Acta Mathematica Hungarica, v. , 2010,
APPROXIMATION OF BANDLIMITED FUNCTIONS BY TRIGONOMETRIC POLYNOMIALS S. NORVIDAS Institute of Mathematics and Informatics, Akademijos 4, Vilnius 08663, Lithuania e-mail:
[email protected]
Abstract. Let σ > 0. For 1 ≤ p ≤ ∞, the Bernstein space Bσp is a Banach space of all f ∈ Lp (R) such that f is bandlimited to σ; that is, the distributional Fourier transform of f is supported in [−σ, σ]. We study the approximation of f ∈ Bσp by finite trigonometric sums X
Pτ (x) = χτ (x) ·
π
ck,τ ei τ kx
|k|≤στ /π
in Lp norm on R as τ → ∞, where χτ denotes the indicator function of [−τ, τ ].
1
Introduction
The Fourier transform of f ∈ L1 (R) is defined here by Z ˆ f (x) = f (t)e−ixt dt. R
We normalize the inverse Fourier transform 1 fˇ(ξ) = 2π
Z f (t)eiξt dt R
so that the inversion formula (c fˇ) = f holds for suitable f . If f ∈ Lp (R) and p > 1, then we understood the Fourier transform fˆ in a distributional sense. Let σ > 0. A function f is called bandlimited to σ ( to [−σ, σ]) if fˆ vanishes outside [−σ, σ]. We denote by Bσp the Bernstein space of all functions f ∈ Lp (R) such that f is bandlimited to σ. The space Bσp is equipped with the norm ——————– Key words and phrases: Fourier transforms, bandlimited functions, entire functions of exponential type, exponential sums, trigonometric polynomials, Bernstein spaces, approximation by trigonometric polynomials. 2000 Mathematics Subject Classification: primary 30D15, secondary 42A10. 1
µZ
∞
kf kp =
¶1/p |f (x)|p dx
,
if 1 ≤ p < ∞
−∞
and kf k∞ = ess sup−∞ 0 such that |f (z)| 6 M e(σ+ε)|z| for all z ∈ C. There are several characterizations of Bσp . For example, a function f belongs to Bσp if and only if f is an entire function satisfying the Plancherel–Polya inequality µZ ∞ ¶1/p p |f (x + iy)| dx ≤ kf kp eσ|y| , (1.1) −∞
where y ∈ R. Note that 1 ≤ p ≤ r ≤ ∞ implies (see [3, p. 49, Lemma 6.6]) Bσp ⊂ Bσr .
(1.2)
Given f ∈ Bσ2 , Martin [9] studied the approximation of f in a suitable fashion by trigonometric polynomials. To be precise, for any τ > 0, set fτ (x) =
N X
π
ck,τ ei τ kx ,
(1.3)
k=−N
where ck,τ is the Fourier coefficient of f ck,τ
1 = 2τ
Z
τ
π
f (t)e−i τ kt dt
−τ
and N = [στ /π]Z . Here, [ω]Z denotes the integer part of a number ω. Define the τ truncated version of (1.3) by ϕf,τ := fτ · χ[−τ,τ ] ,
(1.4)
where χ[−τ,τ ] is the indicator (characteristic) function of [−τ, τ ]. Recall that a function f is said to be strictly bandlimited to [−σ, σ] if it is bandlimited to [−%, %] with 0 ≤ % < σ. It was proved in [9] that if f ∈ Bσ2 is strictly bandlimited to [−σ, σ], then (1.4) converges to f as τ → ∞ in L2 norm on R and on horizontal lines in C. This result was extended 2 in [12] to the Bernstein spaces BK consisting of functions of several variables bandlimited
to a compact set K ⊂ Rn . We now state the main result of this paper. THEOREM 1. Let f ∈ Bσp . If 1 < p < ∞, then lim kf − ϕf,τ kLp (R) = 0.
