NORM DEPENDENCE OF THE COEFFICIENT MAP ON THE WINDOW SIZE 1

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1 NORM DEPENDENCE OF THE COEFFICIENT MAP ON THE WINDOW SIZE Thomas I. Seidman and M. Seetharama Gowda Department of Mathematics and Statistics University of Maryland Baltimore County Baltimore, MD 8, U.S.A ABSTRACT: For sparse exponent sequences (λ k ), satisfying a suitable separation condition defined by an auxiliary sequence ψ, one has a coefficient map C δ giving (c k ) =: c from observation of f = k= c k e iλkt on any arbitrarily small interval [ δ, δ]. In terms of ψ, we estimate the norm of C δ : L [ δ, δ] l, asymptotically as δ. In particular, for (λ k ) k p (p > ) we get a bound which is exponential in (/δ) /(p ), generalizing an earlier result for the case p =. Key Words: nonharmonic Fourier series, coefficient map, norm, window, asymptotic. This has appeared in Math. Scand. 73, pp (994). seidman@math.umbc.edu. This research has been partially supported for this author by the National Science Foundation under grant ECS and by the U.S. Air Force Office of Scientific Research under grant AFOSR-9-8.

2 . Introduction Let λ = {λ k } be a real double sequence (k =, ±, ±,...) and let the collection of all finite sums (.) f = c k e iλ kt k M= M (λ) denote with complex coefficients c k. We will here think of viewing such f through a window ( δ, δ) and determining the coefficients {c k } from this, defining a coefficient map (.) C= C (λ) : f = k c k e iλ kt (c k ) for f M. If we now let M δ = M δ (λ) be the closure of that C extends from M to M δ as a continuous linear operator M in L ( δ, δ), then it is classical (.3) C δ = C δ (λ) : M δ l : f = c k e iλkt (c k ) =: c k= provided the asymptotic density of λ is bounded by δ/π. In this paper, we consider sequences λ satisfying sparsity conditions of the form (.4) λ k+m λ k ψ m (m =,,...) for suitable ψ = {ψ m : m =,,...}. Noting that m/ψ m ensures that C δ (λ) is well defined for all δ >, we then investigate the rapidity with which C δ (λ) as δ. As a by-product of this analysis, we note that our estimates are uniform over the classes of exponent sequences Λ = Λ(ψ) satisfying (.4) for particular admissible sequences ψ. Our results are new in this aspect as well as in the consideration of the asymptotics as δ. It is clear that C δ (λ) is made up of the coefficient functionals γ k : M δ = M δ (λ) C : f c k and that each of these functionals can be represented as (.5) γ k : f c k = f, g k for some g k L ( δ, δ). There is some arbitrariness in the determination of g k since (.5) constitutes an extension of γ k from M δ to all of L ( δ, δ); this also gives an extension C δ of C δ (λ) to L ( δ, δ). Since we are working with exponentials, it is then convenient to construct the Fourier transforms to obtain g k and we actually will work with the adjoint of C δ, (.6) C δ : (a k ) k a k g k : l L ( δ, δ),

