c Pleiades Publishing, Ltd., 2008. ISSN 0001-4346, Mathematical Notes, 2008, Vol. 83, No. 2, pp. 190–200.
c A. A. Zakharova, 2008, published in Matematicheskie Zametki, 2008, Vol. 83, No. 2, pp. 210–220. Original Russian Text


On the Properties of Generalized Frames A. A. Zakharova* Moscow State University Received May 30, 2006; in final form, March 21, 2007

Abstract—In this paper, we introduce the notion of generalized frames and study their properties. Discrete and integral frames are special cases of generalized frames. We give criteria for generalized frames to be integral (discrete). We prove that any bounded operator A with a bounded inverse acting from a separable space H to L2 (Ω) (where Ω is a space with countably additive measure) can be regarded as an operator assigning to each element x ∈ H its coefficients in some generalized frame. DOI: 10.1134/S0001434608010215 Key words: frame, tight frame, integral frame, bounded operator, separable Hilbert space, Lebesgue space, countably additive measure.

INTRODUCTION Frames were first introduced in 1952 by Duffin and Schaeffer in their paper [1] dealing with the study of the exponential system {eiλn t }n∈Z . Definition 1. Let H be a Hilbert space over the field R or C. A system of elements {ϕj }j∈N ⊂ H is called a frame if there exist constants a, b, 0 < a ≤ b < ∞, such that, for all g ∈ H, the following inequalities hold: ∞
|(g, ϕj )|2 ≤ bg2H . ag2H ≤ j=1

If one can take a = b, then the frame is said to be tight. Obviously, a tight frame is not, in general, an orthogonal system (for examples, see [2, p. 100 (Russian transl.)], [3], [4]). Expansions in frame series have properties similar in many respects to those of expansions in orthogonal systems, and they are used in both theoretical studies and applications for signal and image analysis, data compression, and pattern recognition (see, for example, [5, pp. 145–183 (Russian transl.)]). Frames are also widely used in wavelet theory (see [2, p. 95 (Russian transl.)] and, further, [6, pp. 120–129], [7, pp. 109–140 (Russian transl.)]). The continuous analog of discrete frames, the socalled integral frame, was introduced; it is defined as follows (see [7], [8]). Definition 2. Suppose that H is a Hilbert space over the field R or C, and Ω is a space with countably additive measure µ. A system of functions {ϕω }ω∈Ω ⊂ H is called an integral frame if (g, ϕω ) is µ-measurable for all g ∈ H and there exist a, b, 0 < a ≤ b < ∞, such that, for all g ∈ H,  |(g, ϕω )|2 dµ(ω) ≤ bg2H . ag2H ≤ Ω

If Ω = N and, for any k ∈ N, µ(k) = 1, then Definition 2 coincides with Definition 1. A tight integral frame (an integral frame with constants a = b) is an orthogonal-like system. *

E-mail: [email protected].

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Definition 3. A system of elements {eω }ω∈Ω ⊂ H is called an orthogonal-like expansion system in H if any element y ∈ H can be expressed as  y= yω eω dµ(ω), Ω

while the integral is regarded as the proper or improper Lebesgue integral of a where yω = ∞ function with values in H; moreover, in the latter case, ∞there is an exhaustion {Ωk }k=1 of the space Ω (all the Ωk are measurable, Ωk ⊂ Ωk+1 for k ∈ N and k=1 Ωk = Ω), possibly depending on y and called suitable for y, such that the function yω eω is Lebesgue integrable on Ωk and   yω eω dµ(ω) = lim (L) yω eω dµ(ω). y= (y, eω ),

