So reconstruction requires inverting the frame operator which is often difficult or impossible in practice. It follows that for all ϕ H we have
|
|
- Mervin Miller
- 6 years ago
- Views:
Transcription
1 CONSTRUCTING INFINITE TIGHT FRAMES PETER G. CASAZZA, MATT FICKUS, MANUEL LEON AND JANET C. TREMAIN Abstract. For finite and infinite dimensional Hilbert spaces H we classify the sequences of positive real numbers {a n } so that there is a tight frame {ϕ n } for H satisfying: ϕ n = a n, for all n = 1, 2, 3,. In the finite dimensional case we will identify the frames which are closest to being tight (in the sense of minimizing potential enerty) for any sequence {a n }. 1. Introduction If H is a Hilbert space, a sequence {ϕ n } M (M is finite or infinite) is a frame for H if there are constants A, B > 0 so that for all ϕ H, A ϕ 2 ϕ, ϕ n 2 B ϕ 2. If A = B = λ, {ϕ n } M is a λ-tight frame. If λ = 1, it is a Parseval frame; if ϕ n = ϕ m for all 1 n, m M it is a equal-norm frame; and if ϕ n = 1 for all n it is a unit-norm frame. The importance of λ-tight frames is that they allow simple reconstruction of the elements of H. It is known that {ϕ n } M is a frame for H if and only if Sϕ = ϕ, ϕ n ϕ n, is an invertible operator on H called the frame operator. To reconstruct an element ϕ H we write ϕ = SS 1 ϕ = S 1 ϕ, ϕ n ϕ n. So reconstruction requires inverting the frame operator which is often difficult or impossible in practice. It follows that for all ϕ H we have Aϕ, ϕ = A ϕ 2 ϕ, ϕ n 2 = Sϕ, ϕ B ϕ 2 = Bϕ, ϕ. P.G. Casazza and M. Leon were supported by NSF DMS
2 2 CASAZZA, FICKUS, LEON, TREMAIN Hence, AI S BI and so our frame is λ-tight if and only if S = λi. So if {ϕ n } M is a λ-tight frame then for all ϕ H N, ϕ = 1 λ ϕ, ϕ n ϕ n. Frames were introduced in 1952 by Duffin and Schaeffer [12] while they were working on some deep problems nonharmonic Fourier series. Since then, frames have been used extensively in signal/image processing where they are called Gabor frames or Weyl-Heisenberg frames [3, 14, 18, 21]. Recently, many new applications of tight frames have arisen in internet coding [8, 15, 16, 17], wireless communication [20, 22, 23], quantum detection theory [13], and much more. Each new application requires a new class of tight frames. Until recently, it was thought that the class of tight frames was quite sparse and that they probably did not exist for many applications. However, after the introduction of frame potentials by Benedetto and Fickus [2], there was an explosion of new results concerning the construction of tight frames for finite dimensional Hilbert spaces [6, 7, 9, 10]. For a survey of all of these important developments see [5]. The importance of [2] is that it gives a geometric interpretation for equal-norm finite tight frames. Building on the work of Benedetto and Fickus, Casazza, Fickus, Kovačević, Leon and Tremain [6] gave a physical interpretation for finite tight frames along the lines of Columb s law in Physics. This allows us to anticipate results in frame theory by using results from classical Mechanics. In particular, in [6] the authors have identified the finite frames which are the closest to being tight and having the norms of the frame vectors prescribed in advance. In this paper we will extend these results to infinite frames for both finite and infinite dimensional Hilbert spaces. Throughout the paper we will use H to denote a finite or infinite dimensional Hilbert space and H N for an N- dimensional Hilbert space (real or complex). For a background on frame theory we refer the reader to [3, 4, 11, 18] 2. Frame Potentials Recently, Benedetto and Fickus [2] introduced the notion of frame potentials (see also [6]). Definition 2.1. If {ϕ n } M (M finite or infinite) is a frame for H N, the Frame Potential of {ϕ n } M is given by: FP({ϕ n } M ) = M n,m=1 ϕ n, ϕ m 2.
