Uniform error estimates for nonequispaced fast Fourier transforms

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. So far, various window functions have been used and new window functions have recently been proposed. We present novel error estimates for NFFT with compactly supported, continuous window functions and derive rules for convenient choice from the parameters involved in NFFT. The error constant of a window function depends mainly on the oversampling factor and the truncation parameter.

fast multipole method and on the low-rank approximation were presented in [9,22]. After the seminal paper [24], the similarities of the window-based algorithms for NFFT became clear. In the following, we give an overview of the windowbased NFFT used so far. In this construction of NFFT, a window function is applied together with its Fourier transform. This connection is very important in order to deduce error estimates for the NFFT. This made it possible to determine convenient parameters of the involved window function. To develop an NFFT, the necessary Fourier coefficients of the periodized window function can simply be calculated by a convenient quadrature rule. A challenge are error estimations in order to determine the parameters of the window function involved. By C(T) we denote the Banach space of all 1-periodic, continuous functions, where T = R/Z is the torus. The considered window functions depend on some parameters. Assume that N ∈ 2N is the order of the given 1-periodic trigonometric polynomial which values will be computed by NFFT. Let σ > 1 be an oversampling factor such that N 1 := σN ∈ 2 N. For fixed truncation parameter m ∈ N \ {1} with 2m ≪ N 1 , we denote by Φ m,N1 the set of all window functions ϕ : R → [0, 1] with the following properties: • Each window function ϕ is even, has a compact support [−m/N 1 , m/N 1 ], and is continuous on R. Note that for fixed N 1 the truncation parameter m determines the size of the support of ϕ ∈ Φ m,N1 . If a window function ϕ ∈ Φ m,N1 has the form ϕ(x) = 1 ϕ 1 β ϕ 2 (0) ϕ 1 β ϕ 2 (x) , x ∈ R , with β > 0 and convenient functions ϕ 1 , ϕ 2 , then β is a so-called shape parameter of ϕ. Examples of window functions of the set Φ m,N1 are the B-spline window function (5.4), the modified B-spline window function (5.6), the algebraic window function (5.7), the Bessel window function (5.14), the sinh-type window function (5.21), the related sinh-type functions, see Subsection 5.5, the modified cosh-type window function (5.22) and the related cosh-type window functions, see Subsection 5.6. Note that the Kaiser-Bessel window function (see [18, p. 393]) and the Gaussian window function (see [18, p. 390]) are not contained in Φ m,N1 , since these window functions are supported on whole R.
The aim of this paper is a systematic approach to uniform error estimates for NFFT, where a compactly supported, continuous window function ϕ ∈ Φ m,N1 is used. We introduce the C(T)-error constant e σ (ϕ) = sup N ∈2N max n∈IN r∈Z\{0}φ (n + rN 1 ) ϕ(n) e 2πi rN1 · C(T , where I N denotes the index set {−N/2, 1 − N/2, . . . , N/2 − 1}. As shown in Lemma 2, the uniform error of the NFFT with nonequispaced spatial data and equispaced frequencies can be estimated by e σ (ϕ). Analogously in Lemma 4, the error of the NFFT with nonequispaced frequencies and equispaced spatial data is estimated by e σ (ϕ) too. Therefore in the following, we study mainly the behavior of the C(T)-error constant e σ (ϕ). Our main result is Theorem 2, where we describe a general concept for the construction of a convenient upper bound for the C(T)-error constant e σ (ϕ) with a window function ϕ ∈ Φ m,N1 . Applying Theorem 2, we obtain upper bounds for e σ (ϕ) with special window function ϕ ∈ Φ m,N1 . We show that the C(T)-error constant e σ (ϕ) of a window function ϕ ∈ Φ m,N1 depends mainly on the oversampling factor σ > 1 and the truncation parameter m ∈ N \ {1}. Since we are interested in NFFT with relatively low computational cost, the oversampling factor σ ∈ 5 4 , 2 and the truncation parameter m ∈ {2, 3, . . . , 6} are restricted. These parameters σ and m determine the shape parameter β of the window function. For the Bessel window function (5.15), the sinh-type window function (5.21), and the modified cosh-type window function (5.22), a good choice is the shape parameter β = 2πm 1 − 1 2σ . In connection with NFFT, B-spline window functions were first investigated in [6]. In the important application of particle simulation (see [7]), the B-spline window function was also used. Later it became clear that these methods can be interpreted as a special case of the fast summation method, see [19,15] and the references therein. Based on this unified approach, one can use all the other window functions for this application too. The convenient choice of the shape parameter is of special importance, as shown in [14] for the root mean square error of the NFFT. In this paper we suggest four new continuous, compactly supported window functions for the NFFT, namely the algebraic, Bessel, sinh-type, and modified cosh-type window function. The algebraic window function is very much related to the B-spline window function, but much simpler to compute. We show that the Bessel window function (5.14), sinh-type window function (5.21), and modified cosh-type window function (5.22) are very convenient for NFFT, since they possess very small C(T)-error constants with exponential decay with respect to m. It is difficult to design a window function ϕ ∈ Φ m,N1 with minimal C(T)-error constant. We prove that the best error behavior has the modified cosh-type window function (5.22) with the shape parameter β = 2πm 1 − 1 2σ , σ ≥ 5 4 . Further we compare several window functions with respect to the corresponding C(T)-error constants for the NFFT. Based on the error estimates of the sinh-type window function, we are able to extend the error estimates, see [21], to window functions where an analytical expression of its Fourier transform is unknown, see [3,4,5].
We prefer the use of compactly supported, continuous window functions ϕ ∈ Φ m,N1 by following reasons: • As explained in Remark 1, the NFFT with a window function ϕ ∈ Φ m,N1 is simpler than the NFFT with a window function supported on whole R.
• The window functions ϕ ∈ Φ m,N1 presented in Section 5 (and their Fourier transforms) have simple explicit forms and they are convenient as window functions for NFFT.
• Few window functions ϕ ∈ Φ m,N1 (such as Bessel, sinh-type, and modified cosh-type window function) possess low C(T)-error constants with exponential decay with respect to m. The best error behavior has the modified cosh-type window function.
The outline of the paper is as follows. In Section 2 we introduce the basic definitions and develop the error estimates for an NFFT with a general compactly supported, continuous window function. Important tools for the estimation of the Fourier transforms of window functions are developed in Section 3. In Section 4 we present a modified Paley-Wiener Theorem which characterizes the behavior of Fourier transforms of compactly supported functions lying in a special Sobolev space. The main results of this paper are contained in Section 5. Using the uniform norm, we present explicit error estimates for the (modified) B-spline, algebraic, Bessel, sinh-type, and modified cosh-type window functions. Further we show numerical tests so that the C(T)-error constants of the different window functions can be easily compared.

