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1 Oberwolfach Preprints OWP ALEXANDER OLEVSKII AND ALEXANDER ULANOVSKII Interpolation in Bernstein and Paley-Wiener Spaces Mathematisches Forschungsinstitut Oberwolfach ggmbh Oberwolfach Preprints (OWP) ISSN

2 Oberwolfach Preprints (OWP) Starting in 007, the MFO publishes a preprint series which mainly contains research results related to a longer stay in Oberwolfach. In particular, this concerns the Research in Pairs- Programme (RiP) and the Oberwolfach-Leibniz-Fellows (OWLF), but this can also include an Oberwolfach Lecture, for example. A preprint can have a size from - 00 pages, and the MFO will publish it on its website as well as by hard copy. Every RiP group or Oberwolfach-Leibniz-Fellow may receive on request 30 free hard copies (DIN A4, black and white copy) by surface mail. Of course, the full copy right is left to the authors. The MFO only needs the right to publish it on its website as a documentation of the research work done at the MFO, which you are accepting by sending us your file. In case of interest, please send a pdf file of your preprint by to rip@mfo.de or owlf@mfo.de, respectively. The file should be sent to the MFO within months after your stay as RiP or OWLF at the MFO. There are no requirements for the format of the preprint, except that the introduction should contain a short appreciation and that the paper size (respectively format) should be DIN A4, "letter" or "article". On the front page of the hard copies, which contains the logo of the MFO, title and authors, we shall add a running number (0XX - XX). We cordially invite the researchers within the RiP or OWLF programme to make use of this offer and would like to thank you in advance for your cooperation. Imprint: Mathematisches Forschungsinstitut Oberwolfach ggmbh (MFO) Schwarzwaldstrasse Oberwolfach-Walke Germany Tel Fax admin@mfo.de URL The Oberwolfach Preprints (OWP, ISSN ) are published by the MFO. Copyright of the content is hold by the authors.

3 Interpolation in Bernstein and Paley Wiener spaces Alexander Olevskii and Alexander Ulanovskii A.O.: School of Mathematics, Tel Aviv University Ramat Aviv, Israel A.U.: Stavanger University, 4036 Stavanger, Norway Abstract We construct closed sets S of arbitrarily small measure with the property: given any discrete set Λ, every l function on Λ can be interpolated by an L function with spectrum on S. This should be contrasted against Beurling Landau type theorems for compact spectra. Introduction. Spaces. Let S be a closed set in R. We say that a function f defined on (another copy of) R belongs to Bernstein space B S if (i) f L (R); (ii) f is the Fourier transform of a Schwartz distribution F (notation: f = ˆF ), supported by S. The support of F is called the spectrum of f. In other words, a function f belongs to B S if and only if it is bounded and we have f(x) ˆϕ(x) dx = 0, R for every smooth function ϕ(x) whose support is compact and disjoint from S. Endowed with the L norm, B S is a Banach space.

4 We shall also consider the Paley Wiener spaces P W S := {f L (R); f = ˆF, F = 0 on R \ S}, where the spectrum S can be any measurable set. Classical Paley Wiener (Bernstein) spaces are defined when S is a bounded (compact) set. The elements of these spaces are entire functions of finite exponential type. We shall focus on the situation when S is an unbounded closed set, and so the elements of the corresponding spaces do not necessarily admit analytic continuation into the complex plain.. Interpolation. Let Λ = {λ j, j Z} R be a uniformly discrete (u.d.) set, that is inf j k λ j λ k > 0. Given a bounded data c = {c j, j Z}, one wishes to find a function f B S such that f(λ j ) = c j, j Z. () If this is possible for every c l (Z), one says that the interpolation problem is solvable, and Λ is called an interpolation set for B S. Similarly, if for arbitrary data c l (Z) there is a function f P W S satisfying (), one says that Λ is a set of interpolation for P W S. General principles of functional analysis imply that if Λ is a interpolation set for B S, then one can solve interpolation problem () with an additional estimate: f L C c l, () where the constant C does not depend on data c. A similar result is true for the Paley Wiener spaces (see [], p. 9). 3. Density. For every u.d. set Λ one can define its upper uniform density D + #(Λ (a, a + l)) (Λ) := lim max. l a R l A fundamental role of this quantity in the interpolation problem, in the case when S is a single interval, was found by A. Beurling and J-P. Kahane: Beurling proved ([]) that Λ is an interpolation set for B S if and only if D + (Λ) < mes S. π