τ →∞
2
(1.5)
It is easily be checked that (1.5) is no longer true for p = ∞ in general. Indeed, if σ = 1 and f (x) = eix , then fτ (x) =
N X
(−1)k
k=−N
sin τ i π kx eτ . τ − πk
Thus, if τm = π/2 + 2πm, m ∈ Z, then ¯ ¯ ¯ ¯ kf − fτm kL∞ [−τm ,τm ] ≥ ¯(f − fτm )(τm )¯ ≥ =(f − fτm )(τm ) = 1 = kf kL∞ (R) . Approximation by trigonometric polynomials in Bσ∞ was also considered. Recall (see ∞ [2, p. 549]) that a sequence (ψm )∞ 1 , ψm ∈ L (R), is said to converges narrowly to ψ ∈
L∞ (R) if (ψm )∞ 1 converges uniformly to ψ on compact sets of R and limm→∞ kψm kL∞ (R) = kψkL∞ (R) . Lewitan [8] was the first to define the following polynomials f˜τ (x) :=
X
sin2 ( xτ + k) f (x + kτ ) x , τ > 0. ( τ + k)2 k∈Z
(1.6)
It was proved in [8] that if f ∈ Bσ∞ , then f˜τ converges narrowly to f . Krein [6] proposed a new proof of this statement, calculated the degree of approximation, and given several applications of (1.6). H¨ormander [4] considered more general trigonometric polynomials of such type (for more detailed history and references see [1, p.p. 146–152]). Schmeisser [14] proved similar convergence theorems for τ -truncated version of the Lewitan ( LewitanKrein-H¨ormander ) polynomials (1.6) with respect to Lp norms on lines in C parallel to R and with respect to lp norms in the case when f ∈ Bσp , p ∈ [1, ∞). The Lewitan polynomials of several variables were also studied (see [11] for more details and the literature cited there). Several well-known theorems were proved by using (1.6). The proofs are based usual on suitable properties of trigonometric polynomials and on the narrowly convergence of (1.6). For example, Krein [6] extended the Fejer and Riesz theorem on non-negative trigonometric polynomials to any non-negative entire functions of exponential type σ: such an entire function G can be written on R as |g|2 , where g is some entire function of exponential type at most σ/2. Other examples can be found in [4-5], [13] and [15, p. 244]. It turns out that (1.4) as well as the Lewitan polynomials (1.6) converge narrowly. Moreover, the following stronger statement holds. THEOREM 2. Let f ∈ Bσp and 1 < p < ∞. Then lim kf − ϕf,τ kL∞ (R) = 0.
τ →∞
3
(1.7)
2
Preliminaries and proofs
Let Z denote the group of integers and T = (−π, π] = R/(2πZ). We write (1.3) in the form
1 fτ (x) = 2τ
Z
τ
f (t)DN
³π
−τ
τ
´ (x − t) dt,
(2.1)
where DN is the Dirichlet kernel defined by N X
DN (ξ) =
eikξ .
k=−N
Therefore, setting Aτ (f ) = ϕf,τ ,
(2.2)