3 to estimate 3 C δ C δ = C δ. We will be able to treat conditions (.4) for real sequences ψ = {ψ m : m =,,...} for which (.7) < ψ ψ... and /ψ k < Note that this already implies that m/ψ m which precisely corresponds to the condition that Λ have asymptotic density zero. Our paper falls into three parts: First, considering a sequence λ satisfying (.4) subject to (.7), we will apply an important theorem due to Luxembourg and Korevaar ([]; Theorem 3.) which we restate here in a relevant form: THEOREM K-L: Let ω : IR + IR + be nondecreasing with ω(t)/t integrable at. Then, for any δ >, there exists a number Q > and an entire function P ( ) such that: (i) P is of exponential type δ, (ii) P is normalized so P () =, (iii) P (s) e Q e ω( s ) for s IR. Clearly, the constant Q in (iii) will depend on δ so Q = Q(δ) = Q(δ; ω). From this we then obtain an estimate: (.8) C δ Ae Q(δ) for all δ > where A is independent of δ and we use a function ω depending only on ψ. So far, this is only slightly different from the treatment in []. Second, and this is the principal technical innovation of the paper, we extend the analysis of Theorem K-L from that of [], specifically considering the estimation of Q(δ) in (.8) so as to exhibit explicitly its asymptotics as δ as well as the dependence on ψ through ω( ). From (.8), this is precisely what is needed to investigate the asymptotic behavior of C δ. For this estimation, we find it necessary to assume ω Ω where (.9) Ω := { ω : IR + IR + : ω(t) is nondecreasing while ω(t)/t is decreasing and integrable at this strengthens very slightly the hypotheses above for Theorem K-L in requiring that ω(t)/t be decreasing. Third, the combination of the above is applied to obtain specific growth estimates for a number of interesting special cases. In particular, we apply the general analysis to the case ψ m = am p (a >, p > ) which corresponds to λ k ±ak p and obtain for that case the estimate (.) log C δ = O([/δ] /(p ) ) as δ 3 Note that one has the geometric characterization γ k = /[distance from e iλ kt to M δ (λ k )] where λ k = λ\{λ k }. This gives γ k / e iλkt = / δ, showing the uselessness of the crude estimate C δ f = ( ) γ k, f γ k f. k k 3 } ;

4 (i.e., Q(δ) µ[/δ] /(p ) in (.8) for small δ > ; we also have an estimate for µ). We recall that earlier investigation in [4] of the special case (.) λ k = k (k ), λ k = λ k resulted in an estimate log C δ = O(/δ) which was there shown to be sharp (by an example due to Korevaar).. The Interpolation Family Assume that ψ satisfies (.7) and that λ = (λ k ) is in Λ(ψ), i.e., satisfies the separation condition (.4). With ψ we associate the function Ψ given by (.) ( ) Ψ(s) := log + s. m= ψm We will show in the Appendix (Lemma A.) that this function Ψ( ) is, indeed, in Ω. Given λ Λ(ψ), we next define a family of functions (η k ) by first defining (.) µ k (z) := j k ( z λj λ k λ j ) (z C) and then setting (.3) η k (z) := µ k (z)µ k (λ k z) = j k ( z λk λ k λ j ). LEMMA.: (.4) Let ψ, Ψ be as above and define η k for k Z as in (.3). Then one has η k (λ j ) = δ j,k (j, k Z) and each η k ( ) is an entire function of exponential type with (.5) η k (λ k + s) e Ψ( s ) s IR. Proof: For each k and any N >, there is some M = M k,n such that µ k (z) z λ j exp ( z + λ λ < j k N k λ j k ) λ j k >N j λ k [ M( + z ) N+ exp z ] m>n ψ m 4

5 for all z C. This estimate ensures suitable convergence of the product in (.) to have µ k entire. Further, the sum in the exponential can be made arbitrarily small by taking N large since {/ψ m } is summable by assumption whence each µ k (and so each η k ) is of exponential type. The property (.4) is obvious from µ k (λ j ) = δ j,k. Finally, for real s we have η k (λ k + s) ( ) s + j k λ k λ j ( ) + s = e Ψ( s ) m ψm so one has (.5) as desired. Selecting any γ Ω such that e γ is integrable, we take ω = Ψ + γ which is in Ω by Lemma A.; then, fixing δ >, we let P ( ) and Q = Q(δ) be as in Theorem K-L. In terms of this P, we define the family of functions (.6) G k (z) := η k (z)p (z λ k ) (z C). Our first principal result of this section is the following. THEOREM.: We have: (i) Each G k is an entire analytic function of exponential type δ, (ii) For j, k Z we have G k (λ j ) = δ j,k := { for j = k; else }, (iii) Each G k, considered on the reals, is in L (IR) with (.7) (iv) (.8) Each G k is in L (IR) and one has G k (λ k + s) e Q(δ) e γ( s ) G j, G k [4e Q(δ) ] e γ(s) ds e γ(ψm/) for any j = k ± m (i.e., m = k j ). Proof: The assertion (i) follows on combining Lemma. (for η k ) and Theorem K-L with ω = Ψ + γ. As noted in Lemma., we have η k (λ j ) = δ j,k ; hence, since P () =, we have (ii). The estimate (.7) is immediate from (.5) combined with Theorem K-L (iii) so we have (iii). Finally, to prove (iv) we assume, with no loss of generality, that λ j λ k and set λ := (λ j + λ k )/. Note that we then have λ j = λ τ and λ k = λ + τ with τ := (λ k λ j )/ ψ m / by (.4) so γ(τ) γ(ψ m /). Note also that for t λ one has t λ j =: s τ so 5