5 Even earlier, Kahane proved in [5] that for Λ to be an interpolation set for P W S it is necessary that and it is sufficient that D + (Λ) π D + (Λ) < π mes S, (3) mes S. Both results are based on the theory of entire functions. 4. Disconnected spectra. The situation becomes much more delicate for disconnected spectra, in particular when S is a union of two intervals. For the sufficiency part, not only the size but also the arithmetics of Λ is important in that case. On the other hand, using a new approach Landau [6] succeeded to extend the necessity part to the general case: Theorem. [6] Let S be a bounded set. If a u.d. set Λ is an interpolation set for P W S then condition (3) is fulfilled. We shall prove some versions of this result for both Paley Wiener and Bernstein spaces. In fact, it already suffices to know that the delta-functions on Λ admit interpolation by B S or P W S functions (with not very fast growing norms) in order to obtain an estimate from below for the measure of S (see section below). However, for unbounded spectra the situation is completely different. The contrast is most striking for Bernstein spaces: Not only condition (3) is no longer necessary, but there exist universal spectra of arbitrary small measure which deliver positive solution to the interpolation problem for every u.d. Λ: Theorem. For every δ > 0 there is a closed (unbounded) set S, mes S < δ, such that every u.d. set Λ is an interpolation set for the space B S. This theorem will be proved in sec. 3. Such a result cannot hold for Paley Wiener spaces (see Propositions 4. and 4. in sec. 4). However, we show that for a generic Λ there are arbitrarily small spectra S such that Λ is almost a set of interpolation for P W S (Theorem 4.). Results of this paper concerning Paley Wiener spaces were announced in our note [8]. 3

6 Acknowledgment. Part of the present work was done at the Mathematisches Forschungsinstitut Oberwolfach during a stay within the Research in Pairs Programme from April to April 5, 007. The authors appreciate the hospitality of the institute. Compact spectra: Weak interpolation The aim of this section is to show that even a much weaker interpolation property of Λ ensures Beurling Landau type estimates on the measure of S. In what follows we shall denote by f and f the L and L norm of f, respectively... Concentration. Definition: Given a number c, 0 < c <, we say that a linear subspace W of P W S is c-concentrated on a set Q if f(x) dx c f, f W. Q Following Landau we estimate the dimension of a concentrated subspace. Lemma. Given sets S, Q R and a number 0 < c <, let W be a linear subspace of P W S which is c-concentrated on Q. Then dim W (mes Q) (mes S). πc Proof ([6], see (iii) and (iv) on p. 4). Let Q, S R be two sets of finite positive measure. Let A Q and B S denote the orthogonal projections of L (R) onto L (Q) and P W S, respectively: A Q f = Q f, B S f = F S F f, where Q denotes the characteristic function of Q, and Ff is the Fourier transform of f: Ff(x) = ˆf(x) := e itx f(t) dt. π R 4

7 The operator C := A Q B S A Q acts from L (R) into itself. Clearly, C is self-adjoint and positive. It can be written explicitly as Cf(y) = Q (x) Q (y)ˆ S (y x)f(x) dx. π R Since the kernel is square-integrable, C is a compact operator. Denote by l j its eigenvalues arranged in non-increasing order (counting multiplicities). The trace Tr C is equal to the integral of kernel along the diagonal : Tr C := j l j = (mes Q) (mes S). (4) π. One can easily show that the spectrum of operator D := B S A Q B S is identical to the spectrum of operator C. Let W be a linear subspace of P W S of dimension k. The quadratic form (Df, f) on W is given by (A Q f, f) = Q f. If W is c concentrated on Q, then inf (Df, f) c. f W, f = It is well-known that, among all subspaces of dimension k, the greatest value of inf(df, f) on the unit sphere is achieved on a subspace spanned by the first k eigenvectors of D. This means that c l k Tr D/k=Tr C/k. This and (4) prove the lemma. One can show that a similar result holds in several dimensions, too. Remark. The uncertainty principle in Fourier analysis is a statement that a function and its Fourier transform cannot both be concentrated on small sets. A simple corollary of Lemma. is the following variant of uncertainty principle due to Amrein and Berthier []: Let S and Q be any sets of finite measure. There exists C = C(S, Q) such that ( ) f(x) dx C f(x) dx + f(t) dt, f L (R). R R\S.. Interpolation of delta functions. We start with the following R\Q 5