Convenient window functions for NFFT
Letφ : T → [0, 1] be the 1-periodization of ϕ, i.e., (2.1) Note that for each x ∈ R the series (2.1) has at most one nonzero term. Then the Fourier coefficients ofφ read as follows By the properties of the window function ϕ ∈ Φ m,N1 , the 1-periodic functioñ ϕ is continuous on T and of bounded variation over − 1 2 , 1 2 . Then from the Convergence Theorem of Dirichlet-Jordan (see [27,Vol. 1,), it follows thatφ possesses the uniformly convergent Fourier expansioñ Uniform error estimates for nonequispaced fast Fourier transforms 5 Lemma 1 For given ϕ ∈ Φ m,N1 , the series r∈Z c n+rN1 (φ) e 2πi (n+rN1) x is convergent for each x ∈ R and has the sum which coincides with the rectangular rule of the integral Proof. From (2.2) it follows that for all n ∈ Z and x ∈ R it holds Summing the above formulas for ℓ = 0, . . . , N 1 −1 and using the known relation This completes the proof. at M nonequispaced nodes x j ∈ [− 1 2 , 1 2 ), j ∈ I M . Using a window function ϕ ∈ Φ m,N1 , the trigonometric polynomial f is approximated by the 1-periodic function with conveniently chosen coefficients g ℓ ∈ C. The computation of the values s(x j ), j ∈ I M , which approximate f (x j ) is very easy. Since ϕ is compactly supported and thusφ is well-localized, each value s(x j ), j ∈ I M , is equal to a sum of few nonzero terms. The coefficients g ℓ can be determined by discrete Fourier transform (DFT) as follows. The 1-periodic function s possesses the Fourier expansion with the Fourier coefficients In other words, the vector (ĝ k ) k∈IN 1 is the DFT of length N 1 of coefficient vector (g ℓ ) ℓ∈IN 1 such thatĝ In order to approximate f by s, we set Note that the valuesĝ k , k ∈ I N , can be used in an efficient way. Even if (2.3) is only known at finitely many equispaced points of [0, 1], the Fourier coefficients c k (f ), k ∈ I N , can be approximately determined by a fast Fourier transform (FFT). For many window functions ϕ ∈ Φ m,N1 , the Fourier coefficients c k (φ), k ∈ I N , are explicitly known. Then for all r ∈ Z, it holds Uniform error estimates for nonequispaced fast Fourier transforms 7 In particular, we see that c n (s) = c n (f ) for all n ∈ I N and c n (s) = 0 for all n ∈ I N1 \ I N . Substituting k = n + rN 1 with n ∈ I N1 and r ∈ Z, we obtain Let A(T) be the Wiener algebra of all 1-periodic functions g ∈ L 1 (T) with the property k∈Z |c k (g)| < ∞ . Since Therefore we measure the error of NFFT s − f C(T) in the uniform norm. As norm of the 1-periodic trigonometric polynomial (2.3) we use the norm in the Wiener algebra A(T).

