

Rectangular window function
In this Section, we present a simple discontinuous window function which is not convenient for NFFT. Later we will use this discontinuous window function in Remarks 4.6 and 6.8, where we estimate the C(T)-error constants for the standard Kaiser-Bessel window function and the original exp-type window function, respectively. The simplest window function is the rectangular window function Now we show that the rectangular window function (3.1) is not convenient for NFFT. The Fourier transform of (3.1) has the form The discontinuous window function ϕ rect doesn't belong to Φ m,N 1 .
Lemma 3.1 For each n ∈ I N \ {0}, the Fourier series Proof. For fixed n ∈ I N \ {0}, we consider the Fourier series of the special 1 N 1 -periodic function g n (x) := e −2πi nx for x ∈ (0, 1 For n = 0 we have g 0 (x) = 1. Then the rth Fourier coefficient of g n reads as follows for r ∈ Z. By the Convergence Theorem of Dirichlet-Jordan, the Fourier series of g n is pointwise convergent such that for each x ∈ R and n ∈ I N \ {0} This completes the proof.
i.e., the rectangular window function (3.1) is not convenient for NFFT.
By (3.2) we have for n ∈ I N \ {0} and r ∈ Z,φ rect (n+rN 1 ) ϕrect(n) = n n+rN 1 . Thus we obtain by Lemma 3.1 that for x ∈ (0, 1 Analogously, we estimate Consequently the rectangular window function is not convenient for NFFT, since the corresponding C(T)-error constant e σ (ϕ rect ) can be estimated by (3.3).
In the case n = 0 we have f 0 (x) := r∈Z sinc(2πmr) e 2πirN 1 x = 1 . For arbitrary x ∈ R and m ∈ N, it holds obviously sin(2mx) ≤ 2m | sin x| and so 1 − e −2πin/N 1 = 2 sin πn N 1 . We obtain for n ∈ I N \ {0} that Thus for n ∈ I N \ {0} and x ∈ 0, 1 N 1 , we receive In the case n = 0, the above estimate is also true, since This completes the proof.
is positive and decreasing such that Using the scaled frequency w = 2πmv/N 1 , we obtain Proof. Since the sinc-function is even, we consider only the case w ≥ β. For w = β, the above inequality is true, since |sinc 0 − sinc β| ≤ 1 + |sinc β| < 2 . For w > β we obtain it follows that Further we receive by (4.4) that This completes the proof.
In our study we use the following Proof. For arbitrary u ∈ (−1, 1) and r ∈ N, we have Using (4.5), the following series can be estimated by Hence it follows by the integral test for convergence of series that Hence we conclude that We illustrate Lemma 4.2 for some special functions f , which we need later.
Especially for µ = 2, it follows that For the function f (x) = e −ax , x > 0, with a > 0, we obtain by Lemma 4.2 that for each Proof. By (4.3) and Lemma 4.1 we obtain for all frequencies |v| since by (4.7) it holds for all n ∈ I N , By the special choice of b = 2π 1 − 1 2σ , we obtain the above inequality.
From Lemma 4.4 it follows immediately that for all n ∈ I N , (4.10) Then the C(T)-error constant of the continuous Kaiser-Bessel window function (4.1) can be estimated by Proof. By the definition (2.1) of the C(T)-error constant, it holds where it holds (4.2), i.e., Thus from (4.10) it follows the assertion (4.11).