k→∞

Ω

Ωk

If Ω = N, Ωk = {1, 2, . . . , k}, µ(k) > 0 for all k ∈ N, then we obtain the definition of a countable nonnegative orthogonal-like expansion system with ordinary convergence which covers, as a particular case, countable complete orthogonal systems. Thus, the notion of an orthogonal-like system generalizes the notion of an orthogonal system, but, in contrast, an orthogonal-like system does not possess the uniqueness property of the expansion. For more details on the properties of orthogonal-like systems, see [9], [10]. Many properties of discrete frames generalize to the case of integral frames as well as to a still wider class of systems, the so-called generalized frames whose definition will be given later. In this case, generalized orthogonal-like systems whose properties were described in [11], [12] will be tight generalized frames. In particular, the systems of functions defining the Fourier and Hilbert transformations are generalized orthogonal-like systems [12]. Besides the properties of generalized and integral frames, we shall prove that any bounded operator A with a bounded inverse acting from a separable space H to L2 (Ω), can be regarded as an operator assigning to any element x ∈ H its coefficients in some generalized frame {ϕω }ω∈Ω (the corresponding result for generalized orthogonal-like systems was obtained in [13]). 1. DUAL INTEGRAL FRAMES Definition 4. A frame {ψ ω }ω∈Ω ⊂ H is said to be dual to an integral frame {ϕω }ω∈Ω ⊂ H if, for any y ∈ H,  y = (y, ψ ω )ϕω dµ(ω). Ω

Remark. It is not clear from this definition whether the frame {ϕω }ω∈Ω will be dual to the frame {ψ ω }ω∈Ω , i.e., whether the relation  y = (y, ϕω )ψ ω dµ(ω) Ω

holds for any y ∈ H. In the case of a discrete frame one can give an equivalent definition (in the general case, the property presented below follows from Definition 4): a frame {ψn }n∈N ⊂ H is dual to a frame {ϕn }n∈N ⊂ H if, for any x, y ∈ H,
(x, ψn )(ϕn , y). (x, y) = n∈N

This definition implies that if a frame {ψn }n∈N is dual to a frame {ϕn }n∈N , then, similarly, {ϕn }n∈N is dual to {ψn }n∈N . In what follows, we give conditions under which the definitions are equivalent for generalized frames. MATHEMATICAL NOTES

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Consider the frame operator T , acting from H to L2 (Ω) as follows: (T y)ω = (y, ϕω ). Let us calculate the adjoint operator to T :   ∗ ω (T c(ω), y) = (c(ω), T y) = c(ω)(y, ϕ ) dµ(ω) = c(ω)(ϕω , y)dµ(ω); Ω

Ω

hence



∗

c(ω)ϕω dµ(ω).

T c(ω) = Ω

By the lemma from [2, p. 101 (Russian transl.)], the condition a Id ≤ T ∗ T ≤ b Id implies that S = T ∗ T is an invertible operator and its inverse is also bounded. Thus, we have b−1 Id ≤ (T ∗ T )−1 ≤ a−1 Id . Acting by the operator

S −1

=

(T ∗ T )−1

on

{ϕω }ω∈Ω ,

(1)

we obtain the system

{ϕ ω }ω∈Ω = S −1 {ϕω }ω∈Ω . Lemma 1. The system {ϕ ω }ω∈Ω ⊂ H is a dual frame, with constants b−1 , a−1 , to the frame ω {ϕ }ω∈Ω ⊂ H. Proof. Since



∗

c(ω)ϕω dµ(ω),

T c(ω) = Ω

it follows that

 (y, ϕω )ϕω dµ(ω).

Sy = Ω

1. The bounded operator S has a bounded inverse and S ∗ = S; hence S −1 = (S −1 )∗ (see [2, p. 103 (Russian transl.)]). Therefore, (y, ϕ ω ) = (y, S −1 ϕω ) = ((T ∗ T )−1 y, ϕω ), whence





|((T ∗ T )−1 y, ϕω )|2 dµ(ω) = T (T ∗ T )−1 y2 = ((T ∗ T )−1 y, y).

|(y, ϕ  )| dµ(ω) = ω

Ω

2

Ω

Taking (1) into account, we find that {ϕ ω }ω∈Ω ⊂ H is a frame. 2. The frame {ϕ ω }ω∈Ω is dual to {ϕω }ω∈Ω , because   −1 −1 ω ω y = SS y = (S y, ϕ )ϕ dµ(ω) = (y, S −1 ϕω )ϕω dµ(ω). Ω