3 CONSTRUCTING INFINITE TIGHT FRAMES 3 The frame potential is measuring how close a frame is to being orthogonal. In particular, it is shown in [6] that if F is the family of frames with lower frame bound λ then the λ-tight frames are the minimizers of the frame potential over F. This theorem gives us a way to identify tight frames. i.e. They are the minimizers of the frame potential on certain families of frames. Our goal is to identify those families of frames which have minimizers of the frame potential and for which these minimizers must be tight. We will need some standard facts about frames (see [6]). For any frame {ϕ n } M for H N with frame bounds A, B, and frame operator S we have A ϕ n 2 ϕ n, ϕ m 2 B ϕ n 2. Hence, It is known that In particular: Also, implies that A m=1 ϕ n 2 FP({ϕ n } M ) B Trace S = ϕ n 2. ϕ n 2. ATrace S FP({ϕ n } M ) BTrace S. ϕ n 4 ϕ n, ϕ m 2 B ϕ n 2, m=1 ϕ n 2 B, for all n. For a tight frame, S = AI so Trace S = NA and FP({ϕ n } M ) = NA 2. So for a Parseval frame (and hence for an orthonormal basis) FP({ϕ n } M ) = N, and this is the minimal value of the frame potential for any frame for H N with lower frame bound 1 (see [6]).
4 4 CASAZZA, FICKUS, LEON, TREMAIN Relating this to frame potentials, since S 2 is the frame operator for {S 1/2 ϕ n } M we have: Trace S 2 = S 1/2 ϕ n 2 = S 1/2 ϕ n, S 1/2 ϕ n = = Sϕ n, ϕ n = m=1 ϕ n, ϕ m ϕ m, ϕ n m=1 ϕ n, ϕ m 2 = FP({ϕ n } M ). Finally, it follows from the frame inequality if {ϕ n } M is a frame for H N with frame bounds A, B > 0 and P is an orthogonal projection on H N, then {Pϕ n } M is a frame for PH N with frame bounds A, B. In particular, if {ϕ n } M is a λ-tight frame for H N then {Pϕ n } M is a λ-tight frame for P(H). There are two main results from [6] we will need in our work. The first is a result which measures how equally distributed a sequence of nonnegative decreasing sequence of numbers are. Proposition 2.2. Given any sequence {a m } m=1 of real numbers with a 1 a 2 a 3, and any natural number N, there is a unique index 1 d N, such that the inequality (N n)a n > m=n+1 a m holds for all 1 n < d, while the opposite inequality (N n)a n ) holds for d n N. m=n+1 Proposition 2.2 is proved in [6] for finite sequences but the proof works without change for infinite sequences as well. We also need the main results from [6]. First a piece of notation. For a > 0 we let S(a) denote the sphere of radius a centered at the origin in H. For any positive sequence {a m } M m=1 we let S(a 1, a 2,, a M ) denote the Cartesian product of the corresponding sequence of spheres: S(a 1, a 2,, a M ) = S(a 1 ) S(a 2 ) S(a M ). Theorem 2.3. Given a sequence a 1 a 2 a M > 0 and any N M, let d denote the smallest index n for which a 2 n M m=n+1 a2 m N n a m
5 CONSTRUCTING INFINITE TIGHT FRAMES 5 holds (cf. Proposition 2.2). Then, any local minimizer of the frame potential FP : S(a 1, a 2,, a M ) R is of the form {f m } M m=1 = {f m} d 1 m=1 {f m} M m=d, where {f m } d 1 m=1 is an orthogonal set for whose orthogonal complement {f m} M m=d forms a tight frame. 3. Finite Dimensional Hilbert Spaces Here we will show that Theorem 2.3 also holds for infinite sets of vectors on finite dimensional spaces. One way to do this would be to systematically show that all of the main results concerning frame potentials holds for infinite frames for finite dimensional spaces. However, this would be exceptionally cumbersome. So instead, we will derive this case from Theorem 2.3. We first need a lemma. Lemma 3.1. Let {ϕ n } M be a frame for H N with frame bounds A, B and let {ψ n } L, L M satisfy: L and ψ n ϕ n 2 ǫ, n=l+1 ϕ n 2 δ. Then {ψ n } L is a frame for H N with frame bounds ( A ǫ δ) 2 and ( B + ǫ) 2. Proof. For all ϕ H N we have: L ϕ, ψ n 2 = L ϕ, ϕ n + ϕ, ψ n ϕ n 2 L ϕ, ϕ n 2 + L ϕ, ψ n ϕ n 2 B ϕ + ϕ 2 ϕ ( B + ǫ). L ψ n ϕ n 2