8 Theorem. Let S be a compact set and Λ = {λ j, j Z} be a u.d. set. Suppose that for every j Z there is a function f j P W S such that {, k = j f j (λ k ) =, k Z; (5) 0, k j Then (3) holds. sup f j <. (6) j Z Proof.. Recall that if S is a compact set, then the corresponding Paley Wiener space consists of entire functions of exponential type. The following lemma is well known (see [], p. 8): Lemma. Suppose Λ = {λ j, j Z} is a u. d. set, and S R is a compact set. Then there exists C > 0 such that f C j Z f(λ j ), for every f P W S.. Fix a number δ > 0, and set S(δ) := S + [ δ, δ]. (7) Take functions f j P W S satisfying (5) and (6), and set ( ) sin δ(x λj ) g j (x) := f j (x), j Z. δ(x λ j ) Clearly, functions g j satisfy (5), (6) and belong to P W S(δ). 3. Fix any number R, and denote by #(Λ (R r, R + r)) the number of λ j Λ in the interval (R r, R + r). Since Λ is u.d. we have: # (Λ (R r, R + r)) Cr, for all r >, (8) where C > 0 is a constant which does not depend on R. Let us introduce a linear space of functions W r := g(x) = c k g k (x), c k C. k: λ k R <r Clearly, dim W r = #(Λ (R r, R + r)). 6

9 4. We would like to estimate the concentration of W r on the interval (R r rδ, R + r + rδ). Choose any function g = j c jg j W r. Then g(λ j ) = c j when λ j R < r, and g(λ j ) = 0 when λ j R r. By Lemma., there is a constant C = C(S, Λ) > 0 such that g C g(λ j ) = C c j. (9) λ j Λ λ j R <r On the other hand, by Parseval s identity and (6), we have ( ) f j (x) ˆf j (t) dt mes S π π ˆf j C, j Z, S for some C > 0. Observe that x λ j δr whenever λ j (R r, R + r) and x R r + δr. Hence, the last inequality and (8) give g(x) dx = Cr x R r+δr λ j R <r c j x R r+δr λ j R <r x >δr ( ) sin δ(x λj ) c j f j (x) dx δ(x λ j ) ( ) 4 dx C c δx δ 7 r j. λ j R <r This and (9) show that for every ɛ > 0 there exists r ɛ such that g is ( ɛ) concentrated on (R r δr, R + r + δr) for all r r ɛ. 4. It now follows from Lemma., that # (Λ (R r, R + r)) (mes S(δ))(mes (R r δr, R + r + δr)), π( ɛ) whenever r r ɛ. Since this is true for every R, we obtain: D + (Λ) + δ ɛ mes S(δ). π Letting δ 0 and ɛ 0, we conclude that D + (Λ) (π) mes S. Remark. The assumptions of Theorem. admit a geometric interpretation: Set E(Λ) := {e iλt, λ Λ}. 7

10 Conditions (5) and (6) are equivalent to the property that the exponential system E(Λ) is uniformly minimal in L (S), that is the L distance from any element of the system to the span of other elements is greater then some positive constant. Observe that assumptions (5) and (6) are less restrictive then the requirement for Λ to be an interpolation set for P W S. Indeed, consider a classical example: S = [ π, π] and Λ = {λ j, j Z}, λ j := j +, j =,, , j = 0 j, 4 j =,,... The corresponding exponential system E(Λ) is uniformly minimal in L ( π, π) (see [7], Theorem 5). However, Λ is not an interpolation set for L ( π, π) (see the remark following Theorem 5 in [7]). An analog of Theorem. for Bernstein spaces is also true: Theorem. Let S be a compact set, and Λ = {λ j } a u.d. set. Suppose that for every j Z there is a function f j B S satisfying (5) and sup f j <. (0) j Z Then (3) holds. Indeed, suppose functions f j B S satisfy assumptions of Theorem.. Take any positive number δ, and set g j (x) := f j (x) sin δ(x λ j). δ(x λ j ) Then the functions g j belong to P W S(δ), j Z, where S(δ) is defined in (7), and satisfy assumptions of Theorem.. Hence, D + (Λ) (π) mes S(δ). Letting δ 0, we obtain Theorem.. An immediate corollary of Theorem. is a variant of Landau s Theorem. for Bernstein spaces: Corollary Let S be a compact set. If a u.d. set Λ is an interpolation set for B S, then condition (3) is fulfilled..3. Growth of f j and upper density. An estimate similar to (3) holds even if the norms of functions satisfying (5) grow, but not very fast: f j Ce λ j α, j Z, () 8

11 where C > 0 and 0 < α <. We shall show that this and (5) imply estimate (3) in which D + is replaced by the upper density D. Definition: Let Λ be a u.d. set. The upper density of Λ is defined as follows: D #(Λ ( a, a)) (Λ) := lim sup. a a Clearly, we have D (Λ) D + (Λ). Observe that for regularly distributed Λ, for example if Λ = {j +O(), j Z}, these two densities are equal. Theorem.3 Let S R be a compact set, and Λ = {λ j, j Z} a u.d. set. Suppose there exist functions f j P W S, j Z, satisfying (5) and () with some C > 0 and 0 < α <. Then D (Λ) (π) mes S. Proof.. We shall need Lemma.3 Set ϕ(z) := sin(k β z), k β z k k 0 where k 0 N and 0 < β < are some numbers. Then ϕ P W [ σ,σ], where σ = k k 0 k β, and ( ϕ(x + iy) C exp ) C x +β, x R, y, () where C is a constant. Proof. For every a > 0, the function sin az/az is the Fourier transform of a positive constant function on [ a, a]. Hence, ϕ(z) is the Fourier transform of the convolution of such functions. This convolution is concentrated on [ σ, σ], σ = k k 0 k γ, and so ϕ P W [ σ,σ]. Let us denote by δ and δ positive constants such that e δ z sin z z e y δ δ x, z = x + iy C, z. (3) Since sin(a + ib) exp b, for a, b real, we have: sin(k β z) k β z sin(k β z) k 0 k< z +β k 0 k< z +β 9