Ω

Definition 5. The system {ϕ ω }ω∈Ω introduced above is called the dual canonical frame to the integral ω frame {ϕ }ω∈Ω ⊂ H. Denote Ran(T ) = {y ∈ L2 (Ω) : y = Tf for some f ∈ H}. Lemma 2. The following relation holds: Ran(T ) = Ran(T )

and

T ∗ T = Id,

ω ). where T is the frame operator for {ϕ ω }ω∈Ω , i.e., defined by the formula (Ty)ω = (y, ϕ MATHEMATICAL NOTES

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Proof. 1. We have ω ) = (Ty)ω , (T (T ∗ T )−1 y)ω = ((T ∗ T )−1 y, ϕω ) = (y, ϕ and hence T ∗ T = T ∗ T (T ∗ T )−1 = Id . 2. Since T = T (T ∗ T )−1 , it follows that Ran(T ) ⊂ Ran(T ). Also we have T = T(T ∗ T ), whence Ran(T ) ⊂ Ran(T ). Therefore, Ran(T ) = Ran(T ). Theorem 1 (extremal property of the coefficients of the expansion in an integral frame). If, for some c(ω) ∈ L2 (Ω), the following relation holds:  c(ω)ϕω dµ(ω), y= Ω

then



 |(y, ϕ  )| dµ(ω) ≤ ω

|c(ω)|2 dµ(ω);

2

Ω

Ω

moreover, the equality holds if and only if c(ω) = (y, ϕ ω ) a.e. on Ω. (For the case of discrete frames, this theorem was proved in [1]). Proof. As noted above,



∗

c(ω)ϕω dµ(ω),

T c(ω) = Ω

T ∗ c(ω).

i.e., by the condition y =

Since Ran(T ) is closed, we can write

c(ω) = a(ω) + b(ω),

where

a(ω) ∈ Ran(T ) = Ran(T ),

b(ω) ⊥ Ran(T ),

i.e., b(ω) ⊥ a(ω) and c2 = a2 + b2 . Since a(ω) ∈ Ran(T ), there exists an f ∈ H such that  a = Tf

and

 + b; c = Tf

hence  + T ∗ b. y = T ∗ c = T ∗ Tf Recalling that T ∗ T = Id and taking into account the fact that b(ω) ⊥ Ran(T ) implies T ∗ b = 0, we obtain y=f and c(ω) = Ty + b; therefore,



|c(ω)|2 dµ(ω) = Ty2 + b2 >

Ω

 |(y, ϕ ω )|2 dµ(ω) Ω

if b = 0. Proposition 1. Suppose that {ϕω }ω∈Ω is an integral frame in Hilbert space H, c(ω) is a function from the Lebesgue space L2 (Ω) with values in R or C, {Λk }∞ k=1 is a sequence of measurable ω subsets Ω such that lim k→∞ Λk = Ω, and c(ω)ϕ is Lebesgue integrable on Λk . Then, in H, the following limit exists:  c(ω)ϕω dµ(ω). lim (L) k→∞

Λk

Proof. The proof is similar to that of Theorem 4 (the case of tight integral frames) in [9]. MATHEMATICAL NOTES

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2. GENERALIZED FRAMES Definition 6. Suppose that {Hn }∞ n=1 is a system of closed expanding subspaces in H whose union is everywhere dense in H, {eω }ω∈Ω is a system such that any of its elements eω is a sequence {eωn }∞ n=1 of elements of H, and eωn is the orthogonal projection of eωn+1 on Hn . Then {eω }ω∈Ω is a generalized system in H. Definition 7. A generalized system {eω }ω∈Ω is called a generalized orthogonal-like expansion system in H if any element y ∈ Hn can be expressed as  y= yωn eωn dµ(ω), Ω

