Irregular poisson type summation.

Abstract

We present extensions of the classical Poisson summation formula in which the sequence of sampling knots, normally a lattice, can be taken from a relatively wide class of sequences.

Key words and phrases: Poisson summation formula, complete interpolating sequence, amalgam space

2000 A ME Mathematics Subject Classification--41A05, 42A75

1 Introduction

1.1 Background

Suppose f and F are a Fourier transform pair, specifically

F(x)= 1/2[pi][[integral].sup.[infinity].sub.-[infinity]] f([xi])[e.sup.ix[xi]]d[xi]. (1.1.1)

Then, if both f([xi]) and Y(x) are continuous and decay sufficiently rapidly, the classical Poisson summation formula reads

[[infinity].summation over (m=-[infinity])] f(2[pi]m) = [[infinity].summation over (n=-[infinity])] F(n) (1.1.2)

where both sums converge absolutely. More precise statements can be found in various texts on Fourier analysis including [12, 23, 32, 34] and examples of more recent work on the subject can be found in [1, 2, 3, 6, 7, 9, 11, 17, 20, 22].

As is evident from the above citations, the Poisson summation formula is well known and has several variations and interpretations. Most of these exploit a lattice structure or related group theoretic phenomena. Indeed, it is known that formulas of the type

[[infinity].summation over (m=-[infinity])] f([[xi].sub.m]) = c [[infinity].summation over (n=-[infinity])] F([x.sub.n])

are valid only when the sequences {[[xi].sub.m]) and {[x.sub.n]) are appropriate dual lattices; see [9].

On the other hand, replacing f([eta]) by f([xi] + [eta]) in (1.1.2) results in

[[infinity].summation over (m=-[infinity])] f([xi] + 2[pi]m) = [[infinity].summation over (n=-[infinity])] F(n)[e.sup.-in[xi]], (1.1.3)

which may be interpreted in a wider sense than (1.1.2). For instance, as a function of the variable [xi], convergence of the sums may be taken in some mean sense and equality may be valid only almost surely.

1.2 Contents

It is extensions of identity (1.1.3) with which we are primarily concerned. Indeed, the objective of this note is to present such an extensions where the sequence of sampling knots on the right-hand side, the integer lattice Z = {n}, is replaced with a more general sequence [chi] = {[x.sub.n]}. This, of course, also involves appropriate adjustments to the left-hand side as well.

The class of allowable sequences [chi] in our extensions are the so-called complete interpolating sequences for the classical Paley-Wiener space. This includes a very wide range of irregular sampling sequences. In particular, every sufficiently small perturbation of the integer lattice Z is a complete interpolating sequences as is any finite perturbation of Z.

If [chi] is such a sequence and {[[phi].sub.n]([xi])) is a basis which is dual, i.e., biorthogonal, to the Riesz basis of exponentials [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in [L.sup.2]([-[pi], [pi]]), then under suitable restrictions on f our summation formula reads

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.2.1)

Here [[tau].sub.-2[pi]m] f([xi]) = f([xi] + 2[pi]m) and {[B.sub.m]} is a sequence of bounded linear operators on [L.sup.2]([-[pi], [pi]]). In the case [chi] = Z, as expected, (1.2.1) reduces to (1.1.3).

A natural framework for formulas such as (1.1.3) and (1.2.1) is provided by certain amalgam spaces. This is detailed in Section 2. Our formulas, including (1.2.1) and the explicit definition of the operators [B.sub.m], are developed in Section 3. More information concerning complete interpolating sequences, including the examples above, is contained in Section 4.

1.3 Remark

The classical Poisson summation formula has many applications. For example, it provides the justification, in terms of error bounds, for the use of the discrete Fourier transform in lieu of the continuous Fourier transform in certain numerical calculations. This and other examples are detailed in various texts and articles on applied Fourier analysis, such as [5, 30]. It is expected that our formulas will also find meaningful applications.

For the sake of completeness we mention that if, in addition to translations, dilations and modulations are also used, then variants of (1.1.2) which are significantly more complicated than (1.1.3) can be obtained, for instance see [4].

2 Setup

The setting for our extension of (1.1.3) is in certain so-called amalgam spaces. Various amalgam type spaces are well-known and have been studied in many contexts; for example, see [13, 14, 15] and the recent survey article [18] which contains a brief history and a bibliography on the subject. We begin by defining two specific examples of these spaces and viewing (1.1.3) as both an extension and summation procedure.