6 6 CASAZZA, FICKUS, LEON, TREMAIN Similarly, L ϕ, ϕ n 2 L ϕ, ϕ n 2 L ϕ, ψ n ϕ n 2 M ϕ, ϕ n 2 ϕ, ϕ n 2 ǫ ϕ n=l+1 M ϕ, ϕ n 2 M n=l+1 ( A ǫ) ϕ ϕ M ( A ǫ δ) ϕ. ϕ, ϕ n 2 ǫ ϕ n=l+1 ϕ n 2 Now we are ready for the main results of this section. Theorem 3.2. Let a 1 a 2 > 0 be real numbers with a2 n <. Suppose there exists a 1 d < N satisfying Proposition 2.2. Then the frames {ϕ n } which are the closest to being tight (in the sense of minimizing potential energy) with ϕ n = a n are of the form: {c n a n e n } d {ϕ n } n=d+1 where {e i } N i=1 is an orthonormal basis for H N, c i = 1, for all 1 i d, and {ϕ n } n=d+1 is a tight frame for span {e i} N i=d+1. Proof. Fix ǫ > 0, δ > 0 and let {e i } N i=1 be an orthonormal basis for H N. We have assumed that n=d+1 a2 n > Na 2 d. Hence, there is a natural number M 0 > 0 so that for all M M 0 we have n=d+1 a 2 n ()a2 d. Now by Theorem 2.3 for every M M 0 there is a λ M -tight frame {ϕ M n } M for H N of minimum frame potential with respect to the property ϕ M n = a n, for all 1 n M. We may assume our frame has the form: {a n e n } d {ϕ M n } M n=d+1,
7 CONSTRUCTING INFINITE TIGHT FRAMES 7 where {ϕ M n }M n=d+1 is a λ M-tight frame for H = span {e i } M i=d+1 and M n=d+1 λ M = a2 n. By a standard compactness arguement, there are natural numbers M 0 < M 1 < M 2 < so that lim i ϕm i n = ϕ n. First we will show that {ϕ n } n=d+1 is a tight frame for H N d There is a natural number L 0 so that for all L L 0, and all large i we have: and M i n=l+1 L ϕ M i n 2 a 2 n δ n=l ϕ M i n ϕ n 2 ǫ. By Lemma 3.1, {ϕ n } L n=d+1 is a frame for H N d with frame bounds, Mi n=d+1 a2 n ǫ δ, Mi n=d+1 a2 n + ǫ. Letting L yields that {ϕ n } is a frame for H N d with frame bounds n=d+1 a2 n ǫ δ, n=d+1 a2 n + ǫ. Since ǫ, δ > 0 were arbitrary, we have that {ϕ n } is a λ-tight frame for H N with tight frame bound n=d+1 λ = a2 n Next we check that {ϕ n } is of minimal frame potential for those frames {ψ n } for H N with ψ n = a n, for all n = 1, 2,. We proceed by way of contradiction. So assume there is a frame {ψ n } for H N satisfying the above but FP({ψ n } ) FP({ϕ n} ) ǫ. Choose a natural number M so that 2 a 2 m + a 2 n m=m+1 n,m=m+1 a 2 n a2 m < ǫ 4.
8 8 CASAZZA, FICKUS, LEON, TREMAIN Choose M i > M so that Now, FP({ϕ n } M i ) M n,m=1 FP({ψ n } M i ) FP({ψ n} ) FP({ϕ n } ) ǫ ϕ M i n, ϕm i m 2 = FP({ϕ n } M ) + 2 M FP({ϕ n } M ) + 2 m=m+1 n,m=1 a 2 n m=m+1 FP(ϕ n } M i ) + ǫ 4 + ǫ 4 ǫ < FP({ϕ n } M i ). ϕ n, ϕ m 2 ǫ 4. ϕ n, ϕ m 2 + a 2 m + n,m=m+1 n,m=m+1 a 2 na 2 m ϕ n, ϕ m 2 ǫ Since ψ n = ϕ n = a n, for all 1 n M i, we have contradicted the minimality of the frame potential for {ϕ n } M i. Theorem 3.2 yields a classification of infinite tight frames. Theorem 3.3. Given a 1 a 2 > 0 and a Hilbert space H N, the following are equivalent: (1) There is a normalized tight frame {ϕ n } for H N with ϕ n = a n, for all n = 1, 2, 3,. (2) We have that a2 n < and for all 1 d < N, a 2 d n=d+1 a2 n. Proof. (2) (1): Choose a 0 > 0 so that Na 0 > a 2 n. By Theorem 3.3, the frame {ϕ n } n=0 satisfying ϕ n = a n for all n = 0, 1, and which is the closest to being tight for H N+1 is of the form: n=0 {ϕ n } n=0 = {ϕ 0 } {ϕ n }, where {ϕ n } is a tight frame for H N. (1) (2): Now we want to show that whenever {ϕ n } is a tight frame for H N with ϕ n = a n and a 1 a 2 > 0, then {a n } satisfies (1). Fix 1 d < N and choose d k smallest so that dim(span {ϕ n } k ) = d.