yωn

(y, eωn ),

= while the integral is regarded as the proper or improper Lebesgue integral of a where function with values in H. For more details on the properties of generalized orthogonal-like systems, see [11], [12]. Definition 8. A generalized system of functions {ϕω }ω∈Ω ⊂ H is called a generalized frame if there exist a, b, 0 < a ≤ b < ∞, such that, for any y ∈ Hn , all the functions (y, ϕωn ) are measurable and  |(y, ϕωn )|2 dµ(ω) ≤ by2Hn ay2Hn ≤ Ω

for any y ∈ Hn . Example 1. Consider the system of functions 1 ω−x cot , ω ∈ [−π, π], eω (x) = 2π 2 defining the conjugate function  π 1 x−t dt. f (x) = f (t) cot 2π −π 2 If we consider its projections on the subspace 
 n ak cos(kx) + bk sin kx , Hn = k=1

i.e., the subspaces of trigonometric polynomials of degree at most n, then the system {eω }ω∈Ω = {{eωn }∞ n=1 }ω∈Ω ,

where

eωn (x) =

− cos(n + 1/2)(ω − x) + cos((ω − x)/2) , 2π sin((ω − x)/2)

forms a generalized tight frame in the space   H  = f ∈ L2 [−π, π] :

π

 f (t) dt = 0

−π

with constants a = b = 1 (see [15, pp. 528–534]). To simplify the notation, here and elsewhere we denote by eω the sequence {eωn }∞ n=1 , i.e., both an element of the generalized system and the function that “generates” the generalized system and is the limit of this sequence. In the theory of trigonometric series, the following function is considered:  1 π f (t) ∗ dt, f (x) = π −π x − t where the integral is also understood in the sense of the principal value of Cauchy (see [15, pp. 528– 534]). The difference 1/t − cot[t/2]/2 is positive, has a positive derivative on the half-segment (0, π], and attains a maximum value on it at the point π, i.e.,    1 1  − cot t  ≤ 1/π on [−π, 0) ∪ (0, π]. t 2 2  MATHEMATICAL NOTES

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Then, by the inequality f ∗ gp ≤ f p g1 , where ∗ denotes convolution (see [16, p. 9 (Russian transl.)]), we have f ∗ − f 2 ≤ 2f 2 /π. Therefore, the system of functions eω (x) =

1 , [π(ω − x)]

eωn (x) =

− cos(n + 1/2)(ω − x) + 1 , π(ω − x)

ω ∈ [−π, π],

where

is a generalized frame on the same subspace H  with the same system of subspaces Hn with constants a = (1 − 2/π)2 , b = (1 + 2/π)2 . Theorem 2. If {ϕω }ω∈Ω is a generalized frame in H, then, for any y ∈ H, there exists a unique, (up to equivalence) function y(ω) on Ω such that the sequence of functions yωn = (y, ϕωn ) converges to it in the following sense:  |yωn − y(ω)|2 dµ(ω) = 0. lim n→∞ Ω

Proof. For all m, n ∈ N, n ≥ m, and y ∈ H, we have    n m 2 ω ω 2 |yω − yω | dµ(ω) = |(y, ϕn ) − (y, ϕm )| dµ(ω) = |(y, Pn ϕωn ) − (y, Pm ϕωn )|2 dµ(ω) Ω Ω Ω  |(Pn y, ϕωn ) − (Pm y, ϕωn )|2 dµ(ω) ≤ bPn y − Pm y2 , = Ω

 where Pn is the orthogonal projection of H on the subspace Hn . Since ∞ n=1 Hn = H, it follows that Pn y → y as n → ∞. Therefore, for any ε > 0, there exists an N ∈ N such that, for all m, n > N ,  |(y, ϕωn ) − (y, ϕωm )|2 dµ(ω) < ε. Ω

Hence there exists a y(ω) such that



lim

n→∞ Ω

|yωn − y(ω)|2 dµ(ω) = 0.