2.1 The amalgam spaces [l.sup.1]([L.sup.2]) and [l.sup.[infinity]]([L.sup.2])

Let Q = [-[pi], [pi]] = {[xi]: -[pi] [less than or equal to] [xi] [less than or equal to] [pi]} and for a measurable function f defined on Q + 2[pi]m = {[xi]: (2m - 1) [pi] [less than or equal to] [xi] [less than or equal to] (2m + 1)[pi]} let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The amalgam space [l.sup.1]([L.sup.2]) is the class of those measurable functions f on R = (-[infinity], [infinity]) such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is finite.

The amalgam space [l.sup.[infinity]]([L.sup.2]) is its dual. Namely, [l.sup.[infinity]]([L.sup.2]) is the class of those measurable functions f on R = (-[infinity], [infinity]) such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is finite.

2.2 The summation operator [S.sub.Z]

If f is in the amalgam space [l.sup.1]([L.sup.2]), then the left-hand side of (1.1.3) is in [L.sup.2](Q) and the right-hand side is simply its Fourier series.

Indeed,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2.2.1)

is just Minkowski's inequality, which implies that the left-hand side of (1.1.3) may be viewed as a linear mapping of [l.sup.1]([L.sup.2]) onto [L.sup.2](Q) of norm 1. We denote this mapping by [S.sub.Z]; thus,

[S.sub.Z]f([xi]) = [[infinity].summation over (m=-[infinity])] f([xi] + 2[pi]m).

Also note that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.2.2)

follows from the Schwarz inequality and implies that

[l.sup.1]([L.sup.2]) [subset] [L.sup.1](R). (2.2.3)

These observations can be summarized in the following proposition.

2.3 Proposition

If f is in [l.sup.1]([L.sup.2]), then

(i) f is in [L.sup.1] (R) and F(x) is continuous,

(ii) for n = 0, [+ or -] 1, [+ or -] 2, ...,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

are the Fourier coefficients of

[S.sub.Z]f([xi]) = [[infinity].summation over (m=-[infinity])] f([xi] + 2[pi]m),

and

(iii) both sums in (1.1.3) converge in the [L.sup.2](Q) sense and (1.1.3) is valid for almost all [xi] in Q.

2.4 The prolongation operator [E.sub.Z]

In view of 2.2 it is natural to consider the dual or adjoint of [S.sub.Z], the operator [S.sup.*.sub.Z] = [E.sub.Z], as an extension or prolongation operator from [L.sup.2](Q) to [l.sup.[infinity]]([L.sup.2]).

If f [member of] [l.sup.1]([L.sup.2]) and g [member of] [L.sup.2](Q), then setting g([xi]) = 0 whenever [xi] [not member of] Q the routine calculation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

shows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [[tau].sub.[eta]] and [P.sub.m] are the translation and projection operators defined by [[tau].sub.[eta]] g([xi])= g([xi] - [eta]) and [P.sub.m]f([xi]) = [[chi].sub.Q]([xi] - 2[pi]m)f([xi]), respectively. Here [[chi].sub.Q] is the indicator function of the interval Q = [-[pi], [pi]].

Note also that

g([xi]) = [[infinity].summation over (n=-[infinity])] [a.sub.n][e.sup-in[xi]] (2.4.1)

where

[a.sub.n] = 1/2[pi] [[integral].sub.Q] g([xi])[e.sup.in[xi]]d[xi].

The right-hand side of (2.4.1) makes sense for a.e. [xi] in R and defines a continuation of g from the interval Q to all of R. Indeed, we may write

[[infinity].summation over (m=-[infinity])] [P.sub.m][[tau].sub.2[pi]m]g([xi]) = [E.sub.Z]g ([xi]) = [[infinity].summation over (n=-[infinity])] [a.sub.n][e.sup-in[xi]] (2.4.2)

and view [E.sub.Z]g as the periodic continuation of g.

Finally we note that, using the notation established above, the summation operator may be re-expressed as

[S.sub.Z]f([xi]) = [[infinity].summation over (m=-[infinity])] [[tau].sub.2[pi]m][P.sub.m]f([xi]). (2.4.3)

At the moment this formalism may seem unnecessarily pompous but it it gives a hint as to what to expect in what follows.