9 CONSTRUCTING INFINITE TIGHT FRAMES 9 Let P be the orthogonal projection of H N onto [span {ϕ n } k ]. Then both {Pϕ n } and {(I P)ϕ n} are λ-tight frames for their spans. Hence, Pϕ n 2 = λ = (I P)ϕ n 2. d Since (I P)ϕ n = ϕ n for all 1 n k and Pϕ n = 0 for all 1 n k we have, n=d+1 a2 n n=k+1 a2 n n=k+1 Pϕ n 2 = Pϕ n 2 = λ M = (I P)ϕ n 2 d d ϕ n 2 da2 d d d = a2 d. 4. Infinite Dimensional Hilbert Spaces In this section, we will classify the sequence of norms of tight frame vectors for infinite dimensional Hilbert spaces. For our construction we will need a result of Casazza and Leon [10]. Theorem 4.1. Let S be a positive, self-adjoint, invertable operator on H N with an orthonormal basis of eigenvectors {e i } N i=1 and respective eigenvalues λ 1 λ 2 λ N. Fix M N and {a n } M with a 1 a 2 a m. The following are equivalent: (1) There is a frame {ψ n } M for H N with ψ n = a n, for 1 n M having frame operator S with eigenvectors {e n } N and respective eigenvalues {λ n } M. (2) We have k k a 2 n λ n, for all 1 k N. and N a 2 n = λ n.
10 10 CASAZZA, FICKUS, LEON, TREMAIN To simplify our construction, we will single out the main point in the next lemma. Lemma 4.2. Let {e n } be an orthonormal basis for a Hilbert space H, a n > 0 such that sup 1 n< a 2 n A and a2 n =. Fix ǫ > 0, k N, N k N and assume {ϕ n } M k is a λ k -tight frame for H Nk = span 1 n Nk e n with ϕ n = a n and λ k A + k ǫ. Then there exists natural numbers N 2 n k+1 > N k, M k+1 > M k and vectors {ϕ n } M k+1 n=m K +1 with ϕ n = a n and {ϕ n } M k+1 is a λ k+1 - tight frame for H Nk+1 = span 1 n Nk+1 e n with λ k < λ k+1 A + k+1 Proof: Choose N k+1 so that A N k+1 ǫ 2 k+2 and ( A + ǫ ) ( 1 N ) k > A. 2 N k+1 Now choose M k+1 smallest so that We now have k+1 λ k N k + a 2 n > λ k N k + λ k N k + Now choose λ k+1 so that ( λ k + ǫ ) (N 2 k+2 k+1 N k ). ( λ k + ǫ ) M k+1 (N 2 k+2 k+1 N k ) a 2 n ( λ k + ǫ ) (N 2 k+2 k+1 N k ) + a 2 M k+1. λ k+1 N k+1 = M k+1 Now we compute, ( λ k N k + λ k + ǫ ) (N 2 k+2 k+1 N k ) = λ k N k+1 + ǫ 2 (N k+2 k+1 N k ) λ k+1 N k+1. Hence, Also, λ k < λ k + λ k+1 N k+1 λ k N k+1 + a 2 n. ǫ N k+1 2 k+2 (N k+1 N k ) λ k+1. ǫ 2 k+2 (N k+1 N k ) + a 2 M k+1. ǫ. 2 n
11 So, CONSTRUCTING INFINITE TIGHT FRAMES 11 λ k+1 λ k + ǫ 2 k+2 N k+1 N k N k+1 + a2 M k+1 N k+1 A+ k ǫ 2 + ǫ n 2 + ǫ k+2 2 = A+ k+1 ǫ k+2 2 n. We finish by choosing a frame {ϕ n } M k+1 n=m k +1 with ϕ n = a n, for all M k +1 n M k+1 whose frame operator is M k S(ϕ) = (λ k+1 λ k ) ϕ, e k e k + k+1 n=m k +1 λ k ϕ, e k e k. To see that such a frame exists, we must verify that we have the hypotheses of Theorem 4.1. But first, let us observe that this finishes the proof of the lemma since the frame operator S k for {ϕ n } M k satisfies M k S k (ϕ) = λ k ϕ, e k e k, and so the frame operator S k+1 for {ϕ n } M k+1 is S k+1 (ϕ) = (S k + S)(ϕ) M k M k = λ k ϕ, e k e k + (λ k+1 λ k ) ϕ, e k e k + = M k+1 λ k+1 ϕ, e k e k. M k+1 n=m k +1 λ k ϕ, e k e k That is, {ϕ n } M k+1 is a λ k+1 -tight frame. To check the hypotheses of Theorem 4.1, note that by definition we have So, λ k+1 N k+1 = k+1 M k+1 M k a 2 n = M k+1 a 2 n + n=m k +1 a 2 n = λ kn k + M k+1 n=m k +1 a 2 n = λ k+1n k+1 λ k N k = λ k+1 (N k+1 N k ) + (λ k+1 λ k ) N k. The right hand side of this equality is the sum of the eigenvalues for the frame operator S. Also, for 1 m N k we have N k+1 n=m a 2 n AN k+1 ( A + ǫ ) ( 1 N ) k N k+1 2 N k+1 a 2 n.