Corollary 2.1. If {ϕω }ω∈Ω is a generalized frame in H, then, for any y ∈ H,  2 |y(ω)|2 dµ(ω) ≤ by2H . ayH ≤ Ω

Proof. The assertion of the theorem follows from the fact that   n 2 |yω | dµ(ω) → |y(ω)|2 dµ(ω), Ω

n → ∞,

Ω

aPn y2 → ay2 ,

n → ∞,

and from the passage to the limit in the inequalities from Definition 8. Consider the sequence of operators Tn acting from Hn to L2 (Ω) as follows: (Tn f )ω = (f, ϕωn ). By inequalities(1), we have b−1 Id ≤ (Tn∗ Tn )−1 ≤ a−1 Id; hence we obtain the system {ϕ nω }ωω∈Ω = Sn−1 {ϕωn }ω∈Ω , MATHEMATICAL NOTES

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which is the canonical dual frame for {ϕωn }ω∈Ω in the space Hn , where Sn−1 = (Tn∗ Tn )−1 . Note that Sn−1 is a self-adjoint operator (because it is a bounded inverse to the bounded self-adjoint operator Sn ) and, for any yn = Pn y ∈ Hn , there exists an xn = Pn x ∈ Hn such that yn = Sn−1 xn ; moreover, it is unique. Therefore, for any y ∈ H, there exists an x ∈ H such that (y, ϕωn ) = (y, Pn ϕωn ) = (Pn y, ϕωn ) nω ). = (Sn−1 Pn x, ϕωn ) = (Pn x, Sn−1 ϕωn ) = (x, Sn−1 ϕωn ) = (x, ϕ Thus, the sequence (y, ϕ nω ) also converges in L2 (Ω) to some function y(ω) for any y ∈ H. Proposition 2. Suppose that {ϕω }ω∈Ω is a generalized frame in H and {ψnω }ω∈Ω is the dual integral frame to {ϕωn }ω∈Ω in the space Hn for each n. Then, for any y ∈ H,  (y, ψnω )ϕωn dµ(ω); y = lim n→∞ Ω

in particular,

 y = lim

n→∞ Ω

and, for all x, y ∈ H,

(y, ϕ nω )ϕωn dµ(ω)

 (y, x) = lim

n→∞ Ω

(y, ψnω )(ϕωn , x) dµ(ω).

Proof. Since (y, ψnω ) = (Pn y, ψnω ) and  Pn y = (y, ψnω )ϕωn dµ(ω), Ω  (y, ψnω )ϕωn dµ(ω), y = lim Pn y = lim n→∞

n→∞ Ω

where y = limn→∞ Pn y by Lemma 1 from [12], it follows that, for all x, y ∈ H,    ω ω (y, ψn )ϕn dµ(ω), x = lim (y, ψnω )(ϕωn , x) dµ(ω). (y, x) = lim n→∞ Ω

n→∞ Ω

In the study of of the generalized frame {ϕω }ω∈Ω , we shall denote by L2y (Ω) the image of H in the space L2 (Ω) under the mapping y → y(ω), i.e., L2y (Ω) = {c(ω) ∈ L2 (Ω) : there exists a y ∈ H such that c(ω) = y(ω)}. In the general case, L2y (Ω) may not necessarily coincide with L2 (Ω) (for more details, see [12]). Definition 9. A generalized frame {ϕω }ω∈Ω ⊂ H is said to be L2 -measurable if, for any function c(ω) ∈ L2 (Ω) with values in R or C, all the functions c(ω)ϕωn are measurable as functions on Ω with values in H. If a frame is L2 -measurable, then the definitions of duality (see the remark to Definition 4) are equivalent. Therefore, it is of interest to determine when the frame is L2 -measurable. Theorem 3. Suppose that {ϕω }ω∈Ω is a generalized frame in H. Then for this frame to be L2 -measurable, it is necessary and sufficient that the following two conditions hold: 1) for each function c(ω) ∈ L2 (Ω), each measurable subset E ⊂ Ω of finite measure, and each n ∈ N, the function c(ω)ϕωn is almost separable-valued on E; 2) for each y ∈ H and each n ∈ N, the function c(ω)(y, ϕ nω ) is measurable on Ω. MATHEMATICAL NOTES

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The theorem immediately follows from [14, p. 166 (Russian transl.)]. Corollary 2.2. Suppose that {ϕω }ω∈Ω is a generalized frame in H. L2 -measurable, it suffices that H is separable.