we may assume that (1.4) defines on Bσp the one-parametric family (Aτ )τ >0 of bounded linear operators Aτ : Bσp → Lp (R). LEMMA 1. Let p ∈ (1, ∞). Then there exists a constant a(p) > 0 such that kAτ (f )kLp (R) ≤ a(p)kf kp
(2.3)
for every f ∈ Bσp and all τ > 0. PROOF. Let us define uf (t) = f (τ t/π), t ∈ (−π, π]. We will consider uf as an element of Lp (T). It follows from (2.1) and (2.2) that Z π ¯Z π ³ ´ ¯p τ τ ¯ ¯ p p f ξ DN (y − ξ) dξ ¯ dy kAτ (f )kLp (R) = kAτ (f )kLp [−τ,τ ] = p 1+p ¯ 2π π −π −π Z π ¯Z π ¯ p τ τ ¯ ¯ = p 1+p uf (ξ)DN (y − ξ) dξ ¯ dy = p 1+p kuf ∗ DN kpLp (T) , ¯ 2π 2π −π −π
(2.4)
where N = [στ /π]Z . It is well known (see for example [16, p. 26, theorem 2.1.3]) that if 1 < p < ∞, then there exists such c(p) > 0 that kϕ ∗ DN kLp (T) ≤ c(p)kϕkLp (T) for every ϕ ∈ Lp (T) and all N = 0, 1, 2, . . . . If we combine this with (2.4), we get kAτ (f )kLp (R) ≤
c(p) kf · χ[−τ,τ ] kLp (R) 2π
for all σ, τ > 0 and every f ∈ Bσp . Thus we obtain (2.3) with a(p) = c(p)/(2π). Next we estimate the difference sin(ax)/x − αDN (bx) for some a, b and α. LEMMA 2. Let τ, σ > 0 and let N = [στ /π]Z . If 0 ≤ δ < 1, then ³ ´ ¯ 3 + ω π (1 + δ) ¯ ³ ´ ¯ sin σv 2 1 π ¯¯ − DN v ¯< , max ¯¯ |v|≤(1+δ)τ πv 2τ τ 2τ where ω(t) =
1 − cot t. t 4
(2.5)
(2.6)
PROOF. Substituting 2τ y/π for v in ³π ´ sin σv 1 − DN v , πv 2τ τ we get
¢ ¡ ¢! µ ¡ N y sin (2N + 1)y sin σv 1 X i π kv 1 sin 2στ π − − eτ = πv 2τ k=−N 2τ y sin y ¶ µ y − sin 2N y 1 sin 2στ π = − cos 2N y + ω(y) sin 2N y , 2τ y ³³ h i ´ ´ ³³ h i ´ ´ στ µ 2 sin στ − στ ¶ y cos π + στ y π π π 1 Z Z = − cos 2N y + ω(y) sin 2N y , 2τ y
where ω is defined in (2.6). If |v| ≤ (1 + δ)τ , then |y| ≤ π(1 + δ)/2 < π. Thus, to conclude the proof, it remains to note that the function (2.6) is odd on (−π, π) and increases on (0, π). PROOF OF THEOREM 1. First, we claim that if 1 ≤ p < ∞, then Bσ1 is a dense subset of Bσp . Indeed, let f ∈ Bσp , 1 ≤ p < ∞. If 0 < % < 1, then the function f% (x) =
¢ sin2 %x ¡ 2 f (1 − % )x (%x)2
belongs both to Bσp and to Bσ1 . Moreover, as % → 0, these f% tend to f in Lp norm on R. Finally, by (1.2), our claim is proved. Now, by the uniform boundedness principle (the Banach–Steinhaus theorem) and Lemma 1, it suffices to check (1.5) on any function in Bσ1 . Let f ∈ Bσ1 . Fix 0 < δ < 1. In order to prove (1.5), we estimate the Lp (−τ, τ )-norm of the following functions: Z δτ sin σ(x − t) dt, F1 (x) = f (x) − f (t) π(x − t) −δτ µ Z δτ ³π ´¶ sin σ(x − t) 1 F2 (x) = f (t) − DN (x − t) dt, π(x − t) 2τ τ −δτ and
1 F3 (x) = 2τ
Z f (t)DN δτ ≤t≤τ
³π τ
´¶ (x − t) dt,
(2.7) (2.8)
(2.9)
where N = [στ /π]Z . Let us begin with F2 . By Minkowski’s inequality ¯p ´1/p Z d ³Z b ´1/p ³Z b ¯ Z d ¯ ¯ ≤ |u(x, y)|p dx dy u(x, y) dy ¯ dx ¯ c
