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Fourier Series Windowed by a Bump Function

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Fourier Series Windowed by a Bump Function FOURIER SERIES WINDOWED BY A BUMP FUNCTION PAUL BERGOLD AND CAROLINE LASSER Abstract. We study the Fourier transform windowed by a bump function. We transfer Jackson's classical results on the convergence of the Fourier series of a periodic function to windowed series of a not necessarily periodic function. Numerical experiments illustrate the obtained theoretical results. 1. Introduction The theory of Fourier series plays an essential role in numerous applications of contemporary mathematics. It allows us to represent a periodic function in terms of complex exponentials. Indeed, any square integrable function f : R C of period 2π has a norm-convergent Fourier series such that (see e.g. [BN71, Prop.! 4.2.3.]) 1 f(x) = f(k)eikx almost everywhere, k= X−∞ where the Fourier coefficients areb defined according to π 1 ikx f(k) := f(x)e− dx; k Z: 2π π 2 Z− By the classical resultsb of Jackson in 1930, see [Jac94], the decay rate of the Fourier coefficients and therefore the convergence speed of the Fourier series depend on the regularity of the function. If f has a jump discontinuity, then the order of magnitude of the coefficients is (1= k ), as k . Moreover, if f is a smooth function of O j j j j ! 1 period 2π, say f Cs+1(R) for some s 1, then the order improves to (1= k s+1). 2 ≥ O j j In the present paper we focus on the reconstruction of a not necessarily periodic function with respect to a finite interval ( λ, λ). For this purpose let us think of − a smooth, non-periodic real function : R R, which we want to represent by a Fourier series in ( λ, λ). Therefore, we will! examine its 2λ-periodic extension, see − + Figure 1. Whenever ( λ ) = (λ−), the periodization has a jump discontinuity at λ, and thus the Fourier− coefficients6 are (1= k ). An easy way to eliminate these discontinuities at the boundary, is to multiplyO j thej original function by a smooth window, compactly supported in [ λ, λ]. The resulting periodization has no jumps. arXiv:1901.04365v2 [math.CA] 10 Jun 2020 Consequently, one expects faster convergence− of the windowed Fourier sums. The concept of windowed Fourier atoms has been introduced by Gabor in 1946, see [Gab46]. According to [Mal09, chapter 4.2], for (x; ξ) R2 and a symmetric 2 window function g : R R, satisfying g L2( ) = 1, these atoms are given by ! k k R iξy gx,ξ(y) := e g(y x); y R: − 2 Date: June 11, 2020. 2010 Mathematics Subject Classification. 42A16. Key words and phrases. Fourier series; window function; bump function. 1 2 PAUL BERGOLD AND CAROLINE LASSER ψ(x) (Pλψ)(x) Pλ λ λ x λ λ x − − + Figure 1. Effect of the periodization: If ( λ ) = (λ−), then the 2λ-periodic extension produces jump discontinuities− 6 at λ. Consequently, the order of the Fourier coefficients is (1= k ).± O j j The resulting short-time Fourier transform (STFT) of L2(R) is defined as 2 1 iξy (1.1) STFT( )(x; ξ) := ; g 2 = (y)g(y x)e− dy: h x,ξiL (R) − Z−∞ It can be understood as the Fourier transform on the real line of the windowed function g( x). If g is localized in a neighborhood of x, then the same ap- plies to the· windowed• − function. Hence, the spectrum of the STFT is connected to the windowed interval. In particular, Gabor investigated Gaussian windows with respect to the uncertainty principle, see [Chu92, chapter 3.1]. In many engineering applications, windows are discussed in terms of data weighting and spectral leakage. Depending on the type of signal, numerous windows have been developed, see e.g. [Har78, chapter IV]. More recently, in [MRS10] a smooth C1-bump window has been suggested for the analysis of gravitational waves. It is an essential property of this window, that it is equal to 1 in a closed subinterval of its support (plateau). Although the Fourier coefficients of such windows may exceed spectral convergence (faster than any fixed polynomial rate), it is their compact support which limits the order to be at most root exponential and the actual convergence rate depends on the growth of the window's derivatives, see e.g. [Tad07]. For example, in [Boy06] a smooth bump is designed such that the order of the windowed Fourier coefficients is root-exponential (at least for the saw wave function), wheres in [Tan06] we find a non-compactly supported window, for which