For this frame to be

Lemma 3. If {ϕω }ω∈Ω is a generalized L2 -measurable frame in H, then, for any c(ω) ∈ L2 (Ω), ω there exists an exhaustion {Ωk }∞ k=1 of the space Ω such that, for each n ∈ N, the function c(ω)ϕn is Lebesgue integrable on Ωk . Proof. The sets Ωk are chosen in the same way as in the proof of Lemma 3 from [12]. Theorem 4. Suppose that {ϕω }ω∈Ω is a generalized frame in Hilbert space H; c(ω) is a function from the Lebesgue space L2 (Ω) with values in R or C; for each n ∈ N, {Λnk }∞ k=1 is a sequence of measurable subsets Ω such that lim k→∞ Λk = Ω; and c(ω)ϕωn is Lebesgue integrable on Λnk . Then the limit  c(ω)ϕωn dµ(ω) lim lim (L) n→∞ k→∞

Λk

exists in H. Proof. By Proposition 1, the limit

 y n = lim (L) k→∞

Λn k

c(ω)ϕωn dµ(ω).

exists for each n ∈ N. Since, for m ≤ n, the following relation holds:   n ω c(ω)Pm (ϕn ) dµ(ω) = lim (L) Pm (y ) = lim (L) k→∞

k→∞

Λn k

and y n 2 ≤ a−1



|(y n , ϕ nω )|2 dµ(ω) ≤ a−1

Λn k

c(ω)ϕωm dµ(ω) = y m



Ω

|c(ω)|2 dµ(ω) < ∞ Ω

for any n ∈ N, it follows that, by Lemma 1 from [12], the sequence y n is convergent. Theorem 5. Suppose that {ϕω }ω∈Ω is a generalized L2 -measurable frame in Hilbert space H, then, for any c(ω), which is a function from the Lebesgue space L2 (Ω), the element  c(ω)ϕωn dµ(ω) y = lim n→∞ Ω

exists in H. This theorem is a consequence of Lemma 3 and Theorem 4. Theorem 6 (extremal property of the coefficients of the expansion in a generalized frame). Suppose that {ϕω }ω∈Ω is a generalized frame in H. If, for some function cn (ω) → c(ω) ∈ L2 (Ω),  cn (ω)ϕωn dµ(ω), y = lim n→∞ Ω

then the following inequality holds:   2 | y (ω)| dµ(ω) ≤ |c(ω)|2 dµ(ω); Ω

Ω

here the equality holds if and only if c(ω) = y(ω) a.e. on Ω. MATHEMATICAL NOTES

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Proof. Suppose that Tn : Hn → L2 (Ω) acts so that (Tn y)ω = (y, ϕωn ). Then  ∗ Tn c(ω) = lim c(ω)ϕωn dµ(ω), n→∞ Ω

Tn∗ → T ∗ , and Tn∗ cn (ω) → T ∗ c(ω), because (Tn∗ cn (ω), y) − (T ∗ c(ω), y) = (cn (ω), Tn y) − (c(ω), T y)   = (cn (ω) − c(ω))Tn y dµ(ω) + c(ω)(Tn y − T y) dµ(ω) Ω Ω  c(ω)(Tn y − T y) dµ(ω) → 0. ≤ cn (ω) − c(ω)2 Tn y2 + Ω

Hence, by the condition, we have



y = lim

n→∞ Ω

cn (ω)ϕωn dµ(ω) = T ∗ c(ω).

