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Growth Estimates for Sine-Type-Functions and Applications to Riesz Bases of Exponentials

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Growth Estimates for Sine-Type-Functions and Applications to Riesz Bases of Exponentials Growth estimates for sine-type-functions and applications to Riesz bases of exponentials Alexander M. Lindner Abstract We present explicit estimates for the growth of sine-type-functions as well as for the derivatives at their zero sets, thus obtaining explicit constants in a result of Levin. The estimates are then used to derive explicit lower bounds for exponential Riesz bases, as they arise in Avdonin's Theorem on 1/4 in the mean or in a Theorem of Bogm´er, Horv´ath,Jo´oand Seip. An application is discussed, where knowledge of explicit lower bounds of exponential Riesz bases is desirable. 1 Introduction Consider the function sin(π ). From the triangle inequality, it follows that · π z sin πz e j= j z C j j ≤ 8 2 and 2π 1 e− π z − e j= j sin πz z C : z 1: 2 ≤ j j 8 2 j= j ≥ 1 in( ) Furthermore, the integers are the zero set of sin(π ), and ( e · )n Z is · p2π 2 an orthonormal basis for L2( π; π). Motivated by this, Levin asked for functions having similar properties− to the sine function, such that their zero sets give rise to more general bases of exponentials. In [12], he invented sine-type-functions. We have the following Definition 1.1 (a) An entire function F is of exponential type at most σ (σ > 0), if for any " > 0 there exists an A" > 0 such that (σ+") z F (z) A" e j j z C: j j ≤ · 8 2 1 sine-type-functions 2 If there is σ > 0, such that F is of exponential type at most σ, then F is called of exponential type. (b) Let σ > 0. An entire function F of exponential type is called σ-sine- type-function, if there are positive constants C1;C2 and τ such that σ y σ y C1 e j j F (x + iy) C2 e j j x; y R : y τ: (1) · ≤ j j ≤ · 8 2 j j ≥ We shall say that F has growth constants (C1;C2; τ): A large class of sine-type-functions was established by Sedleckiˇi [19], see also Young [24, Ch. 4, Sec. 5, Problem 3]. iλn( ) 2 In [12], Levin proved that (e · )n Z will be a Schauder basis for L ( σ; σ) 2 − if (λn)n Z is the zero set of a σ-sine-type-function and if it is separated. In [5], 2 iλn( ) 2 Golovin showed that (e · )n Z will then even be a Riesz basis for L ( σ; σ). Here we have the following definitions:2 − Definition 1.2 A sequence (λn)n Z of complex numbers is called separated, if there is δ > 0 such that 2 λn λm δ n; m Z : n = m: j − j ≥ 8 2 6 The constant δ is called a separation constant. Definition 1.3 Let σ > 0 and (λn)n Z be a sequence of complex numbers. iλn( ) 22 iλn( ) Then (e · )n Z is a Riesz basis for L ( σ; σ), if (e · )n Z is complete in L2( σ; σ) and2 if there are positive constants− A and B such2 that − σ 2 2 iλkx 2 A ck Z ck e dx B ck (2) X j j ≤ σ X ≤ X j j − holds for all finite sequences (ck) of complex scalars. The constants A and B are called lower and upper bounds of the Riesz basis. Riesz bases of exponentials allow representations of functions as infinite series. They also give rise to irregular sampling series. For the correspond- ing truncation error, explicit estimates for the bounds of the Riesz basis are needed. This will be discussed in Section 3. To obtain the result of Levin and Golovin mentioned above, Levin [12, 13] showed that a sine-type-function has the following properties. The proofs can be found in Levin [12], [13], [14, Lectures 6, 17, 22]. sine-type-functions 3 Theorem 1.4 Let F be a σ-sine-type-function with zero sequence (λn)n Z, counted according to multiplicity, and ordered such that 2 λn λn+1 n Z: (3) < ≤ < 8 2 Then holds: (a) M > 0 : λn+1 λn M n Z: (4) 9 j − j ≤ 8 2 (b) For any positive number r, there exists some positive integer Sr such that for every t R, 2 card n Z : r λn t < r Sr: (5) f 2 − ≤ < − g ≤ (c) If we put 1 z/λ ; if λ = 0 ( n n Gλn (z) := − 6 z; if λn = 0; then there is a constant CF = 0 such that 6 F (z) = CF lim Gλn (z) z C: (6) · R Y 8 2 !1 n Z: λn R 2 j |≤ The product converges