Remark 1
The NFFT with a window function of the set Φ m,N1 is simpler than the NFFT with Kaiser-Bessel or Gaussian window function ϕ, since both window functions are supported on whole R. For such a window function ϕ, an additional step in the NFFT is necessary, where the 1-periodic function (2.4) is approximated by the 1-periodic well-localized function whereψ is the 1-periodization of the truncated window function (see [18, pp. 378-381]) Thus the NFFT with a window function ϕ supported on whole R requires also a truncated version of ϕ. In this case, the error of the NFFT is measured by In [18, p. 393], it is shown that the error of the NFFT with the Kaiser-Bessel window function can be estimated by We will see in 6 that special window functions ϕ ∈ Φ m,N1 possess a similar error behavior as the Kaiser-Bessel window function.
We say that the window function ϕ ∈ Φ m,N1 is convenient for NFFT, if the fulfills the condition e σ (ϕ) ≪ 1 for conveniently chosen truncation parameter m ≥ 2 and oversampling factor σ > 1. Later in Theorem, 2 we will show that under certain assumptions on ϕ ∈ Φ m,N1 the value e σ,N (ϕ) is bounded for all N ∈ N. This C(T)-error constant is motivated by techniques first used in [24] and later also in [5]. G. Steidl [24] has applied this technique for error estimates of NFFT with B-spline and Gaussian window functions, respectively. .
This completes the proof.
Thus the condition e σ (ϕ) ≪ 1 with (2.7) means that each exponential e 2πi n · , n ∈ I N , Uniform error estimates for nonequispaced fast Fourier transforms 9 can be approximately reproduced by a linear combination of shifted window functionsφ(· + ℓ N1 ) with ℓ ∈ I N1 . In other words, the equispaced shiftsφ(·+ ℓ N1 ) with ℓ ∈ I N1 are approximately exponential reproducing. For each node x j ∈ [− 1 2 , 1 2 ), j ∈ I M , the linear combination has only few nonzero terms, since the support of ϕ is very small for large N 1 . If we replace exp(2πinx j ) for each n ∈ I N and j ∈ I M by the approximate value (2.8), we compute approximate values of in the form mainly by DFT. This is the key of the NFFT with nonequispaced spatial data and equispaced frequencies. Special window functions ϕ ∈ Φ m,N1 which are convenient for NFFT will be presented in Section 5.
Then the error of NFFT with nonequispaced spatial data and equispaced frequencies can be estimated by Proof. From (2.5) it follows that Note that c n (φ) =φ(n) > 0 for n ∈ I N by assumption ϕ ∈ Φ m,N1 . Then by Hölder's inequality we obtain that for all x ∈ T it holds Hence we get (2.9).
Remark 2 Let λ ≥ 0 be fixed. We introduce the 1-periodic Sobolev space H λ (T) of all 1-periodic functions f : T → C which are integrable on [0, 1] and for which For λ = 0, we have H 0 (T) = L 2 (T). Then the Sobolev embedding theorem (see [23, p. 142 Applying the Cauchy-Schwarz inequality, we obtain for λ > 1 2 that Using the Riemann zeta function ζ(2λ) := ∞ k=1 1 k 2λ , λ > 1 2 , we obtain the following inequality Thus under the assumptions of Lemma 3, the error of NFFT with nonequispaced spatial data and equispaced frequencies can be estimated by Note that Uniform error estimates for nonequispaced fast Fourier transforms 11 2.2 NFFT with nonequispaced frequencies and equispaced spatial data The NFFT with nonequispaced frequencies and equispaced spatial data or transposed NFFT evaluates the exponential sums for arbitrary given coefficients f j ∈ C and nonequispaced frequencies x j ∈ − 1 2 , 1 2 , j ∈ I M . Assume that the window function ϕ ∈ Φ m,N1 is convenient for NFFT. Introducing the 1-periodic function the Fourier coefficients of g read as follows Using the trapezoidal rule, we approximate c k (g) by Note that c k (g), k ∈ I N , can be efficiently computed by FFT, for details see [18, p. 382]. Then the results of this NFFT with nonequispaced frequencies and equispaced spatial data are the values It is interesting that the same C(T)-error constant (2.6) appears in an error estimate of the NFFT with nonequispaced frequencies and equispaced spatial data too.
. For given f j ∈ C and nonequispaced frequencies x j ∈ − 1 2 , 1 2 , j ∈ I M , we consider the exponential sums (2.10) and the related approximations (2.11). Then the error of NFFT with nonequispaced frequencies and equispaced spatial data can be estimated by Proof. For each k ∈ I N we have From Hölder's inequality and Lemma 2 it follows that This completes the proof.