12 exp y k +β k 0 k< z +β e C y, z = x + iy, where C is a constant. Now, using the right-hand inequality in (3), we get for every z C, z + k +β 0, that sin(k γ z) k β z C exp δ x k +β k z +β k z +β ( C exp ) C x +β, y, where C is a constant. This and the previous inequality give ().. Fix a small number δ > 0 and pick up k 0 in Lemma.3 so large that σ δ, i.e. the function ϕ P W [ δ,δ]. Also, assume that β < satisfies α < /( + β). The rest of the proof is pretty similar to the proof of Theorem.. Without loss of generality, we may assume that λ 0 = 0. Take functions f j P W S satisfying (5) and (), and set g j (x) := f j (x)ϕ(x λ j ), j Z, where ϕ is a function from Lemma.3. It is clear that g j P W S(δ), where S(δ) is defined in (7), and that (5) is true for every j Z. 3. Since Λ is u.d., we have # (Λ ( r, r)) Cr, (4) for some C > 0 and all r >. Set W r := g(x) = c k g k (x) : c k C. λ k <r Clearly, dim W r = #(Λ ( r, r)). 4. Let us estimate the concentration of W r on the interval ( r rδ, r + rδ). Take any function g W r. Since g(λ j ) = 0 when λ j r, by Lemma., we have: g C c j. (5) λ j <r 0

13 Further, by () and Parseval s identity, we have ( ) f j (x) ˆf j (t) dt mes S π π ˆf j Ce rα, S for some C > 0 and all j, λ j < r. Observe also that x λ j δr whenever λ j < r and x r+δr. This, (4) and the last inequality give g(x) dx = c j f j (x)ϕ(x λ j ) dx x r+δr x r+δr Cre rα c j λ j <r λ j <r x >δr ϕ(x) dx. Recall that α < /( + β), so that () gives: re rα ϕ(x) dx 0, r. x >δr Hence, by (5), we see that for every ɛ > 0 there exists r ɛ such that g is ( ɛ) concentrated on ( r δr, r + δr) for all r r ɛ. 4. It now follows from Lemma., that # (Λ ( r, r)) This gives: D (Λ) = lim sup r (mes S(δ))(mes ( r δr, r + δr)), r r ɛ. π( ɛ) # (Λ ( r, r)) r + δ mes S(δ). π( ɛ) Letting δ 0 and ɛ 0, we conclude that D (Λ) (π) mes S. By using the same argument which deduces Theorem. from Theorem., one can easily extend Theorem.3 to Bernstein spaces: Theorem.4 Let S R be a compact set, and Λ = {λ j, j Z} a u.d. set. Suppose that there exist functions f j B S, j Z, satisfying (5) and f j Ce λ j α, j Z, (6)

14 for some C > 0 and 0 < α <. Then D (Λ) (π) mes S. Remark. The growth restrictions () and (6) can be replaced by any non quasi-analytic growth. However, we do not know if they can be omitted at all..4. A counterexample. The results of last subsection are not true for the density D +. We shall show this only for the Paley Wiener case. Theorem.5 For every ρ > there exist a u.d. set Λ, D + (Λ) ρ, and functions f j P W [ π,π], j Z, which satisfy (5) and () with some C > 0, 0 < α <. Proof.. Fix any numbers k 0 N and 0 < β <, and let ϕ be the function in Lemma.3. Let us show that for every r 0 we have ( ϕ(x + ri) C exp ) C x +β log( + x ), x R, (7) where C > 0 depends only on r. For simplicity, we shall assume that r =. From the left-hand inequality of (3), we obtain: k x+i +β sin(k β (x + i)) k β (x + i) exp δ x + i ) exp ( C x + i +β, where C is a constant. Further, since we have k 0 k< x+i +β sin(a + ib) eb e b k 0 k< x+i +β b, a R, b > 0, sin(k β (x + i)) k β (x + i) exp ( k +β ) exp ( k +β ) x + i k β This and the previous estimate prove (7). k +β k x+i +β ( ) x+i /(+β). x + i