locally uniformly on C. (d) σ z C0 > 0 : F (z) C0 e j= j z C: 9 2 j j ≤ 2 · 8 2 (e) For any " > 0 there exists m" > 0 such that σ z F (z) m" e j= j z C : dist (z; λn : n Z ) ": (7) j j ≥ · 8 2 f 2 g ≥ (f) If (λn)n Z is separated, then there exist positive constants C3 and C4 such that2 C3 F 0(λn) C4 n Z: (8) ≤ j j ≤ 8 2 Properties (a) and (b) refer to the sequence (λn)n Z, and are usually easy to check. We also remark that by the Hadamard Factorization2 Theorem, the zero-set of any sine-type-function must be countable infinite. Furthermore, the exponential type σ is related with the numbers Sr via the following Lemma by Horv´athand Jo´o[6, Lemma 2]: sine-type-functions 4 Lemma 1.5 Let F be a σ-sine-type function with zero sequence (λn)n Z. Then it holds 2 1 lim max card n Z : r λn t < r = r r t R !1 2 f 2 − ≤ < − g 1 σ lim min card n Z : r λn t < r = : r r t R π !1 2 f 2 − ≤ < − g In particular, if Sr denotes the best constant occuring in (5), then S S σ inf r = lim r = : r>0 2r r 2r π !1 Suppose that F is a σ-sine-type-function which has growth constants (C1;C2; τ). Then it is easy to show that possible choices for C20 and C4 are στ C0 := C2 e ; 2 · 2 στ+σδ=2 C4 := C2 e δ · · (using the maximum principle for C20 and Levin's proof for C4). However, Levin's proof for the existence of the sequence (m")">0 was indirect. He thus obtained no explicit values for m", in terms of the growth constants, of σ and of constants describing the sequence (λn)n Z, e.g. a separation constant or the constants M and S appearing in (a) and2 (b). Since he derived the ex- istence of C3 using mδ=2 (where δ is the separation constant), he neither did obtain an explicit expression for C3. In this paper, we shall obtain explicit estimates for m" and for C3. This will be done in the next section, where we also give examples to discuss the goodness of these estimates. In [11, Theorem 2], Katsnel'son proved a generalization of Levin and Golovin's Theorem, which was further generalized by Avdonin [1, Theorem 2]. The latter proved, that if the deviation of (λn)n Z from the zero set of 2 iλn( ) a sine-type-function is less than "1/4 in the mean", then (e · )n Z forms a Riesz basis for L2( π; π). However, while it is easy to give an explicit2 ex- pression for an upper− bound, Avdonin's proof for the lower bound rested on property (f) of Theorem 1.4. Thus, its existence was proved indirectly and he did not obtain an explicit expression for the lower bound, in terms of the restricting data. Using our results on sine-type-functions from Section 2 en- ables us to give an explicit lower bound for Avdonin's Theorem on 1/4 in the sine-type-functions 5 mean. This will be done in Section 3, along with applications for Riesz bases of exponentials and their bounds. We shall also give explicit lower bounds for a Theorem of Bogm´er–Horv´ath–Jo´o{Seip[2, Theorem 3], [20, Theorem 2.3] and for a Theorem of Duffin–Schaeffer [4, Theorem I]. For special cases, explicit estimates have already been given in [15], [21]. Here, we shall obtain lower bounds for the Theorems in full generality. Lower bounds for the The- orem of Duffin–Schaeffer in full generality (i.e., for complex sequences) have also been obtained by Voß [22, 23]. The motivation for our studies is that for practical applications of bases of exponentials, knowledge of the occuring bounds is essential (cf. Section 3). To obtain such bounds, explicit estimates for sine-type-functions turn out to be a good tool. It should be noted that our results are aimed to provide first estimates for the occuring theorems. While the derived estimates for the sine-type- function seem to be quite good, our values for Avdonin's theorem are too small to be valuable in practice so that further research would be desirable. To conclude this section we want to mention the characterization of ex- ponential Riesz bases by Pavlov [18, Theorem 1]: Theorem 1.6 Let (λn)n Z be a sequence of complex numbers satisfying 2 0 < c1 λn c2 n Z ≤ = ≤ 8 2 iλn( ) with two positive constants c1 and c2.
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