Auxiliary estimates
In our study we use later the following Proof. For −1 < u < 1, r ∈ N, and µ > 1 we have Using (3.1), the series can be estimated as follows Hence it follows by the integral test for convergence that For fixed µ ≥ 0, the µth Bessel function of first kind is defined by For fixed µ ≥ 0, the µth modified Bessel function of first kind is defined by For the properties of Bessel functions we refer to [1, pp. 355-478] and [25]. In particular, these Bessel functions possess the following asymptotic behaviors for x → ∞ (see [1, pp. 364, 377]), Here we are interested in explicit error estimates for NFFT with compactly supported, continuous window function. For this purpose, we need explicit bounds for the Bessel functions instead of the asymptotic formulas (3.2) and (3.3).
Lemma 6 For fixed µ > 1 2 and all x ≥ 0, it holds where 2 π is the best possible upper bound. In particular for µ = 1 and x ≥ 6, we have For µ = 5 2 and x ≥ 6, we have Proof. The inequality (3.4) was shown in [13]. For µ = 1, the inequality (3.4) means that Thus for all x ≥ 6 we have with the constant Similarly, one can show the inequalities (3.5) and (3.6).
Hence the function f (x) := √ 2πx e −x I µ (x) is strictly increasing on (0, ∞).  The main tool of this approach is the study of the Fourier transformφ(v) of ϕ ∈ Φ m,N1 for |v| → ∞. A rapid decay ofφ is essential for a small C(T)error constant (2.6). From Fourier analytical point of view, it is very interesting to discuss the relation to the known Theorem of Paley-Wiener (see [17, pp. 12-13] Proof. Let ϕ ∈ H k 0 (I) be given, i.e., D j ϕ ∈ L 2 (I), j = 0, . . . , k, and For arbitrary z = x + i y ∈ C and t ∈Ī we have and hence Since by the Schwarz inequality we have ϕ ∈ L 1 (I) too. Since D j ϕ ∈ L 2 (I) for j = 1, . . . , k, we have D j ϕ ∈ L 1 (I) analogously to (4.3).
Obviously, f is an entire function, because for each z ∈ C and j ∈ N it holds Using (4.1), repeated integration by parts applied to the function ϕ gives for j = 1, . . . , k the equalities From (4.2) and (4.5) it follows that for all z ∈ C it holds with the positive constant By (4.4) with j = k, we have for x ∈ R the equalities such that x k f (x) ∈ L 2 (I) by the Theorem of Plancherel. Now we assume that an entire function f with the properties 1. and 2. is given.
By the original Theorem of Paley-Wiener, the function vanishes for almost all x ∈ R \ I such that ϕ| I ∈ L 2 (I). For t ∈ R, from f (t) ∈ L 2 (R) and Each function in L 2 (R) generates a tempered distribution. By the differentiation property of the Fourier transform of tempered distributions we conclude that the jth weak derivative of ϕ exists almost everywhere and that Now we have to show that D j ϕ| R\I = 0. From property 2. of the entire function f it follows that the entire function (2πiz) j f (z) with j = 1, . . . , k fulfills the inequality (2πiz) j f (z) = |2πz| j |f (z)| ≤ γ k (ϕ) |2πz| j (1 + |2πz|) k e 2πm |Imz|/N1 ≤ γ k (ϕ) e 2πm |Imz|/N1 for all z ∈ C. Applying the original Theorem of Paley-Wiener to the entire function (2πiz) j f (z), we see that vanishes almost everywhere in R \ I. Since ψ j (x) = (D j ϕ)(x) = 0 almost everywhere, we have (D j ϕ)(x) = 0 for almost all x ∈ R \ I. Hence we have D j ϕ| I ∈ L 2 (I) for j = 1, . . . , k such that ϕ| I ∈ H k (I). This completes the proof.
Unfortunately, the assumption ϕ ∈ H k 0 (I) is too strong for the most popular window function.