In particular, y = T ∗ y(ω). Define T : H → L2 (Ω) so that nω ). (Ty)ω = lim (y, ϕ n→∞

Further, we can apply the arguments from Theorem 1 if we show that T ∗ T = Id and Ran(T ) = Ran(T ). The first assertion is obvious from the definition of the operators T ∗ and T. Hence, using the arguments from Lemma 2, we immediately obtain Ran(T ) = Ran(T ). Remark. Note that the system {ψ ω }ω∈Ω , where each element of the system ψnω is the dual frame to the element ϕωn of the generalized frame {ϕω }ω∈Ω in the space Hn , is not, generally, a generalized system ω from Hn+1 onto Hn ) in H (because the element ψnω is not necessarily the orthogonal projection of ψn+1 ω and, therefore, cannot be called a dual generalized frame to {ϕ }ω∈Ω . Let us determine when a generalized frame is integral or discrete (the corresponding results for generalized tight frames were obtained in [12]). Theorem 7. Suppose that {ϕω }ω∈Ω is a generalized L2 -measurable frame in the space H (with system of subspaces {Hn }∞ n=1 and indices from Ω). The whole frame (almost the whole) is an integral frame in the space H with ω ranging over all (almost all) values of Ω in the sense that, for each n ∈ N and each (almost each) ω ∈ Ω, the element ϕωn is the orthogonal projection ϕω from H onto Hn if and only if, for each (almost each) ω ∈ Ω, sup ϕωn  < +∞.

n∈N

Proof. Necessity. If, for some ω ∈ Ω, ϕωn are the orthogonal projections of ϕω from H onto Hn for all n ∈ N, then sup ϕωn  ≤ ϕω  < +∞.

n∈N

Sufficiency. By Lemma 1 from [12], there exist ϕω = lim ϕωn n→∞

and

ϕωn = Pn (ϕω )

for each (almost each) ω ∈ Ω. Let us prove that {ϕω }ω∈Ω is an integral frame. By Theorem 4, for any y ∈ Hn , y(ω) = lim (y, ϕωm ) = (y, ϕωn ) = (y, ϕω ) m→∞

for each (almost each) ω ∈ Ω. Since the union of the spaces Hn is everywhere dense in H, it follows that, for any y ∈ H, the equality y(ω) = (y, ϕω ) holds. Then, by Theorem 5, the system {ϕω }ω∈Ω is an integral frame. MATHEMATICAL NOTES

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Theorem 8. Suppose that {ϕω }ω∈Ω is a generalized L2 -measurable frame in the space H (with system of subspaces {Hn }∞ n=1 and indices from Ω). The whole frame (almost the whole) is a discrete frame in the space H with ω ranging over all (almost all) values of Ω in the sense that, for each n ∈ N and each (almost each) ω ∈ Ω, the element ϕωn is the orthogonal projection ϕω from H onto Hn if and only if, for each (almost each) ω ∈ Ω the inequality µ(ω) > 0 holds. Proof. The necessity is obvious: for any discrete frame, for each (almost each) ω ∈ Ω we have µ(ω) > 0.

Sufficiency. The subsystem {ϕω }µ(ω)>0 is a generalized frame with ω ∈ Ω such that µ(ω) > 0. Theorem 9. Suppose that H is a separable Hilbert space, Ω is a space with measure µ, L2 (Ω) is the Lebesgue space over Ω, and A is a bounded linear operator with bounded inverse acting from H to L2 (Ω) (i.e., there exist a, b, 0 < a ≤ b < ∞, such that, for any x ∈ H, ax ≤ Ax ≤ bx.) Then there exists a system {ϕω }ω∈Ω which is a generalized frame in H such that, for any x ∈ H, the following relation holds: A(x) = (L2 ) lim (x, ϕωn ). n→∞

(For the case of tight generalized frames, the corresponding theorem was proved in [13]). Proof. Suppose that {xn }n∈N is a complete orthonormal system in H (which exists, because the space is separable) and Hn is the linear hull of the elements x1 , . . . , xn . Define ϕωn =

n


Axk (ω)xk ;

k=1

hence, obviously,

ϕωn

∈ Hn and A(xk ) =

(xk , ϕωn )