15 . Set k=0 ψ(z) := η(z)η( z), η(z) := k=0 j j= k=0 ( z ). 4 j + k Observe that j z ( ) j 4 j + k exp ( ) z j z exp < e j j, 4 j + k 4 j whenever j > z. This gives: η(z) e j j ( + z ) j log z C ( + z ) log Ce C log (+ z ), j > z j z from which is follows that ψ(z) Ce C log (+ z ), (8) where C is a constant. We also need to estimate from below the derivative ψ at the zero points ±(4 j + k) of ψ. Pick up numbers J N and K, 0 K < J. We have 0 k<j,k K Further, since 4J + K 4 J + k = 0 k<j,k K ( ) J K k 4 J + k e CJ. 4 J + J 4 J 4 j + k, j < J, 4 J + j, j > J, 0 k < j, 4 j we get: j 4J + K 4 j + k ( ) j 4J + J C. 4 j j>j j,j J k=0 This and the previous estimate show that η (4 J + K) Ce CJ, J N, 0 K < J, where C is a constant. Since η( x) for x 0, the same estimate holds for ψ (4 J + K). Since ψ is even, we obtain: ψ (λ) Ce C log λ, for all λ, ψ(λ) = 0. 3

16 Now, choose a number ρ > and set f(z) := ψ(ρz)ϕ( π (z + i)), σ where ϕ is a function from Lemma.3, and σ is its type. The last inequality and and (7) imply: f (λ) Ce C λ +β log(+ λ ), for all λ, ψ(ρλ) = 0. (9) Moreover, from (8) and (), we see the type of f is equal to π and that f L (R). Hence, f P W [ π,π], by the Paley Wiener theorem. Let Λ be the zero set of ψ(ρx) : Λ := Γ ( Γ), Γ := j= { 4j ρ,..., 4j + j }. ρ It is clear that D + (Λ) = ρ. Set λ 0 = 4, and numerate the elements of Λ = {λ j, j Z} in increasing order. Set f j (z) := f(z) f (λ j )(z λ j ), j Z. Then functions f j P W [ π,π] and satisfy (5). We have: f j f λj + f (λ j ) + f(x) f (λ j )(x λ j ) λ j dx. Using the Cauchy integral formula, one can estimate from above the integral in the right hand-side by ( f (λ j ) sup f(z) ). z λ j = We conclude that f j C f (λ j ), where C depends only on f. Now, by (9), we see that assumption () is also fulfilled for each α < satisfying α > /( + β). 4

17 3 Unbounded spectra: interpolation in Bernstein spaces The goal of this section is to construct universal interpolation spectra S of arbitrarily small measure: Theorem 3. For every δ > 0 there is a closed (unbounded) set S, mes S < δ, such that every set Λ which has no finite limit points is an interpolation set for the space B S. This result is a refinement of Theorem. in which we relax assumption of uniform discreteness of Λ. 3.. Lemmas. Lemmas 3. For every N there exists a set S(N) ( N, N), mes S(N) =, such that N N e itx sin Nx dt Nx C, x R, (0) N S(N) where C > 0 is an absolute constant independent on N. Proof.. Fix an integer N, and let M j, j =,..., N, be any even numbers satisfying Set M N 4, M j+ N M j, j =,..., N. () Ω(j, k) := j + k M j + M j (, ], Ω( j, k) := Ω(j, k), S(j, k) := j + k M j + N M j (, ], S( j, k) := S(j, k), where j =,..., N and k =,..., M j /. Set One can check that M j / k= S(N) := N j = M j / k= S(j, k). Ω(j, k) = (j, j], j =,..., N, 5

18 and that mes S(N) = N.. For simplicity, throughout the proof we denote by C different positive constants. Since sin Nx Nx = N e itx dt, N N to prove (0) it suffices to show that N e itx dt N e itx dt < C, x R. () N S(N) 3. Assume first that x M l, for some l N. Using (), we have for j l, j N, that e itx dt N e itx dt = sin(x/m j ) x This and () give: Ω(j,k) N sin(x/n M j ) x {l t N} e itx dt N N j =l M j / k= S(j,k) Cx M 3 j CM l M 3 j {l t N} S(N) CM l M 3 j Clearly, this proves () for x M., k =,..., M j. e itx dt C. (3) 4. Assume now that x > M l, for some l N. Then, clearly, we have e itx dt = sin(l )x x. (4) M l Also, { t l } N {l t l } S(N) 6 e itx dt