c
a
(2.10)
a
(see [15, p. 592, inequality (12)]), we have ³Z
τ
p
|F2 (x)| dx −τ
´ p1
µZ τ ¯ ³π ´¯p ¶ p1 1 ¯ ¯ sin σ(x − t) − DN (x − t) ¯ dx |f (t)|dt. ≤ ¯ π(x − t) 2τ τ −δτ −τ Z
δτ
5
Since |x − t| ≤ τ (1 + δ), it follows from (2.5) that ¡π ¢ Z δτ ³Z τ ´ p1 3 + ω (1 + δ) 2 |F2 (x)|p dx ≤ |f (t)| dt. 1−1/p (2τ ) −τ −δτ
(2.11)
Now we consider F1 . The function sin σx/πx belongs to each Bσq with 1 < q ≤ ∞. In addition, we have fˆ = χ[−σ,σ] . Therefore, if f ∈ B p , where 1 ≤ p < ∞, then σ
Z
∞
f (x) =
f (t) −∞
sin σ(x − t) dt π(x − t)
for x ∈ R (see [3, p. 50, theorem 6.11]). Applying this representation to f in (2.7) and using (2.10), we have ³Z
µZ τ ¯ ¯p ¶ p1 sin σ(x − t) ¯ ¯ |F1 (x)|p dx ≤ ¯ ¯ dx |f (t)| dt π(x − t) −τ |t|≥δτ −τ Z ° sin σv ° ° ° ≤° |f (t)| dt. ° πv Lp (R) |t|≥δτ τ
´ p1
Z
A similar argument shows that Z Z ´ p1 ³Z τ 1 ³ τ ¯¯ ³ π ´¯¯p ´ p1 p |F3 (x)| dx ≤ |f (t)| dt. ζ ¯ dζ · ¯DN 2τ −τ τ −τ δτ ≤t≤τ
(2.12)
(2.13)
It is known (see, for example, [10, p. 153]) that if 1 < p < ∞, then there exists 0 < α(p) < ∞ such that kDN kLp (T) ≤ α(p)(2N + 1)1−1/p . Therefore, since N ≤ στ /π, it follows from (2.13) that ³Z τ ´ p1 ³ 2σ 1 ´1−1/p Z p |F3 (x)| dx ≤ α(p) |f (t)| dt. + π τ −τ δτ ≤t≤τ
(2.14)
We have F1 + F2 − F3 = f − fτ . Hence, combining our estimates (2.11), (2.12), and (2.14) we obtain lim kf · χ[−τ,τ ] − ϕτ,f kLp (R) = lim k(f − fτ )χ[−τ,τ ] kLp (R) = 0.
τ →∞
τ →∞
Finally, the triangle inequality gives us (1.5). Theorem 1 is proved. Recall that if Q is a trigonometric polynomial on T of degree at most n, then kQkL∞ (T) ≤ 2n1/p kQkLp (T)
(2.15)
for each 1 ≤ p < ∞ (see [10, p. 495]). An analogous inequality holds in Bσp (see, for example, [15, p. 233]). Namely, if f ∈ Bσr1 and 1 ≤ r1 ≤ r2 ≤ ∞ , then 1
kf kr2 ≤ 2σ r1 6
− r1
2
kgkr1 .
(2.16)
We will need the following Bernstein inequality ³ σ|x − y| ´ |F (x) − F (y)| ≤ 2 sin kF k∞ , 2
(2.17)
where F ∈ Bσ∞ , x, y ∈ R, and σ|x − y| ≤ π (see [15, p. 213]). PROOF OF THEOREM 2. Let f ∈ Bσp , 1 < p < ∞. By (1.5), there exists a constant L(f ) > 0 such that sup kϕf,τ kLp (−τ,τ ) ≤ L(f ) < ∞.
(2.18)
τ >0
Set uf (t) = fτ
¡τ ¢ t , t ∈ (−π, π]. We will consider uf as an element of Lp (T). Then (2.15) π
and (2.18) imply that µ
1 p
kfτ kL∞ (R) = kfτ kL∞ ([−τ,τ ]) = kuf kL∞ (T) ≤ 2N kuf kLp (T)
πN =2 τ
1
1
¶ p1
kfτ kLp (−τ,τ )
1
≤ 2σ p kfτ kLp (−τ,τ ) = 2σ p kϕf,τ kLp (−τ,τ ) ≤ 2σ p L(f ). Therefore, using (2.16) in the case where r1 = p and r2 = ∞, we have that there is M (f ) > 0 such that sup kf − fτ kL∞ (R) ≤ M (f ) < ∞.