3 Poisson type summation formulas

Our extensions of (1.1.3) are based on appropriate generalizations of the summation and prolongation operators [S.sub.Z] and [E.sub.Z] defined in Section 2. Because [E.sub.Z] is intuitively easier to use, we begin with its generalization.

3.1 The prolongation operator [E.sub.[chi]]

Recall the fact that [E.sub.Z]f is simply the periodic extension of a function f defined on Q to one defined on R. If f is in [L.sup.2](Q), then this extension or prolongation can be described via its expansion in terms of the complete orthonormal system of functions [{[e.sup.-in[xi]}.sub.n[member of]Z] as in (2.4.2). Now suppose one wants to obtain a similar type prologation of [L.sup.2](Q) by using a sequence of frequencies, [chi] = [{[x.sub.n]}.sub.n[member of]Z], other than the integers. It is clear that at the very least the corresponding sequence of exponentials [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] should form some sort of nice basis for [L.sup.2](Q). In what follows we consider the case when [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a Riesz basis for [L.sup.2](Q).

Recall that a basis {[f.sub.n]} of a Hilbert space H is a Riesz basis if and only if there are positive constants [c.sub.0] and [c.sub.1] such that for every linear combination f = [[summation].sub.n], [a.sub.n][f.sub.n] in H the bounds

[c.sup.2.sub.0][summation over (n)] [[absolute value of [a.sub.n]].sup.2] [less than or equal to] [[parallel]f[parallel].sup.2.sub.H] [less than or equal to] [c.sub.1] [summation over (n)] [[absolute value of [a.sub.n]].sup.2] (3.1.1)

are valid. Such a basis may be alternatively defined as the image of a complete orthonormal system under an invertible bounded linear transformation, i.e. [f.sub.n] = T[u.sub.n] for all n where {[u.sub.n]} is a complete orthonormal system and T is an invertible bounded linear transformation. For other equivalent definitions and further information on Riesz bases, see [8, 10, 16, 24, 29, 33]. In the case that H is [L.sup.2](Q) and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], n = 0, [+ or -] 1, [+ or -] 2, ..., the corresponding frequencies [chi] = [{[x.sub.n]}.sub.n[member of]Z] are associated to a special interpolation problem in the classical Paley-Wiener space of entire functions of exponential type, e.g., see [25]. We refer to such sequences X as complete interpolating sequences, in what follows, abbreviated as CIS. For examples of CISs see Section 4 and the references mentioned there.

Let [chi] = [{[x.sub.n]}.sub.n[member of]Z] be a CIS so that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], is a Riesz basis for [L.sup.2](Q) and [{[[phi].sub.n]([xi])}sub.n[member of]Z] is the corresponding dual or biorthogonal Riesz basis. Namely, the functions [{[[phi].sub.n]([xi])}sub.n[member of]Z] satisfy

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3.1.2)

Every function f in [L.sup.2](Q) may be uniquely expressed as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.1.3)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3.1.4)

For each m in Z consider the linear transformation [A.sub.m] which maps [L.sup.2](Q) onto itself and is defined by the formula

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where the coefficients {[a.sub.n]} are uniquely determined by f via (3.1.3). Because [([e.sup.-[ix.sub.n][integral]}.sub.n[member of]Z] is a Riesz basis the linear transformations [A.sub.m] are well defined, bounded, and invertible. Indeed, see 3.6 below; it is not difficult to see that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.1.5)

where [c.sub.0] and [c.sub.1] are the Riesz basis constant as in (3.1.1).

Note that [A.sub.m] = [A.sup.m.sub.1] for all integers m; in particular, [A.sub.0] = I, the identity operator. Also note that [A.sub.m] = I for all m in the case [chi] = Z.

Next observe that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

so that in view of (3.1.5) we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.1.6)

where, as in (3.1.5), [c.sub.0] and [c.sub.1] are constants from relation (3.1.1) associated with the Riesz basis [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In view of (3.1.6) the right-hand side of (3.1.3), initially defined for [xi] in the interval Q, is well-defined for almost all real [xi] and is a function in the amalgam space [l.sup.[infinity]]([L.sup.2]). We define the right-hand side of (3.1.3) as the prolongation of f and denote it by [E.sub.[chi]]f. This prolongation can also be defined in terms of the operators [A.sub.m]. Thus,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.1.7)

where [[tau].sub.n] and [P.sub.m] are, respectively, the translation and projection operators defined in Section 2.

We summarize these observations in what follows.