12 12 CASAZZA, FICKUS, LEON, TREMAIN ( = A + ǫ ) (N k+1 N k ) λ k+1 (N k+1 N k ). 2 Also, for N k + 1 m N k+1 we have N k+1 n=m a 2 n (N k+1 m)a (N k+1 m)λ k+1. This shows that the hypotheses of Theorem 4.1 are satisfies. Theorem 4.3. Let H be an infinite dimnsional Hilbert space and a n > 0, for all n = 1, 2, 3,. The following are equivalent: (1) There is a frame {ψ n } for H with ψ n = a n, for all n = 1, 2, 3,. (2) For every ǫ > 0, there is a λ-tight frame {ϕ n } for H with ϕ n = a n, for all n = 1, 2, 3, and sup 1 n< a 2 n λ sup 1 n< a 2 n + ǫ. (3) The sequence {a n } is bounded and a2 n =. Moreover, if {ψ n } is any λ 1-tight frame for H with ψ n = a n, for all n = 1, 2, 3,, then λ 1 sup 1 n< a 2 n. Finally, in general we cannot find a λ-tight frame for H with ϕ n = a n, for all n = 1, 2, 3, and satisfying λ = sup 1 n< a 2 n. Proof: (2) (1): Obvious. (1) (3): For any frame {ϕ n } with upper frame bound B we have that ϕ n B for all n = 1, 2,. Hence, {a n } is bounded. Assume {ϕ n} is any frame for H with frame bounds A, B. We proceed by way of contradiction. If ϕ n 2 <, then choose M so that ϕ n 2 < A. n=m Let P be the orthogonal projection onto span {ϕ n } M 1. Since H is infinite dimensional, Then {(I P)ϕ n } is a frame for its span with frame bounds A, B. But (I P)ϕ n = 0 for 1 n M. Hence, (I P)ϕ n 2 ϕ n 2 A. So the Bessel bound of the frame {(I P)ϕ n } is less than A and so the upper frame bound (and hence the lower frame bound) is less than A, which is a contradiction. (3) (2): Fix ǫ > 0. By induction on Lemma 4.2, there are sequences of natural numbers N 1 < N 2 < N 3 < and M 1 < M 2 < M 3 < and vectors ϕ n H for n = 1, 2, 3, satisfying: (1) ϕ n = a n, for all n = 1, 2, 3,. (2) {ϕ n } M k is a λ k-tight frame for H Nk with λ λ k λ + k ǫ and 2 n λ 1 λ 2 λ 3. n=m
13 It follows that CONSTRUCTING INFINITE TIGHT FRAMES 13 lim λ k = B λ + k Also, if we fix ϕ = M k a ne n, then M k ϕ, ϕ n 2 = lim k ǫ 2 n = λ + ǫ. ϕ, ϕ n 2 = lim k λ k ϕ 2 = B ϕ 2. Since this equality holds on a dense subspace of H, it follows that it holds on H. That is, for all ϕ H, ϕ, ϕ n 2 = B ϕ 2. So {ϕ n } is a B-tight frame for H with λ B λ + ǫ. For the moreover part of the theorem, we observe that if {ψ n } is a λ 1-tight frame for H, then ψ n 2 λ 1, for all n = 1, 2, 3,. Hence, λ 1 sup 1 n< a 2 n. Finally, (and this simple example was communicated to us by D.R. Larson) assume a 1 = 1 2 and a n = 1 for all n 2. Suppose, by way of contradiction, there is a Parseval frame {ψ n } for H with ψ n = a n for all n = 1, 2, 3,. Since ψ n = 1, for all n 2 it follows that ψ n span k n ψ k, for all n 2. Hence, ψ 1 span 2 k ψ k, and so {ψ n } has an optimal lower frame bound of and optimal upper frame bound of 1, and so is not a tight frame. 1 2 References [1] N.I. Akhiezer and I.M. Glazman. Theory of Linear Operators in Hilbert spaces, Volume 1. Frederick Ungar Publisher, [2] J.J. Benedetto and M. Fickus. Finite normalized tight frames. Advances in Computational Math, vol. 18 Nos. 2-4 (2003) [3] P.G. Casazza. Modern tools for Weyl-Heisenberg (Gabor) frame theory. Adv. in Imag. and Electron Physics, Vol. 115 (2000) [4] P.G. Casazza. the art of frame theory. Taiwanese Journ. of Math. (4)2 (2000) [5] P.G. Casazza. Custom building finite frames. Preprint. [6] P.G. Casazza, M. Fickus, J. Kovačević, M. Leon and J.C. Tremain, A physical interpretation for finite frames, Preprint. [7] P.G. Casazza, M. Fickus, J. Kovačević, M. Leon and J.C. Tremain, Representations of frames. Preprint. [8] P.G. Casazza and J. Kovačević. Equal norm tight frames with erasures. Advances in Computational Math. vol. 18 Nos. 2-4 (2003) [9] P.G. Casazza and M. Leon. Existence and construction of finite tight frames. Preprint. [10] P.G. Casazza and M. Leon. Frames with a given frame operator. Preprint. [11] O. Christensen, An introduction to frames and Riesz bases. Birkh auser, [12] R.J. Duffin and A.C. Schaeffer. A class of nonharmonic Fourier series. Trans. Amer. Math. Soc., Vol 72 (1952)