19 N mes ({l t l } S) =. (5) These estimates and (3) prove () for M < x M. Assume that x > M l, for some l 3. Since N e itx dt = N sin x/(n M j ) x N, x > M l, M l S(j,k) by (), we obtain: N { t l } S(N) l e itx dt From this, (4) and (5) we get e itx dt N { t l } j= { t l } S M j / k= N M l N M l M l C. e itx dt C, x > M l, l 3. Now, this and (3) imply () for x > M l, l 3, which completes the proof of Lemma 3.. Lemma 3. For every ɛ > 0 there is a compact S = S ɛ and a function g = g ɛ B S such that: (i) mes S ɛ; (ii) g = g(0) = ; (iii) g(x) < ɛ for x > ɛ. In addition, S can be chosen disjoint from any given segment. This follows from Lemma 3.. Indeed, fix an ɛ > 0, and choose N in Lemma 3. so large that mes S(N) < ɛ, sin Nx/Nx < ɛ/ when x > ɛ and C/N < ɛ/, where C is the constant in (0). Given a segment I, set S := R + S(N), where R is any number such that R + S(N) I =. Set g(x) := N e itx dt = N eirx e itx dt. S S(N) Conditions (ii) and (iii) follow immediately from Lemma Proof of Theorem 3.. Suppose 0 < δ <, and take any sequence ɛ(j) > 0, j Z, such that ɛ(j) < δ. (6) j Z 7

20 Fix a sequence of disjoint compacts S ɛ(j), mes S ɛ(j) < ɛ(j), tending to infinity and satisfying the conditions of Lemma 3., such that the set S := j Z S ɛ(j). is closed. Let Λ = {λ j, j Z} be a set without finite limit points. Taking if necessary a subsequence of ɛ(j), we may assume that d j := inf k:k j λ k λ j > ɛ(j), j Z. (7) Conditions (ii) and (iii) of Lemma 3. allow one to define a sequence of functions f j B S such that each function f j has a compact support, = f j = f j (l j ), and By (7), we see that f j (x) < ɛ(j), for x l j d j. (8) f j (λ k ) < ɛ(j), j k. (9) Consider a linear operator A : l l defined as { fj (λ A := (a(j, k)) j,k Z, a(j, k) = k ), j k f j (λ j ), j = k From (6) and (9) one can see that A = sup k a(j, k) = sup k f j (λ k ) <, j so that the operator A + I, I is the identity operator, is invertible in l. Take an arbitrary data c = {c j } l, and denote by b = {b j } l the solution of the equation (A + I)b = c. j k Set f(x) := j Z b j f j (x). 8

21 By (6), (7) and (8), we see that the series converges uniformly on every finite interval, and sup f(x) ( + δ) b l <. x R Also, it is clear that f satisfies the interpolation condition f(l j ) = c(j), j Z. Now let ϕ be any smooth test function supported by a segment disjoint from S. Let ɛ(r) 0 and N(R) as R, be some functions satisfying f(x) b j f j (x) < ɛ(r), x R. j N(R) Since each f j has a compact support which lies in S, then ( ˆϕ, f j ) = 0, j Z, and so we have ( ˆϕ, f) = ( ˆϕ, f b j f j ) ɛ(r) ϕ(x) dx+ x R ( + δ) b l j N(R) x >R ˆϕ(x) dx 0, R. Hence, ( ˆϕ, f) = 0, which shows that f B S Remark on sampling sets. We finish this section by the following remark. Definition. A set Λ is called a sampling set for B S if there is a constant C > 0 such that for every f B S. f C sup f(λ), λ Λ Similarly, one may define sampling set for P W S spaces. When S is a single interval, the sampling sets for B S were completely characterized by Beurling in terms of so-called lower uniform density D (L) (see [3]), by the following condition: D (L) > π 9 mes S.

22 For the disconnected compacts S no such metrical characterization may exist. However, one can construct a set of critical density which serves as B S sampling set for every compact of given measure. More precisely, the following is true: Theorem 3. There is a set Λ = {j + O(), j Z}, which is a sampling set for the Bernstein space B S, for every compact S of measure < π. This is a consequence of Theorem 3 in [9] (see details in [0]), where such a universal sampling set was constructed for Paley Wiener spaces. In order to deduce the proposition above from Theorem 3 in [9], one needs Lemma 3.3 Suppose there exists δ 0 > 0 such that a u.d. set Λ is a sampling set for P W S(δ), 0 < δ δ 0, where S(δ) is defined in (7). Then Λ is a sampling set for B S. Indeed, assume that the conclusion of Lemma 3.3 does not hold: for every ɛ > 0 there exists f B S such that sup x f(x) =, and f(λ) < ɛ for every λ Λ. Pick up a point x 0 such that f(x 0 ) /, and set g(x) := f(x) sin δ(x x 0) δ(x x 0 ) P W S(δ), where δ δ 0. Let σ > 0 be such that S(δ) [ σ, σ]. Since g(x) for all x, the Bernstein inequality says that sup f (x) σ. Using this and inequality g(x 0 ) /, one can easily deduce that g C, where C depends only on σ. On the other hand, since Λ is u.d., we have g(λ) ɛ sin δ(λ x 0 ) δ(λ x 0 ) Cɛ, λ Λ λ Λ where C depends only on the infimum of distances between elements of Λ. However, since this can be done for every ɛ > 0, we see that Λ is not a sampling set for P W S(δ), which is a contradiction. In a sharp contrast to Theorem 3., the following claim is true: Corollary 3. A sampling set for the closed (unbounded) spectrum S, constructed in the proof of Theorem 3., must be dense in R. 0