Remark 4.6 As in [9] and [4], we consider also the standard Kaiser-Bessel window function with the shape parameter β = mb = 2πm (1 − 1 2σ ), σ > 1, N ∈ 2 N, and N 1 ∈ 2 N. Further we assume that m ∈ N \ {1} fulfills 2m ≪ N 1 . This window function possesses jump discontinuities at x = ± m N 1 with very small jump height I 0 (β) −1 , such that (4.13) is "almost continuous". The Fourier transform of (4.13) is even and reads by [14, p. 95] as followsφ 2π ) is positive and decreasing such that we can estimate for all n ∈ I N , Then from Lemma 4.4 and (3.4) it follows that max n∈I N r∈Z\{0}φ Consequently, we obtain the following estimate of the C(T)-error constant of (4.13) For σ ≥ 5 4 , we sustain by (4.12) that
For a fixed oversampling factor σ ≥ 5 4 , the C(T)-error constant of (5.1) decays exponentially with the truncation parameter m ≥ 2. On the other hand, the computational cost of NFFT increases with respect to m (see [16, pp. 380-381]) such that m should be not too large. For σ = 2 and m = 4, we obtain e σ (ϕ sinh ) ≤ 3.7 · 10 −6 .

Continuous exp-type window function
For fixed shape parameter β = bm with m ∈ N \ {1}, b = 2π 1 − 1 2σ , and oversampling factor σ ≥ 5 4 , we consider the continuous exp-type window function Obviously, ϕ cexp ∈ Φ m,N 1 is a continuous window function. Note that a discontinuous version of this window function was suggested in [3,4]. A corresponding error estimate for the NFFT was proved in [4], where an asymptotic value of its Fourier transform was determined for β → ∞ by saddle point integration. We present new explicit error estimates for fixed shape parameter β of moderate size.
In the following, we present a new approach to an error estimate for the NFFT with the continuous exp-type window function (6.1). Unfortunately, the Fourier transform of (6.1) is unknown analytically. Therefore we represent (6.1) as sum where the Fourier transform of ψ is known and where the correction term ρ has small magnitude |ρ|. We choose m −ρ(0) = e −β 2 8.06 · 10 −5 3 7.24 · 10 −7 4 6.51 · 10 −9 5 5.85 · 10 −11 6 5.25 · 10 −13 Since ρ has small absolute values in the small supportĪ, the Fourier transformρ is small too and it holds |ρ Substituting t = N 1 x/m, we determine the Fourier transform For simplicity, we introduce the scaled frequency w := 2πmv/N 1 such that where I 1 denotes the modified Bessel function and J 1 the Bessel function of first order. Therefore we consider as correction term of (6.4). Now we estimate the integral by complex contour integrals. Then the stronger form of Cauchy's Integral Theorem (see [5]) provides with the contour integrals Note that I 3 (w) is the complex conjugate of I 1 (w) such that |I 3 (w)| = |I 1 (w)|.
The line segment C 2 can be parametrized by z = t + i, t ∈ [−1, 1] such that and hence Then we have e −β we obtain the estimate A parametrization of the line segment C 1 is z = −1 + i t, t ∈ [0, 1], such that For w ≥ β > 0, we split I 1 (w) into the sum of two integrals Since 2 ≤ | √ 2i + t| ≤ 4 √ 5, t ∈ [0, 1] , the integral I 1,0 (w) is bounded in magnitude by Above we have used the simple inequality 1 − e −x ≤ x for x ≥ 0.
Finally we estimate the integral I 1,1 (w) as follows it follows that Thus we receive for w ≥ β, This completes the proof.

is bounded by
Thus we receive for all n ∈ I N and r ∈ Z \ {0, ±1}, Hence for all n ∈ I N , we obtain by (4.6) that For all the other v = n ± N 1 = − N 2 + N 1 , n ∈ I N , we have such that by (6.3),ψ(n ± N 1 ) reads as follows Since can be small for n ∈ I N , we estimate the Bessel function J 1 (x), x ≥ 0, by Poisson's integral (see [22, p. 47]) By (6.11), this estimate of |ψ(n ± N 1 )| is valid for all n ∈ I N . This completes the proof. Now for arbitrary n ∈ I N , we have to estimate the series r∈Z\{0} |ρ(n + rN 1 )| .