6 τ s τ and γ( τ s ) γ(τ); for t λ we set s := t λ k τ and γ( τ + s ) γ(τ). Thus, using (.7), G j, G k = τ G j (t) G k (t) dt = λ + λ G j (λ j + s) G k (λ k [τ s]) ds + τ [ τ e Q(δ) e Q(δ) e γ(ψm/) [ G j (λ j + [τ + s]) G k (λ k + s) ds ] e γ( s ) e γ(τ) ds + e γ(τ) e γ( s ) ds e γ( s ) ds which is just (.8). In particular, for j = k this shows G k L (IR). Depending on the choice of γ( ), this construction will determine the constant Q(δ) of (.7) as a function of δ >. Also depending on the choice of γ( ), but now not on δ, we set A := [ ] (.9) e γ(s) ds e γ() + e γ(ψm/). π Noting that (.7) gives ψ m cm for some c > so γ(ψ m /) γ(cm), we may compare the sum to the integral e γ(cs) ds and observe that the integrability of e γ ensures finiteness of A. Our other principal result of this section is the following. THEOREM.3: m= τ ] Let λ Λ(ψ) for some sequence ψ satisfying (.7). Then for any δ > the coefficient map C δ = C δ (λ) defined by (.3) satisfies (.) C δ Ae Q(δ) (δ > ) with Q(δ) as in (.7) and A as in (.9), independent of δ. Proof: The argument is here quite similar to that in [4]. We use the Fourier transform F : L (IR) L (IR) given by (.) F : g G with G(z) = e izt g(t) dt and note that, with a factor of π, this is an isometric isomorphism: (.) g, g := g(t) g(t) dt = π G(t) G(t) dt = G, G. π By Theorem. (iv), each G k is in L (IR) so there exist functions g k L (IR) with G k = Fg k (k Z). By the Paley-Wiener Theorem [3], since Theorem. (i) gives each G k entire and of exponential type δ, the support of each g k is contained in the window [ δ, δ]. 6

7 Initially, let us consider f M (so f = k c k e iλ kt is a finite sum) and view this through the window as f L ( δ, δ). Then, for each k Z, noting that supp (g k ) [ δ, δ], f, g k := = j δ δ j c j By Theorem. (ii) we thus have, as in (.5), c j e iλ jt g k (t) dt e iλ jt g k (t) dt = j G k (λ j )c j. (.3) c k = f, g k (k Z, f M). Now consider the Gramian matrix G with entries g j, g k. Since we continue to consider the (fixed) function f M as a finite sum, we may take G to be a finite matrix, considering only the indices k for which c k ; thus there are no convergence problems but we seek estimates independent of this restricted index set. As a Gramian matrix, G is positive definite so the l -induced matrix norm G is just the largest eigenvalue of G. Hence, (.4) G G := max { j k g j, g k } since G is itself the l -induced matrix norm. Thus we have a k g k = g j, g k a j a k G a k L ( δ,δ) j,k for (finite) vectors a = (a k ) l. Hence, using (.3), c = k c k = k f, g k c k = f, k c k g k f k c k g k f ( G c ) / so for f M we have the estimate (.5) c l ( G ) / f L ( δ,δ). We now use (.) and (.8) to estimate G from (.4). Fixing j, we consider k Z and set m := k j so g j, g k = π G j, G k 4e Q(δ) Summing over k Z then gives g j, g k A e Q(δ) k 7 [ ] e γ(s) ds e γ(ψm/). π