3.2 Proposition

Suppose [chi] = [{[x.sub.n]}.sub.n[member of]Z] is a CIS. If f is in [L.sup.2](Q) then f enjoys the representation (3.1.3) and the mapping of f into [E.sub.[chi]]f defined via (3.1.7) is a bounded linear transformation from [L.sup.2](Q) into [l.sup.[infinity]]([L.sup.2]) whose norm depends only on the constants associated with the Riesz basis [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

3.3 The summation operator [S.sub.[chi]]

Suppose [chi] = {[x.sub.n]} is a CIS. In view of the case [chi] = Z, the summation operator [S.sub.[chi]] should simply be the dual or adjoint of the prolongation operator [E.sub.[chi]] and map [l.sup.1]([L.sup.2]) into [L.sup.2](Q). To obtain an explicit expression, take f and g [l.sup.1]([L.sup.2]) and [L.sup.2](Q), respectively, and write

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [A.sup.*.sub.m] is the adjoint of [A.sub.m]. So it is clear that the operator [S.sub.[chi]] should be defined as

[S.sub.[chi]]f([xi]) = [[infinity].summation over (m=-[infinity]] [A.sup.*.sub.m] [tau]-2[pi]m [P.sub.m]f([xi]).

Note that the mapping f to [S.sub.[chi]]f maps the amalgam space [l.sup.1]([L.sup.2]) into [L.sup.2](Q) linearly with norm no greater than [c.sub.1]/[c.sub.0], where the constants [c.sub.0] and [c.sub.1] are the constants associated with the Riesz basis of exponentials [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] via (3.1.1).

The right-hand side of (3.3.1) can be re-expressed in a somewhat more efficiently by noting that

[tau]-2[pi]m [P.sub.m]f([xi]) = [P.sub.0][tau]-2[pi]mf([xi]),

setting

[B.sub.m] = [A.sup.*.sub.m] [P.sub.0],

and writing

[S.sub.[chi]]f([xi]) = [[infinity].summation over (m=-[infinity])] [B.sub.m][tau]-2[pi]mf([xi]). (3.3.2)

[S.sub.[chi]]f([xi]) can also be expressed as

[S.sub.[chi]]f([xi]) = [[infinity].summation over (n=-[infinity])] F([x.sub.n]) [[pi].sub.n]([xi]) (3.3.3)

for almost all [xi] in the interval Q, where {[[pi].sub.n]} is the Riesz basis dual to {[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]} in [L.sup.2](Q) and F and f are a Fourier transform pair (1.1.1). Identity (3.3.3) follows from

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and the fact that {[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]} and {[[pi].sub.n]} are dual Riesz bases for [L.sup.2](Q).

Formulas (3.3.2) and (3.3.3) are essentially our analogue of Poisson's summation formula (1.1.3), which we summarize as follows.

3.4 Proposition

Suppose [chi] = {[x.sub.n]}n[member of]Z] is a CIS and f is in [l.sup.1]([L.sup.2]). Then the following holds:

(i) f is in [L.sup.1](R) and F(x) is continuous.

(ii) The summation operator [S.sub.[chi]] defined by (3.3.2) is a bounded linear transformation from [l.sup.1]([L.sup.2]) into [L.sup.2](Q) whose norm depends only on the constants associated with the Riesz basis {[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]}n[member of]Z.

(iii) For n = 0, [+ or -] 1, [+ or -] 2, ...,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(iv) We have the summation formula

[[infinity].summation over (m=-[infinity])] [B.sub.m][tau]-2[pi]m f([xi]) = [S.sub.[chi]] f([xi]) = [[infinity].summation over (m=-[infinity])] F([x.sub.n]) [[phi].sub.n]([xi]) (3.4.1)

where [{[[phi].sub.]([xi])}.sub.n[member of]Z] is the Riesz basis dual to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Both sums converge in the [L.sup.2](Q) sense and formula is valid for almost all [xi] in Q.

3.5 Remarks

(a) If g is in [L.sup.2](Q) then [B.sub.m]g = [A.sup.*.sub.m]g and there is an explicit formula for [A.sup.*.sub.m]g in terms of {[[phi].sub.n]}. Namely, since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and {[[pi].sub.n]} are dual Riesz bases for [L.sup.2](Q), if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

So if

g([xi]) = [summation over (n)] [b.sub.n] [[phi].sub.n]([xi])

then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(b) In the case [chi] = Z we have that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] so that [A.sub.m] = [A.sup.*.sub.m] = [B.sub.m] is the identity on [L.sup.2](Q) and (3.4.1) reduces to (1.1.3). The fact that the class of CISs is sufficiently rich to make identity (3.4.1) interesting should be evident from the examples in Section 4.

