Fourier Series Windowed by a Bump Function
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(2020) 26:65
Fourier Series Windowed by a Bump Function Paul Bergold1 · Caroline Lasser1 Received: 14 January 2019 / Revised: 11 June 2020 © The Author(s) 2020
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. Keywords Fourier series · Window function · Bump function Mathematics Subject Classification 42A16
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. [1, Prop. 4.2.3.]) f (x) =
∞
f (k)eikx almost everywhere,
k=−∞
where the Fourier coefficients are defined according to 1 f (k) := 2π
π
−π
f (x)e−ikx dx, k ∈ Z.
Communicated by Arieh Iserles.
B
Paul Bergold [email protected] Caroline Lasser [email protected]
1
Zentrum Mathematik, Technische Universität München, München, Germany 0123456789().: V,-vol
65
Page 2 of 28
Journal of Fourier Analysis and Applications
(2020) 26:65
(Pλ ψ)(x)
ψ(x)
Pλ
−λ
λ
x
−λ
λ
x
Fig. 1 Effect of the periodization: If ψ(−λ+ ) = ψ(λ− ), then the 2λ-periodic extension produces jump discontinuities at ±λ. Consequently, the order of the Fourier coefficients is O(1/|k|) (Color figure online)
By the classical results of Jackson in 1930, see [14], 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 O(1/|k|), as |k| → ∞. Moreover, if f is a smooth function of period 2π , say f ∈ C s+1 (R) for some s ≥ 1, then the order improves to O(1/|k|s+1 ). 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 ψ : R → R, which we want to represent by a Fourier series in (−λ, λ). Therefore, we will examine its 2λ-periodic extension, see Fig. 1. Whenever ψ(−λ+ ) = ψ(λ− ), the periodization has a jump discontinuity at λ, and thus the Fourier coefficients are O(1/|k|). An easy way to eliminate these discontinuities at the boundary, is to multiply the original function by a smooth window, compactly supported in [−λ, λ]. The resulting periodization has no jumps. Consequently, one expects faster convergence of the windowed Fourier sums. The concept of windowed Fourier atoms has been introduced by Gabor in 1946, see [8]. According to [17, Chap. 4.2], for (x, ξ ) ∈ R2 and a symmetric window function g : R → R, satisfying g L 2 (R) = 1, these atoms are given by gx,ξ (y) := eiξ y g(y − x),
y ∈ R.
The resulting short-time Fourier transform (STFT) of
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