(2.19)
τ >0
If g ∈ Bσp and 1 ≤ p < ∞, then limt→∞ g(t) = 0 (see [7, p. 150]). Hence, in order to prove (1.7), it suffices to prove that limτ →∞ kf − ϕf,τ kL∞ (−τ,τ ) = 0. Assume to the contrary that there exist a constant a > 0 and a sequence τm → ∞ such that kf − ϕf,τm kL∞ (−τm ,τm ) ≥ a, m ∈ N. Therefore, we can choose xm ∈ [−τm , τm ] so that |(f − fτm )(xm )| = |(f − ϕf,τm )(xm )| ≥ a > 0,
(2.20)
m ∈ N. Now we will estimate kf − fτm kLp (−τm ,τm ) . Fix δ ∈ (0, 1). Assume without loss of generality that τm ≥ π/σ for all m ∈ N. Then for at least one of the intervals [xm − πδ/σ, xm ] and [xm , xm + πδ/σ] is contained in [−τm , τm ]. We denote such interval by Im . For any y ∈ Im , we take this y together with x = xm and F = f − fτm in (2.17). Therefore, if we recall (2.19) and (2.20), we obtain |(f − fτm )(y)| ≥ |(f − fτm )(xm )| − 2M (f ) sin
σ|y − xm | ≥ a − πδM (f ). 2
(2.21)
Also, (2.19) and (2.20) clearly imply a ≤ M (f ). Thus, substituting a/(2πM (f )) for δ in (2.21), we have Z
Z
τm
p
|(f − fτm )(y)|p dy ≥
|(f − fτm )(y)| dy ≥ −τm
Im
ap+1 2p+1 σM (f )
for all m ∈ N. This contradicts (1.5) and Theorem 2 is proved. Acknowledgments. The author thanks the referee for valuable suggestions which helped to improve this paper. 7
References [1] N. Y. Achieser, Theory of Approximation, Dover Publications. Inc. (New York, 1992). [2] E. Hewitt and K.A. Ross, Abstract Harmonic Analysis II, Springer-Verlag (BerlinHeidelberg, 1997). [3] J.R. Higgins, Sampling Theory in Fourier and Signal Analysis: Foundations, Clarendon Press (Oxford, 1996). [4] L. H¨ormander, A new proof and a generalization of an inequality of Bohr, Math. Scand., 2 (1954), 33–45. [5] L. H¨ormander, Some inequalities for functions of exponential type, Math. Scand., 3 (1955), 21–27. [6] M.G. Krein, On the representation of functions by Fourier–Stieltjes integrals, (Russian) Uch. Zap. Kuibysh. Gos. Ped. Uchit. Inst., 7 (1943), 123-147. [7] B.Ya. Levin, Lectures on Entire Functions, Transl. Math. Monogr., 150, Amer. Math. Soc. (1996). ¨ [8] B.M. Lewitan, Uber eine Verallgemeinerung der Ungleichungen von S. Bernstein und H. Bohr, Dokl. Akad. Nauk., 15 (1937), 169–172. [9] R. Martin, Approximation of Ω–bandlimited functions by Ω–bandlimited trigonometric polynomials, Sampl. Theory Signal Image Process., 6(3) (2007), 273–296. [10] G.V. Milovanovi´c, D.S. Mitrinovi´c and Th. M. Rassias, Topics in Polynomials: Extremal problems, inequalities, Zeros, World Scientific Publishing Co., Inc. (River Edge, NJ, 1994). [11] S. Norvidas, Approximation of entire functions by exponential polynomials, Lith. Math. J., 34(4) (1994), 415–421. [12] S. Norvidas, Approximation of bandlimited functions by finite exponential sums, Lith. Math. J., 49(2) (2009), 185–189. [13] Q.I. Rahman and G. Schmeisser, Lp inequalities for entire functions of exponential type, Trans. Amer. Math. Soc., 320 (1990), 91–103. [14] G. Schmeisser, Approximation of entire functions of exponential type by trigonometric polynomials, Sampl. Theory Signal Image Process., 6(3) (2007), 297–306. [15] A.F. Timan, Theory of Approximation of Functions of a Real Variable, Dover Publications, Inc. (New York, 1994). [16] R.M. Trigub and E.S. Bellinsky, Fourier Analysis and Approximation of Functions, Kluwer Academic Publishers (Dordrecht, 2004).
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