14 14 CASAZZA, FICKUS, LEON, TREMAIN [13] Y. Eldar and G.D. Forney, Jr. Optimal tight frames and quantum measurement. Preprint. [14] H. G. Feichtinger and T. Strohmer, eds. Gabor Analysis and Algorithms - Theory and Applications. Birkh user, Boston (1998). [15] V.K. Goyal and J. Kovacević. Optimal multiple description transform coding of Gaussian vectors. In Proc. Data Compr. Conf. Snowbird, UT, March 1998, [16] V.K. Goyal, J. Kovacević, and J.A. Kelner. Quantized frame expansions with erasures. Journal of Appl. and Comput. Harmonic Analysis, 10 (3), (2001) [17] V.K. Goyal, J. Kovacević, and M. Vetterli. Quantized frame expansions as courcechannel codes for erasure channels. In Proc. Data Compr. Conf., Snowbird, UT (1999) [18] K. Gr ochenig. Foundations of time-frequency Analysis. Birkh auser, Boston (2001). [19] D. Han and D.R. Larson. Frames, bases and group representations. Memoirs AMS, Providence, RI, [20] B. Hassibi, B. Hochwald, A. Shkrollahi, and W. Sweldens. Representation theory for high-rate multiple-antenna code design. IEEE trans. Inform. Th. 47 (6) (2001) [21] C.E. Heil and D. Walnut. Continuous and discrete wavelet transforms. SIAM Rev. 31 (4) (1989) [22] B. Hochwald, T. Marzetta, T. Richardson, W. Sweldens, and R. Urbanke. Systematic design of unitary space-time constellations. Preprint. [23] T. Strohmer. Approximation of dual Gabor frames, window decay, and wireless communications. Appl. Comp. Harm. Anal., 11 (2) (2001) Casazza, Leon, Tremain, Department of Mathematics, University of Missouri- Columbia, Columbia, MO 65211, Fickus, Department of Mathematics, Cornell University, Ithaca, NY address: pete,mleon,janet@math.missouri.edu, fickus@polygon.math.cornell.edu
2 PETER G. CASAZZA, MANUEL T. LEON In this paper we generalize these results to the case where I is replaced by any positive selfadjoint invertible op
FRAMES WITH A GIVEN FRAME OPERATOR PETER G. CASAZZA, MANUEL T. LEON Abstract. Let S be a positive self-adjoint invertible operator on an N-dimensional Hilbert space H N and let M N. We give necessary and
More informationUniversity of Missouri Columbia, MO USA
EXISTENCE AND CONSTRUCTION OF FINITE FRAMES WITH A GIVEN FRAME OPERATOR PETER G. CASAZZA 1 AND MANUEL T. LEON 2 1 Department of Mathematics University of Missouri Columbia, MO 65211 USA e-mail: casazzap@missouri.edu
More informationA FRAME THEORY PRIMER FOR THE KADISON-SINGER PROBLEM
A FRAME THEORY PRIMER FOR THE KADISON-SINGER PROBLEM PETER G. CASAZZA Abstract. This is a primer on frame theory geared towards the parts of the theory needed for people who want to understand the relationship
More informationA NEW IDENTITY FOR PARSEVAL FRAMES
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000 000 S 0002-9939(XX)0000-0 A NEW IDENTITY FOR PARSEVAL FRAMES RADU BALAN, PETER G. CASAZZA, DAN EDIDIN, AND GITTA KUTYNIOK
More informationA DECOMPOSITION THEOREM FOR FRAMES AND THE FEICHTINGER CONJECTURE
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000 000 S 0002-9939(XX)0000-0 A DECOMPOSITION THEOREM FOR FRAMES AND THE FEICHTINGER CONJECTURE PETER G. CASAZZA, GITTA KUTYNIOK,