Example 1
The triangular function ϕ(x) := 1 − N1 m |x|, x ∈ I, belongs to the Sobolev space H 1 0 (I), but doesn't belong to H 2 0 (I), since it holds ϕ ± m N1 = 0 and ϕ ′ ± m N1 = 0. In this case, we obtain that for z ∈ C, Thus Theorem 1 results in k = 1. Otherwise we observe a quadratic decay of the Fourier transformφ, since for x ∈ R \ {0}, Hence we will present a better method in the following Theorem 2.

Special window functions
In this section, we determine upper bounds of the error constant (2.6) for various special window functions ϕ ∈ Φ m,N1 by two methods. If the series For the algebraic, Bessel, sinh-type, and modified cosh-type window functions, we use the following argument. By Hölder's inequality it follows from (2.6) that Thus we show that the minimum of allφ(n) for relatively low frequencies n ∈ I N is equal toφ(N/2) and that the series with certain constants c 1 > 0, c 2 > 0, and µ > 1.

Then the constant e σ,N (ϕ) is bounded for all
Further, the C(T)-error constant e σ (ϕ) of the window function (5.2) has the upper bound Proof. Note that it holds By the scaling property of the Fourier transform, we havê For all n ∈ I N and r ∈ Z \ {0, ±1}, we obtain and hence From Lemma 5 it follows that for fixed u = n N1 ∈ − 1 2σ , 1 2σ , For all n ∈ I N , we sustain Thus we estimate for each n ∈ I N , Since m |n| N1 ≤ m 2σ for all n ∈ I N , we obtain Thus we see that the constant e σ,N (ϕ) can be estimated by an upper bound which depends on m and σ, but does not depend on N . We obtain Consequently, the C(T)-error constant e σ (ϕ) has the upper bound (5.3).
Uniform error estimates for nonequispaced fast Fourier transforms 21

B-spline window function
We start with the popular B-spline window function (see [6,24]). Assume that N ∈ 2 N and σ > 1 with N 1 = σN ∈ 2 N are given. We consider the B-spline window function where M 2m denotes the centered cardinal B-spline of even order 2m with m ∈ N. For m = 1, we obtain the triangular window function. Using the three-term recursion , m = 2, 3, . . . , Note that M 2m (0) > 0 for all m ∈ N, since it holds M 2m (x) > 0 for each x ∈ (−m, m). As known, the Fourier transform of (5.4) (see [18, p. 452]) has the formφ where sinc Ifφ B is the 1-periodization (2.1) of ϕ B , then the Fourier coefficients ofφ B read as follows Note that c k (φ B ) > 0 for all k ∈ I N . By (2.5) we see that Now we estimate the C(T)-error constants for NFFT. Applying the special structure of the Fourier coefficients c k (φ B ) and Lemma 5, we obtain a good upper bound (5.1) of the C(T)-error constant by this method. For a proof of the following result see [24].