8 for each j so G (Ae Q(δ) ). Combining this with (.5) gives for all f M. By the density of C δ f = c Ae Q(δ) f M in M δ, this gives precisely the desired estimate (.). 3. The Mollifier Our next object is to re-examine Theorem K-L so as to introduce the mollifier P ( ) with a reasonably explicit estimate for Q = Q(δ). To this end, given ω Ω we set (3.) v(s) = ω(s) s, dq = s dv. Note that the definition (.9) of Ω ensures that q is an unbounded increasing function of s and that ω()/ as. For each in (, ) we can then set (3.) δ() := + ω() + ω(s) s ds = + dq s. LEMMA 3.: Fix ω Ω and let δ( ) be defined by (3.). Then δ() is nonincreasing on (, ) and δ() as so for each δ > there exists an := (δ) such that δ() δ. Further, fixing δ >, there is a sequence (a j ) such that (3.3) (3.4) a j δ() δ [a j ] ω( s ) a j s s for s >. Proof: Deferred to the Appendix. We can now state our revised form of Theorem K-L, including the estimate of Q(δ). THEOREM 3.: For any δ >, define P (z) by (3.5) P (z) := cos(a j z) j= (z C), using the sequence (a j ) of Lemma 3.. Then P ( ) is an even entire function of exponential type δ with P () =. Further, one has (3.6) P (s) e Q(δ) e ω( s ) 8

9 for all s IR, where Lemma 3. is used to define (3.7) Q(δ) := / + ω((δ)). Proof: We know that cos( ) is even and entire of exponential type. By (3.3), it follows that P is a well-defined even, entire function of exponential type δ. Observing that [ ] cos s exp s for s, it follows from (3.4) that for s > we have P (s) { cos(a j s) : a j s } exp s a j e / e ω( s ). a j s Since P (s) e ω() e ω( s ) for any s, we have (3.6) for all s IR. 4. Examples We now specialize our work to treat some particular cases more explicitly. In each case, we take ω = ( + ε)ψ, i.e., γ := εψ. A principal point, here, is that the asymptotics of Q(δ) as δ are (almost) determined by the asymptotics of ψ m as m. In the first two examples, we also note the convenience of taking ψ m = ψ(m) for a suitable function ψ( ), giving an integral version of (.) for the asymptotically correct determination of Ψ( ). EXAMPLE : We first suppose ψ(x) = ax p (a >, p > ); when p = this is the case considered in [4]. It is easily seen that ψ := {ψ m } satisfies (.7). To simplify the explicit computation of various quantities, we deal with the integral version of (.), namely, Ψ(s) := log ( = log [ = s /p a /p where (cf., e.g., [] p.4) β(p) = (3.), δ() = + ω() = ( ) + s dx ψ(x) ) + s dx a x p ( log + u p ) ] du =: β(p) s /p π. Now, let ε > and let ω(s) := ( + ε)ψ(s). From a /p sin π p ω(s) + + ( + ε)β(p)/p = + ϑ /q 9 s + ds ( + ε)β(p)s /p ds s

10 where ϑ := ϑ(ε, p) = ( + q) ( + ε) β(p) with pq = p + q. Since δ() ( + ε)ϑ/ /q for large, we see that (δ) [( + ε)ϑ/δ] q for large, i.e., for small δ. By (3.7), Thus, (.) becomes (4.) Q(δ) := + ω() + ( + ε)β(p) ( ) (+ɛ)ϑ q/p δ. C δ Ae / exp [ B (/δ) q/p] where, with a corresponding constant A, B > B is arbitrary with ( ) + q q/p ( ) q B := β(p)ϑ q/p = q π. a sin π/p Since q =, we have the promised estimate (.). p p EXAMPLE : We next consider sequences which are even more sparse: ψ(x) = ce βx with c, β >, indicating how various quantities can be computed. We now have Ψ(s) = = β σ = β log ( ) + s ψ(x) log( + e r ) dr log( + e r ) dr + β β [log s] as s. dx = β σ ( ) log + s β dx c e βx (where s c = e σ, r = βx σ) log( + e r ) dr Asymptotically, ω(s) ( + ε) [log β s], so one has δ() 8(+ε) [log β ] by a simple computation and δ() (/) /p for arbitrary p > and large. Hence (δ) /δ p for small δ. Thus, Q(δ) := + ω() + ( + β ε)p [log /δ] and one has, therefore, (4.) C δ A exp [ B(log /δ) ] for any B > B := /β and a suitable constant A. EXAMPLE 3: In this example, we consider the ultimate asymptotic sparsity: a finite sequence {λ j } of L + distinct real numbers. Taking these in increasing order and setting c := min{ λ k λ j : j k} >, we then automatically have the condition (.4) with ψ m := mc for m =,..., L and (formally) ψ m := for m > L, giving ( ) L Ψ(s) := log + s = 4L log s + O(). m= c m