An x(ω) = (x, ϕωn )

and

for all k ≤ n. Then  2 An x = |(x, ϕωn )|2 dµ(ω); Ω

hence the

ϕωn

constitute an integral frame in Hn , i.e.,  2 |(x, ϕωn )|2 dµ(ω) ≤ bx2 . ax ≤ Ω

By construction, ϕωn is the orthogonal projection of ϕωn+1 on Hn . Thus, {ϕω }ω∈Ω is a generalized frame in H. Suppose that Pn is the orthogonal projection operator from H onto Hn . Then, for any x ∈ H, the following relations hold: Ax = A lim Pn x = lim A(Pn x) = lim (Pn x, ϕωn ) = lim (x, ϕωn ). n→∞

n→∞

n→∞

n→∞

The theorem is proved. ACKNOWLEDGMENTS The author wishes to express gratitude to Professor T. P. Lukashenko for posing the problem and valuable discussions of the subject matter of the paper, as well as for reading the paper and making useful remarks. This work was supported by the Russian Foundation for Basic Research (grant no. 05-01-00192). MATHEMATICAL NOTES

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REFERENCES 1. R. J. Duffin and A. C. Schaeffer, “A class of nonharmonic Fourier series,” Trans. Amer. Math. Soc. 72 (2), 341–366 (1952). 2. I. Daubechies, Ten Lectures on Wavelets (SIAM, Philadelphia, PA, 1992; RKhD, Moscow–Izhevsk, 2001). 3. S. Verblunsky, “Some theorems on F. A.-series,” Rend. Circ. Mat. Palermo (2) 3 (1), 89–105 (1954). 4. D. Han and D. R. Larson, Frames, Bases, and Group Representations, in Mem. Amer. Math. Soc. (Amer. Math. Soc., Providence, RI, 2000), Vol. 147 (697). 5. S. Mallat, A Wavelet Tour of Signal Processing (Academic Press, Boston, MA, 1999; Mir, Moscow, 2005). 6. Ch. Chui, An Introduction to Wavelets (Academic Press, Boston, MA, 1992; Mir, Moscow, 2001). 7. Ch. Blatter, Wavelets: A Primer (Peters, Ltd., Natick, MA, 1998; Tekhnosfera, Moscow, 2004). 8. A. A. Zakharova, “Integral Riesz systems and their properties,” Vestnik Moskov. Univ. Ser. I Mat. Mekh., No. 6, 28–33 (2004) [Moscow Univ. Math. Bull., No. 6, 29–33 (2004)]. 9. T. P. Lukashenko, “Coefficients with respect to orthogonal-like decomposition systems,” Math. USSR-Sb. 188 (12), 57–72 (1997) [Russian Acad. Sci. Sb. Math. 188 (12), 1783–1798 (1997)]. 10. T. P. Lukashenko, “Orthogonal-like nonnegative expansion systems,” Vestnik Moskov. Univ. Ser. I Mat. Mekh., No. 5, 27–31 (1997) [Moscow Univ. Math. Bull No. 5, 30–34 (1997)]. 11. T. P. Lukashenko, “Generalized expansion systems similar to orthogonal ones,” Vestnik Moskov. Univ. Ser. I Mat. Mekh., No. 4, 4–8 (1998) [Moscow Univ. Math. Bull, No. 5, 30–34 (1998)]. 12. T. P. Lukashenko, “Properties of generalized decomposition systems similar to orthogonal systems,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 10, 33–48 (2000) [Russian Math. (Iz. VUZ), No. 10, 30–44 (2000)]. 13. T. Yu. Semenova, “Existence and equivalence of generalized orthogonal-like systems,” Vestnik Moskov. Univ. Ser. I Mat. Mekh., No. 3, 10–15 (2001) [Moscow Univ. Math. Bull., No. 3, 9–13 (2001)]. 14. N. Dunford and J. T. Schwartz, Linear Operators: General Theory (Interscience Publ., New York–London, 1958; Inostr. Lit., Moscow, 1962), Vol. 1. 15. N. K. Bari, Trigonometric Series (Fizmatgiz, Moscow, 1961) [in Russian]. 16. E. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces (Princeton Univ. Press, Princeton, NJ, 1971; Mir, Moscow, 1974).

MATHEMATICAL NOTES

Vol. 83 No. 2 2008