By (6.5) and Lemma 6.1, we obtain for any v ∈ R \ {0}, Thus we obtain that The inequalities (4.6), (4.8), and (4.9) imply that Thus we obtain the following Lemma 6.6 Let N ∈ 2N and σ ≥ 5 4 be given, where Hence from Lemmas 6.5 and 6.6 it follows that max n∈I N r∈Z\{0}φ Using Lemma 6.4, we obtain by the following Theorem 6.7 Let N ∈ 2N and σ ≥ 5 4 be given, where N 1 = σN ∈ 2N. Further let m ∈ N with 2m ≪ N 1 , β = bm, and b = 2π 1 − 1 2σ . Then the C(T)-error constant of the continuous exp-type window function (6.1) can be estimated by In other words, the continuous exp-type window function (6.1) is convenient for NFFT.
Note that for σ ∈ 5 4 , 2 and m ≥ 2, it holds by (6.19), In order to compute the Fourier transformφ of window function ϕ ∈ Φ m,N 1 , we approximate this window function by numerical integration. In our next numerical examples we apply the following method. Since the window function ϕ ∈ Φ m,N 1 is even and supported We evaluate the last integral using a global adaptive quadrature [20] forφ(k), k = 0, . . . , N . In general, this values can be precomputed, see [10,2].
We split (6.20) in the form with the window functions (6.1) and (3.1). Then the Fourier transform of (6.20) reads as followsφ , v ∈ R , By Lemma 6.4 it follows that Using (6.18) and (3.4), we estimate for all n ∈ I N , Thus we obtain Thus the discontinuous window function (6.20) possesses a similar C(T)-error constant as the continuous exp-type window function (6.1).
Obviously, ϕ cosh ∈ Φ m,N 1 is a continuous window function. Note that recently a discontinuous version of this window function was suggested in [3, Remark 13]. But up to now, a corresponding error estimate for the related NFFT was unknown. Now we show that the C(T)-error constant e σ (ϕ cosh ) can be estimated by a similar upper bound as e σ (ϕ cexp ) in Theorem 6.7. Thus the window functions (6.1) and (7.1) possess the same error behavior with respect to the NFFT.
In the following, we use the same technique as in Section 6. Since the Fourier transform of (7.1) is unknown analytically, we represent (7.1) as the sum where the Fourier transform of ψ 1 is known and where the correction term ρ 1 has small magnitude |ρ 1 |. We choose x ∈ R \ I and x ∈ R \ I .

Conclusion
In this paper, we prefer the use of continuous, compactly supported window functions for NFFT (with nonequispaced spatial data and equispaced frequencies). Such window functions simplify the algorithm for NFFT, since the truncation error of NFFT vanishes. Further, such window functions can produce very small errors of NFFT. Examples of such window functions are the continuous Kaiser-Bessel window function (4.1), continuous exp-type window function (6.1), sinh-type window function (5.1), and continuous coshtype window function (7.1) which possess the same support and shape parameter. For these window functions, we present novel explicit error estimates for NFFT and we derive rules for the convenient choice of the truncation parameter m ≥ 2 and the oversampling parameter σ ≥ 5 4 . The main tool of this approach is the decay of the Fourier transform ϕ(v) of ϕ ∈ Φ m,N 1 for |v| → ∞. A rapid decay ofφ is essential for small error constants. Unfortunately, the Fourier transform of certain window function ϕ, such as (6.1) and (7.1), is unknown analytically. Therefore we propose a new technique and split ϕ into a sum of two compactly supported functions ψ and ρ, where the Fourier transformψ is explicitly known and where |ρ| is sufficiently small. Further, it is shown that the standard Kaiser-Bessel window function and original exp-type window function which have jump discontinuities with very small jump heights at the endpoints of their support, possess a similar error behavior as the corresponding continuous window functions. In summary, the C(T)-error constant of the continuous/standard Kaiser-Bessel window function is of best order O m e −2πm √ 1−1/σ . For the sinh-type, continuous/original