11 We now have ω(s) 4L( + ε) log s so δ() 6L( + ε) log for large. Hence, as in the previous example, (δ) /δ p for arbitrary p > and small δ so Q(δ) := + ω() + 4p( + ε)l log /δ. One therefore has algebraic growth in /δ for the norm in this case: (4.3) C δ Aδ ν for any ν > ν := 4L with a corresponding constant A. 5. Appendix LEMMA A.: Let ψ be any increasing positive sequence satisfying (.7). Then (.) defines a function Ψ on IR + such that (5.) (i) Ψ is continuous and unbounded on [, ) with Ψ() =, (ii) Ψ is C and (strictly) increasing on IR +, (iii) Ψ(s)/s is decreasing on (, ), (iv) Ψ(s)/s is integrable at, (v) e Ψ is integrable on IR +. Proof: Since ψ = ψ k as k so log(+/ψ) /ψ, we see from (.7) that the sum in (.) is well defined and finite for each s. Further, each term in that sum is (strictly) increasing in s and continuous. By the Weierstrass M-test, the series converges uniformly on any closed and bounded interval in IR + so Ψ(s) is continuous. Similarly, Ψ = 4 s (s +ψm) which is finite by (.7) and positive on IR +. Thus we have (5.-i,ii). To see (iii), we observe that Ψ(s)/s ρ([s/ψ m ] ) = ψm with ρ(u) := log(+u) and note that ρ is strictly decreasing for u >. u To get (iv), we observe that Ψ(s)/s will be integrable at if and only if the series { (/s ) log( + s /ψk) ds} is summable. From the identity log( + u ) = tan u log( + u ), u u

12 we get (/s ) log ( ) + s ψ ds = π ψ ( ψ tan ψ + log + ) ψ Using (.7) and (i), we get (iv). Statement (v) is obvious. Finally, we provide the promised proof of Lemma 3.. Proof [of Lemma 3.]: As already noted, the definition (.9) of Ω ensures that q is increasing, so the right hand side of (3.) is (strictly) decreasing to as by the integrability of ω/s. Thus, δ( ) is invertible with (δ) defined for (small) δ >. Now, fixing δ > and so = (δ), we use 4 (3.) to define. (5.) a j := /q (z j ) with z j := q() + j/ for j =,,.... An integral comparison, noting that the function /q ( ) is decreasing and that z j+ z j /, gives a j = + q (z j ) + dz q() q (z) which precisely gives (3.3) on using (3.) for z = q(s). For s >, we now note that j where j = j (s, ) is the smallest j for which a j s ; hence, z j q( s ) <. An argument similar to the above then gives which is just (3.4). a j s [a j ] = [q (z j )] j q( s ) = v( s ) /s z j dz [q (z)] dz [q (z)] s (since q (z) s for s z z j ) References [] J.B. Conway, Functions of One Complex Variable, Springer, NY, 973. [] W.A.J. Luxemburg and J. Korevaar, Entire functions and Müntz Szász type approximation, Trans. Amer. Math. Soc. 57 (97), pp [3] W. Rudin, Real and Complex Analysis (3 rd ed.), McGraw-Hill, NY, Although (3.) only defines q( ) to within an additive constant, the formula (5.) suffices to specify (a j ).

13 [4] T.I. Seidman, The coefficient map for certain exponential sums, Nederl. Akad. Wetensch. Proc. Ser. A, 89 (= Indag. Math. 48), pp (986). [5] T.I. Seidman, How violent are fast controls? Math. of Control, Signals, Syst. (988), pp