Modified B-spline window function
Let σ ≥ 1, m ∈ N \ {1}, N ∈ 2 N, and N 1 = σN be given. The approach to the B-spline window function (5.4) can be generalized to the modified B-spline window function (see [14]) where M 2b denotes the centered cardinal B-spline of order 2b ∈ N \ {1, 2}, i.e., b ∈ { 3 2 , 2, 5 2 , . . .}. Assume that N 1 b ∈ 2 N and that m ∈ N fulfills the conditions m < 2σb and b = m. Using the three-term recursion (5.5), we find M 3 (0) = 3 4 , M 5 (0) = 115 192 . Obviously, it holds ϕ mB ∈ Φ m,N1 , where the Fourier transform of (5.6) (see [18, p. 452]) reads as followŝ Ifφ mB denotes the 1-periodization of (5.6), then the Fourier coefficients of ϕ mB have the following form Note that c k (φ mB ) > 0 for all k ∈ I N . Let f be an arbitrary 1-periodic trigonometric polynomial (2.3) which we approximate by the 1-periodic function with conveniently chosen coefficients g ℓ ∈ C. Then s possesses the Fourier expansion with the Fourier coefficients Uniform error estimates for nonequispaced fast Fourier transforms Thus the vector (ĝ k ) k ∈ I N 1 b is equal to the DFT of length N 1 b of the vector (g ℓ ) ℓ∈I N 1 b and we haveĝ k+rN1b =ĝ k for all k ∈ I N1b and r ∈ Z. In order to approximate f by s, we choosê Thus we see that c k (s) = c k (f ) for all k ∈ I N and c k (s) = 0 for all k ∈ I N1b \I N . Substituting k = n + rN 1 b with n ∈ I N1b and r ∈ Z, we obtain Then it follows by Hölder's inequality that Theorem 4 Let σ ≥ 1, N ∈ 2 N, 2b ∈ N \ {1, 2}, and N 1 b ∈ 2 N with N 1 = σN . Further let m ∈ N with 2b > m and b = m be given. Then the C(T)-error constant of NFFT with the modified B-spline window function (5.6) can be estimated by i.e., the modified B-spline window function (5.6) is convenient for NFFT.
Proof. Now we estimate max n∈IN r∈Z\{0} For n = 0 and r ∈ Z \ {0} we have For n ∈ I N \ {0} and r ∈ Z \ {0} we obtain since for n ∈ I N it holds Using Lemma 5, we conclude that with 2σb ≥ 3.

Algebraic window function
For fixed shape parameter β = 3m with m ∈ N \ {1} and for an oversampling factor σ > π 3 , we consider the algebraic window function Theorem 5 Let N ∈ 2N and σ > π 3 , where N 1 = σN ∈ 2N. Further let m ∈ N \ {1} and β = 3m be given. Then the C(T)-error constant of the algebraic window function (5.7) can be estimated by i.e., the algebraic window function (5.7) is convenient for NFFT.
Thus each factor of the infinite product (5.10) is positive and decreasing for v ∈ 0, m 2σ . Hence by (5.9) and (5.10), the Fourier transformφ 0,alg (v) is positive and decreasing for v ∈ 0, m 2σ and it holdŝ By (3.6) we know that for |v| ≥ m 2 it holds .
A proof of Theorem 7 can be found in [21]. The proof based mainly on the knowledge of the analytical Fourier transform. In [21] we consider in addition two related window functions, namely the continuous exp-type window function as well as the continuous cosh-type window function The main drawback for the numerical analysis of the exp-type/cosh-type window function is the fact that an explicit Fourier transform of this window function is unknown. Therefore we split the continuous exp-type/cosh-type 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. Note that ϕ exp and was first suggested in [4,2] and a discontinuous version of ϕ cosh was suggested in [4, Remark 13].
Note that the main idea to consider these modified window functions, comes from the Fourier transform of the Kaiser-Besser window, see [19,Remark 1]. The modified sinh-type window function ϕ msinh is used in the NFFT package [11], and gives very good error bounds.

Conclusion
In this paper, we present explicit error estimates for the NFFT with continuous, compactly supported window function ϕ ∈ Φ m,N1 . Such window functions simplify the algorithm for NFFT, since the truncation error of the NFFT vanishes. Using the C(T)-error constant of ϕ, we propose a unified approach to error estimates of the NFFT with nonequispaced spatial data and equispaced frequencies as well as of the NFFT with nonequispaced frequencies and equispaced spatial data. Further we discuss the connection with a modified Paley-Wiener theorem. We present two techniques to find upper bounds of the C(T)-error constant. The second method which uses the scaling structure of the window function ϕ, shows that the constant e σ,N (ϕ) is bounded for all N ∈ 2N. We see that e σ (ϕ) depends essentially on the decay of the Fourier transformφ(v) for |v| → ∞ and the positive size ofφ(v) for small frequencies. For the (modified) B-spline, algebraic, Bessel, sinh-type, and modified cosh-type window functions, we present new explicit upper bounds of the corresponding C(T)-error constants. Here we use the fact that the Fourier transforms of these window functions are explicitly known. It is remarkable that the C(T)-error constants of Bessel, sinh-type, and modified cosh-type window function decay exponentially with respect to m. Numerical experiments verify the different behavior of the C(T)-error constants for these window functions, see Figure 5.1. We point out that the modified cosh-type, exp-type, and sinh-type window functions give the best error constants, see Figure 5.2. The modified sinh-type window function, see Section 5.6, is used in the NFFT package [11].