23 Indeed, suppose a set Λ satisfies Λ (a, b) =, for some a < b. By the construction in the proof of Theorem 3., for every ɛ > 0 and c R there exists f B S such that f(c) = and f(x) < ɛ for x c > ɛ. By choosing c = (a + b)/ and ɛ < (b a)/, we see that for every ɛ > 0 there exists a function f B S satisfying f > (/ɛ) f l (Λ), so that Λ is not a sampling set for B S. 4 Unbounded spectra: interpolation in Paley Wiener spaces In the previous section we have seen that there exist universal closed sets S of arbitrarily small measure such that for arbitrary Λ and c the interpolation problem () can be solved by functions f B S. In this section we obtain somewhat similar results for the Paley Wiener spaces. Observe, however, that the universality is not possible to achieve in this case: Proposition 4. No set of measure < π admits interpolation of any δ function on Z by f P W S. Proof. Given a set S R of finite measure, set ( [ π, S π := (S + πj)) π]. j Z Now, suppose there exists f P W S such that f(0) = and f(j) = 0, j Z \ {0}. Denote by F L (S) the inverse Fourier transform of f, and set F π (t) := j Z F (t + πj). Observe that π F π (t) dt = π R F (t) dt mes S F <, so that the function F π is a.e. finite. Also, we see that π {, j = 0 e ijt F π (t) dt = e ijt F (t) dt = f(j) = 0, j 0 π R

24 This implies F π (t) = π for a.e. t [ π, π], and so supp F π = [ π, π]. However, clearly, suppf π S π. We conclude that mes S π. Proposition 4. shows that some restrictions on Λ are necessary in order to get interpolation by P W functions with small spectra. We consider a generic situation: elements of Λ are supposed to be rationally-independent. In this case, we construct closed spectra which admit interpolation of any function on Λ with a certain weak decay condition. Theorem 4. Let a u.d. set Λ = {λ j, j Z} be linearly independent over rationales (mod π). Then for every δ > 0 there is is a set S, a union of some of intervals centered at integers, such that: (i) mes S < δ; (ii) for every sequence c = {c j } satisfying c j ( + j β ) < j Z with some β >, there is a function f P W S satisfying (). Remark. One can see from the proof below that the assumption (ii) can be relaxed by replacing it with j Z c j w j <, where the {w j } l (Z) is a fixed positive sequence. However, assumption (ii) cannot be replaced by c l. Indeed, consider a random sequence of interpolation knots: Λ := {n + ξ(n)}, (30) where ξ j, j Z, are independent random variables uniformly distributed on ( a, a), 0 < a < /. Clearly, Theorem 4. is applicable in this case: Almost surely, there is a closed set S of arbitrary small measure which admits interpolation in P W S of every sequence c satisfying (ii). On the other hand, we have Proposition 4. Let Λ be a random set in (30). With probability, there is no set S of measure < π such that Λ is an interpolation set for P W S.

25 Observe that we don t know if there exists Λ = {j + O(), j Z} which is a set of interpolation for P W S for some (unbounded) S, mes S < π. Proposition 4. shows that if such a set Λ does exist then it should be an exception. Proof of Proposition 4.. It is well-known that the interpolation property in a Hilbert space is equivalent to certain inequalities (see [], p.9). In our case, a set Λ = {λ j, j Z} is a set of interpolation for P W S if and only if there is a positive constant A such that c j e iλ jt dt A c j, (3) S j j for every finite sequence {c j }. Let ξ j be random sequence defined above. Fix an integer N. Clearly, we have with probability one that for every ɛ > 0 there exists k = k(ɛ, N) such that ξ k+j ξ k < ɛ, for all j, j N. Fix an element of the underlying probability space such that the latter is true, and assume that S R is such that the set Λ in (30) is a set of interpolation for P W S. By (3), we have for every c N,..., c N that N c S j e ijt N dt = lim c ɛ 0 S j e i(k+j+ξ k+j)t N dt A c j. j= N j= N j= N Since this is true for every N, we see that Z is also a set of interpolation for P W S. By Proposition, we conclude that mes S π. Proof of Theorem 4.. Throughout the proof we shall denote by C different positive constants.. Without loss of generality we may assume that β <. Set S := j Z S j, S j := ( M j 4β j, M j + 4β j ) (M j 4β j, M j + 4β j ), where β j := γ + j β, 3