we obtain true exponential decay. We note that Boyd and Tanner focus on an optimal choice of window parameters in order to obtain the best possible approximation results. We investigate the convergence speed of Fourier series windowed by compactly supported bump functions with a plateau. The properties of these bump windows will allow an effortless transfer of Jackson's classical results on the convergence of the Fourier series for smooth functions. The main new contributions of this paper can be found in Theorem 3.3 and Theorem 4.6, respectively. In the first one we show that pointwise multiplication (in the time domain) by a window with plateau yields smaller reconstruction errors in the interior of the plateau, compared to those windows without plateau. We complement this result by a lower error bound for the Hann window, a member of the set of cosα functions. In Theorem 4.6 we connect the decay rate of windowed Fourier coefficients to a new bound for the variation of windowed functions, which is based on the combination of two main ingredients: the Leibniz product rule and a bound for intermediate derivatives due to Ore. FOURIER SERIES WINDOWED BY A BUMP FUNCTION 3 1.1. Outline. We start by recalling basic properties of the Fourier series for func- tions of bounded variation in 2. Afterwards, in 3 we present the windowed trans- form, see Proposition 3.2, andx estimate the reconstructionx errors in Theorem 3.3. In 4 we introduce the Cs-bumps and transfer the results of chapter 3 to this class. Asx a special candidate of C1-bumps, we consider the Tukey window in 4.1. Finally, we present numerical experiments in 5, that underline our theoreticalx results and illustrate the benefits of bump windows.x 2. Functions of bounded variation and their Fourier series 2.1. Functions of bounded variation. We denote by BVloc the set of functions f : R R, which are locally of bounded variation, that is of bounded variation on every! finite interval. In particular, we assume that such functions are normalized for any x in the interior of the interval of definition, see [BN71, 0.6], by x 1 + 1 f(x) = f(x ) + f(x−) = lim f(x + t) + lim f(x t) : 2 2 t 0+ t 0+ − ! ! We recall that a function of bounded variation is bounded, has at most a countable set of jump discontinuities, and that the pointwise evaluation is well-defined. 2.2. The classical Fourier representation. Any 2π-periodic function f BVloc has a pointwise converging Fourier series, see [BN71, Prop. 4.1.5.]. Let us2 transfer this representation to an arbitrary interval of length 2λ: Lemma 2.1. (Fourier series of the periodization) Suppose that BVloc as well as λ > 0 and t R. Then, 2 2 ik π x (x) = c (k)e λ ; x (t λ, t + λ); 2 − k X2Z where the coefficients c (k) are given by t+λ 1 ik π x c (k) := (x)e− λ dx; k Z: 2λ t λ 2 Z − For the proof of Lemma 2.1 and our subsequent analysis, we will use a translation, a scaling and a periodization operator. For the center t R and a scaling factor a > 0, we introduce: 2 T : BV BV ; (T )(x) := (x + t); t loc ! loc t S : BV BV ; (S )(x) := (ax): a loc ! loc a For the period half length λ > 0, we set P : BV BV ; λ loc ! loc (x); if x ( λ, λ); 2 − (Pλ )(x) := 81 + > ( λ ) + (λ−) ; if x = λ. <2 − > Proof. Consider the 2π-periodic: function f = PπSλ/πTt . Then, it follows from Lemma A.1 that f BV and therefore 2 loc 1 ikx f(x) = f(k)e ; x R: 2 k= X−∞ b 4 PAUL BERGOLD AND CAROLINE LASSER The Fourier coefficients of f are given by π π 1 ikx 1 ikx f(k) = f(x)e− dx = Sλ/πTt (x)e− dx 2π π 2π π Z− Z− t+λ b 1 ik π (x t) = (x)e− λ − dx: 2λ t λ Z − Consequently, for all x (t λ, t + λ) we obtain 2 − ik π x (x) = T tSπ/λf (x) = c (k)e λ : − k X2Z 2.3. The classical result of Jackson. In general, even if is a smooth function, the periodic extension f = PπSλ/πTt BVloc has jump discontinuities at π. Let V (f) < denote the total variation2 of f. Then, by [Edw82, chapter 2.3.6],± 1 1 k c (k) = k f(k) V (f); for all k Z: j · j j · j ≤ 2π 2 Hence, the coefficients are (1= k ). Moreover, the rate of the coefficients transfers O j jb to an estimate for the reconstruction errors. For an arbitrary function f BVloc of period 2π let us introduce the partial Fourier sum 2 n ikx Snf(x) := f(k)e ; n 1; x R: ≥ 2 k= n X− Our analysis relies on the following classicalb result by Jackson on the convergence of the Fourier sum, see [Jac94, chapter II.3, Theorem IV]: Proposition 2.2.
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