(c) In general the members of the dual basis, {[[phi].sub.n]}, need not be continuous so that the terms in the sum on the right hand side of (3.4.1) need not make sense for certain values of [xi]. On the other hand there are many cases where these members are continuous so that with additional restrictions on f the pointwise interpretation of (3.4.1) is indeed possible.

3.6 Remarks

To see inequalities (3.1.5) and (3.1.6), suppose f is in [L.sup.2](Q) and enjoys representation (3.1.3). Then, because [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a Riesz basis in [L.sup.2](Q),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [c.sub.0] and [c.sub.1] are the constants, from relation (3.1.1), associated with the Riesz basis [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Similarly

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This proves (3.1.5).

To see (3.1.6), note that if [xi] is in the interval Q + 2[pi]m we may write

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

so that (3.1.6) follows from (3.1.5).

4 Appendix

Basic material on the classical Paley-Wiener space and CISs can be found, for example, in [25, 33]. CISs have been completely characterized in terms of zeros of entire functions of exponential type in [31], see also [19, 27]. Several examples of specific CISs are given in the subsection below.

Formula (3.4.1) has already been implicitly applied to show that certain piecewise polynomial splines converge to entire functions of exponential type, see [26, 28]. Other applications should follow. For example, under appropriate scaling this formula should also be useful in approximating the Fourier transform in terms of irregular samples.

4.1 Examples of CISs

(a) Any sequence [chi] = [{[x.sub.n]}.sub.n[member of]Z] such that

[absolute value of [x.sub.n] - n] [less than or equal to] r < 1/4

is a CIS. This was established in [21].

(b) If N is a positive integer, 0 [less than or equal to] [[alpha].sub.k] < ... < [[alpha].sub.N] < N, and

NZ + [[alpha].sub.k] = [{Nm + [[alpha].sub.k]}.sub.m[member of]Z],

then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is a CIS; see [25, Lectures 22 and 23].

(c) If [[chi].sub.1] is a CIS and [[chi].sub.2] is the sequence which results after any perturbation of a finite number of terms, then [[chi].sub.2] is also a CIS. For example, replacing {1, 2,..., N} in the integer lattice Z by any mutually distinct collection of numbers {[x.sub.1], ..., [x.sub.n]} such that [x.sub.k] [not member of] Z \ {1, 2, ..., N}, k = 1, ..., N, and, if necessary, reordering the result gives rise to a CIS. In particular, if 0 < [x.sub.1] < [x.sub.2] < ... [x.sub.N] < + 1, then

..., -2, -1, 0, [x.sub.1], [x.sub.2], ... [x.sub.N], N + 1, N + 2, ...

is a CIS. This is an elementary consequence of the characterization mentioned above.

References

[1] L. Auslander and Y. Meyer, A generalized Poisson summation formula, Appl. Comput. Harmon. Anal., 3(4), 372-376, 1996.

[2] M. G. Beaty and 3. R. Higgins, Aliasing and Poisson summation in the sampling theory of Paley-Wiener spaces, J. Fourier Anal. Appl., 1(1), 67-85, 1994.

[3] J. J. Benedetto and G. Zimmermann, Sampling multipliers and the Poisson summation formula, J. Fourier Anal Appl., 3(5), 505-523, 1997.

[4] R. P. Boas, Jr., Poisson's summation formula in [L.sup.2], J. London Math. Soc., 21, 102-105, 1946.

[5] W. L. Briggs and V. E. Henson, The DFT. An Owner's Manual for the Discrete Fourier Transform, SIAM, Philadelphia, 1995.

[6] P. L. Butzer and R. L. Stens, The Poisson summation formula, Whittaker's cardinal series and approximate integration. in CMS Conf. Proc., 3, AMS, Providence, 1983, 19-36.

[7] P. L. Butzer and A. Gessinger, The approximate sampling theorem, Poisson's sum formula, a decomposition theorem for Parseval's equation and their interconnections. Ann. Numer. Math., 4, 143-160, 1997.

[8] O. Christensen, An Introduction to Frames and Riesz Bases, Applied and Numerical Harmonic Analysis, Birkhauser, Boston, 2003.