More informationDecompositions of frames and a new frame identity
Decompositions of frames and a new frame identity Radu Balan a, Peter G. Casazza b, Dan Edidin c and Gitta Kutyniok d a Siemens Corporate Research, 755 College Road East, Princeton, NJ 08540, USA; b Department
More informationA BRIEF INTRODUCTION TO HILBERT SPACE FRAME THEORY AND ITS APPLICATIONS AMS SHORT COURSE: JOINT MATHEMATICS MEETINGS SAN ANTONIO, 2015 PETER G. CASAZZA Abstract. This is a short introduction to Hilbert
More informationarxiv:math/ v1 [math.fa] 14 Sep 2003
arxiv:math/0309236v [math.fa] 4 Sep 2003 RANK-ONE DECOMPOSITION OF OPERATORS AND CONSTRUCTION OF FRAMES KERI A. KORNELSON AND DAVID R. LARSON Abstract. The construction of frames for a Hilbert space H
More informationbuer overlfows at intermediate nodes in the network. So to most users, the behavior of a packet network is not characterized by random loss, but by un
Uniform tight frames for signal processing and communication Peter G. Casazza Department of Mathematics University of Missouri-Columbia Columbia, MO 65211 pete@math.missouri.edu Jelena Kovacevic Bell Labs
More informationReal Equiangular Frames
Peter G Casazza Department of Mathematics The University of Missouri Columbia Missouri 65 400 Email: pete@mathmissouriedu Real Equiangular Frames (Invited Paper) Dan Redmond Department of Mathematics The
More informationMULTIPLEXING AND DEMULTIPLEXING FRAME PAIRS
MULTIPLEXING AND DEMULTIPLEXING FRAME PAIRS AZITA MAYELI AND MOHAMMAD RAZANI Abstract. Based on multiplexing and demultiplexing techniques in telecommunication, we study the cases when a sequence of several
More informationLinear Algebra and its Applications
Linear Algebra and its Applications 434 (011) 1893 1901 Contents lists available at ScienceDirect Linear Algebra and its Applications journal homepage: www.elsevier.com/locate/laa Robustness and surgery
More informationApproximately dual frames in Hilbert spaces and applications to Gabor frames
Approximately dual frames in Hilbert spaces and applications to Gabor frames Ole Christensen and Richard S. Laugesen October 22, 200 Abstract Approximately dual frames are studied in the Hilbert space
More informationOn Riesz-Fischer sequences and lower frame bounds
On Riesz-Fischer sequences and lower frame bounds P. Casazza, O. Christensen, S. Li, A. Lindner Abstract We investigate the consequences of the lower frame condition and the lower Riesz basis condition
More informationWEYL-HEISENBERG FRAMES FOR SUBSPACES OF L 2 (R)
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 129, Numer 1, Pages 145 154 S 0002-9939(00)05731-2 Article electronically pulished on July 27, 2000 WEYL-HEISENBERG FRAMES FOR SUBSPACES OF L 2 (R)
More informationDensity results for frames of exponentials
Density results for frames of exponentials P. G. Casazza 1, O. Christensen 2, S. Li 3, and A. Lindner 4 1 Department of Mathematics, University of Missouri Columbia, Mo 65211 USA pete@math.missouri.edu
More informationBANACH FRAMES GENERATED BY COMPACT OPERATORS ASSOCIATED WITH A BOUNDARY VALUE PROBLEM
TWMS J. Pure Appl. Math., V.6, N.2, 205, pp.254-258 BRIEF PAPER BANACH FRAMES GENERATED BY COMPACT OPERATORS ASSOCIATED WITH A BOUNDARY VALUE PROBLEM L.K. VASHISHT Abstract. In this paper we give a type
More informationReal, Tight Frames with Maximal Robustness to Erasures
Real, Tight Frames with Maximal Robustness to Erasures Markus Püschel 1 and Jelena Kovačević 2,1 Departments of 1 ECE and 2 BME Carnegie Mellon University Pittsburgh, PA Email: pueschel@ece.cmu.edu, jelenak@cmu.edu