26 the sequence M j will be specified in step 4, and γ is any small positive number such that mes S < δ. In what follows we also assume that γ is so small that S j S k =, for j k.. Set Λ k := (Λ λ k ) \ {0}, k Z. The independence condition on Λ implies, by Kronecker s theorem, that for every N > 0 the subgroup {mλ (mod π), λ Λ k [ N, N], m Z} is dense in the l dimensional torus, l being the number of elements in Λ k [ N, N]. Hence, the l numbers cos(mx), x Λ k [ N, N], can be made as small as we like by choosing appropriate M N. 3. Set g j (x) := cos (M j (x λ j )) ( ) 4 sin γj (x λ j ). γ j (x λ j ) Clearly, the spectrum of g j belongs to S j, and we have g j (λ j ) =, (3) and g j C γ j C ( + j β), j Z. (33) 4. Since Λ is uniformly discrete, there is a constant C > 0 such that ( sin γj (λ k λ j ) γ j (λ k λ j ) ) 4 γ 4 j C, k j, k, j Z. (k j)4 Take any small number ɛ > 0, and let N j be so large that we have ( ) 4 sin γj (λ k λ j ) γ j (λ k λ j ) ɛ ( + j )(k j), λ k λ j N j, k j. By Step, the first factor in the definition of g j can be made arbitrarily small for 0 λ k λ j < N j. We shall choose M j N such that ɛ cos M j (λ k λ j ) ( + j ) max{(k j), λ k λ j < N j }, 4

27 for all k j such that λ k λ j < N j. This and the previous estimate give ɛ g j (λ k ), k j, k, j Z. ( + j )(j k) One may check that this estimate implies j Z,j k 5. Given a sequence c = {c j, j Z}, set c β := g j (λ k ) Cɛ + k 3. (34) j= c j ( + j β). Let lβ denote the weighted space of all sequences c, c β <. Define a linear operator R : lβ l β as follows: Re j := k= g j (λ k )e k e j, j Z. Using (3), we have R c j e j β = ( ) c j g j (λ k ) e k β = j Z k Z j Z,j k c j g j (λ k ) ( + k β ) k Z j Z,j k ( ) c g j (λ k ) β ( + k β ). + j β k Z j Z,j k Since β <, we see from (34) that R j Z c j e j β Cɛ c β, for some constant C > 0. Choose ɛ so small that the norm of operator R in lβ is less than. It follows that the operator T := I+R is invertible in lγ, where I is the identity operator. We conclude that 5

28 for every c lβ the interpolation problem () has a solution f given by f(x) := b j g j (x), {b j } = T c lβ. j Z Recall, that the spectrum of g j belongs to the set S j defined in step. Since S j and S k are disjoint for j k, it follows that the functions g j and g k are orthogonal in L (R). Using this and (33), we see that f = j Z b j g j C b β <. Recall again that g j P W Sj P W S, j Z. We conclude that f P W S. References [] Amrein, W. O.; Berthier, A. M. On support properties of L p - functions and their Fourier transforms. J. Functional Analysis 4 (977), no. 3, [] Beurling, A. Interpolation for an interval in R. In: The collected Works of Arne Beurling, Vol., Harmonic Analysis. Birkhauser, Boston, 989. [3] Beurling, A. Balayage of Fouirer-Stiljetes trasnforms. In: The collected Works of Arne Beurling, Vol., Harmonic Analysis. Birkhauser, Boston, 989. [4] Beurling, A., Malliavin, P. On the closure of characters and the zeros of entire functions. Acta Math., 8 (967), [5] Kahane, J.-P. Sur les fonctions moyenne-périodiques bornées. Ann. Inst. Fourier, 7, 957, [6] Landau, H. J. Necessary density conditions for sampling and interpolation of certain entire functions. Acta Math. 7, 967, [7] Redheffer, R.M., Young, R.M. Completeness and basis properties of complex exponentials. Trans. Amer. Math. Soc. 77 (983), no., 93. 6

29 [8] Olevskii, A., Ulanovskii, A. Interpolation by functions with small spectra. C. R. Math. Acad. Sci. Paris 345 (007), no. 5, [9] Olevskii, A., Ulanovskii, A. Universal sampling of band-limited signals. C. R. Math. Acad. Sci. Paris 34 (006), no., [0] Olevskii, A., Ulanovskii, A. Universal sampling and interpolation of band-limited signals. To be published in GAFA. [] Young, R.M. An introduction to Nonharmonic Fourier Series. Academic Press