[9] A. Cordoba, La formule sommatoire de Poisson. C. R. Acad. Sci. Paris Set. I Math., 306(8), 373-376, 1988.

[10] I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, 61, SIAM, Philadelphia, 1992.

[11] R. J. Duffin and H. F. Weinberger, On dualizing a multivariable Poisson summation formula, J. Fourier Anal. Appl., 3(5), 487-497, 1997.

[12] H. Dym and H. P. McKean, Fourier Series and Integrals, Academic, New York, 1972.

[13] H. Feichtinger, Banach convolution algebras of Wiener type and interpolation, in Operators, Proc. Conf. Budapest 38, Coll. Math. Soc. Janos Bolyai, 509-524, 1980.

[14] H. Feichtinger, Generalized amalgams, with applications to Fourier transform, Canadian J. Math., 42, 395-409, 1990.

[15] J. J. F. Fournier and J. Stewart, Amalgams of [L.sup.p] an [l.sup.q], Bull. Amer. Math. Soc., 13, 1-21, 1985.

[16] I. C. Gohberg and M. G. Krein, Introduction to the Theory of Linear Non-self-adjoint Operators, Translations of Mathematical Monographs, 18, AMS, Providence, R.I. 1969.

[17] K. Grochenig, An uncertainty principle related to the Poisson summation formula, Studia Math., 121, 87-104, 1996.

[18] C. Hell, An introduction to weighted Wiener amalgams, in Wavelets and Their Applications, M. Krishna, R. Radha, and S. Thangavelu, eds., Allied, New Delhi, 183-216, 2003.

[19] S. V. Hrushchev, N. K. Nikol'skii, and P. S. Pavlov, Unconditional bases of exponentials and of reproducing kernels, in Complex Analysis and Spectral Theory, V. P. Havin and N. K. Nikol'skii (eds.), Lecture Notes in Mathematics, 864, Springer-Verlag, Berlin, 214-335, 1981.

[20] A. Jerri, An extended Poisson summation formula for the generalized integral transforms and aliasing error bound for the sampling theorem, J. Appl. Anal., 26, 199-221, 1988.

[21] M. J. Kadets, The exact value of the Paley-Wiener constant, Doki. Akad. Nauk SSSR, 155(6), 1243-1254, 1964.

[22] J.-P. Kahane and P.-G. Lemarie-Rieusset, Remarques sur la formule sommatoire de Poisson, Studia Math., 109, 303-316, 1994.

[23] Y. Katznelson, An Introduction to Harmonic Analysis, Wiley, New York, 1968.

[24] H. O. Kim and J. K. Lim, New Characterizations of Riesz Bases, App. Comp. Harm. Anal., 4, 222-229, 1997.

[25] B. Ya. Levin, Lectures on Entire Functions, Translation of mathematical monographs, 150, American Math. Society, Providence, 1996.

[26] Yu. Lyubarskii and W. R. Madych, The recovery of irregularly sampled band limited functions via tempered splines, J. Functional Analysis, 155, 201-222, 1994.

[27] Yu. Lyubarskii and K. Seip, Complete interpolating sequences for Paley-Wiener spaces and Muckenhoupt's (Ap) condition, Revista Math. Iber., 13(2), 361-376, 1997.

[28] W. R. Madych, Summability of Lagrange type interpolation series. J. Anal. Math. 84, 207-229, 2001.

[29] Y. Meyer, Wavelets and Operators, Cambridge Univ. Press, Cambridge, U. K., 1992.

[30] F. Natterer, The Mathematics of Computerized Tomography, Teubner, Stuttgart, 1986.

[31] B. S. Pavlov, The basis property of a system of exponentials and the condition of Muckenhoupt, Dokl. Acad. Nauk SSSR, 247(1), 37-40, 1979.

[32] E. M. Stein and G. Weiss, Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, Princeton, N. J., 1971.

[33] R. M. Young, An introduction to Nonharmonic Fourier Series, Academic Press, New York, 1980.

[34] A. Zygmund, Trigonometric Series Second edition, Cambridge, London 1968.

Yu. Lyubarskii *

Math. Dept., NTNU 

Trondheim, Norway

[e-mail]

W. R. Madych

Math. Dept,, University of Connecticut 

Storrs, CT 06269-3009, USA

[e-mail]

* Yurii Lyubarskii is partially supported by Norwegian Research Council grants 177355/V30, 10323200.