More informationSpanning and Independence Properties of Finite Frames
Chapter 1 Spanning and Independence Properties of Finite Frames Peter G. Casazza and Darrin Speegle Abstract The fundamental notion of frame theory is redundancy. It is this property which makes frames
More informationg-frame Sequence Operators, cg-riesz Bases and Sum of cg-frames
International Mathematical Forum, Vol. 6, 2011, no. 68, 3357-3369 g-frame Sequence Operators, cg-riesz Bases and Sum of cg-frames M. Madadian Department of Mathematics, Tabriz Branch, Islamic Azad University,
More informationDecomposition of Riesz frames and wavelets into a finite union of linearly independent sets
Decomposition of Riesz frames and wavelets into a finite union of linearly independent sets Ole Christensen, Alexander M. Lindner Abstract We characterize Riesz frames and prove that every Riesz frame
More informationApproximately dual frame pairs in Hilbert spaces and applications to Gabor frames
arxiv:0811.3588v1 [math.ca] 21 Nov 2008 Approximately dual frame pairs in Hilbert spaces and applications to Gabor frames Ole Christensen and Richard S. Laugesen November 21, 2008 Abstract We discuss the
More informationTHE KADISON-SINGER PROBLEM AND THE UNCERTAINTY PRINCIPLE
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000 000 S 0002-9939(XX)0000-0 THE KADISON-SINGER PROBLEM AND THE UNCERTAINTY PRINCIPLE PETER G. CASAZZA AND ERIC WEBER Abstract.
More informationFrame Diagonalization of Matrices
Frame Diagonalization of Matrices Fumiko Futamura Mathematics and Computer Science Department Southwestern University 00 E University Ave Georgetown, Texas 78626 U.S.A. Phone: + (52) 863-98 Fax: + (52)
More informationC -Algebra B H (I) Consisting of Bessel Sequences in a Hilbert Space
Journal of Mathematical Research with Applications Mar., 2015, Vol. 35, No. 2, pp. 191 199 DOI:10.3770/j.issn:2095-2651.2015.02.009 Http://jmre.dlut.edu.cn C -Algebra B H (I) Consisting of Bessel Sequences
More informationFRAME DUALITY PROPERTIES FOR PROJECTIVE UNITARY REPRESENTATIONS
FRAME DUALITY PROPERTIES FOR PROJECTIVE UNITARY REPRESENTATIONS DEGUANG HAN AND DAVID LARSON Abstract. Let π be a projective unitary representation of a countable group G on a separable Hilbert space H.
More informationj jf, S K cf = j K c j jf, f H.
DOI 10.1186/s40064-016-2731-2 RESEARCH New double inequalities for g frames in Hilbert C modules Open Access Zhong Qi Xiang * *Correspondence: lxsy20110927@163.com College of Mathematics and Computer Science,
More informationarxiv:math/ v1 [math.fa] 5 Aug 2005
arxiv:math/0508104v1 [math.fa] 5 Aug 2005 G-frames and G-Riesz Bases Wenchang Sun Department of Mathematics and LPMC, Nankai University, Tianjin 300071, China Email: sunwch@nankai.edu.cn June 28, 2005
More informationApplied and Computational Harmonic Analysis
Appl. Comput. Harmon. Anal. 32 (2012) 139 144 Contents lists available at ScienceDirect Applied and Computational Harmonic Analysis www.elsevier.com/locate/acha Letter to the Editor Frames for operators
More informationFilter Bank Frame Expansions With Erasures
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL 48, NO 6, JUNE 2002 1439 Filter Bank Frame Expansions With Erasures Jelena Kovačević, Fellow, IEEE, Pier Luigi Dragotti, Student Member, IEEE, and Vivek K Goyal,
More informationSubsequences of frames
Subsequences of frames R. Vershynin February 13, 1999 Abstract Every frame in Hilbert space contains a subsequence equivalent to an orthogonal basis. If a frame is n-dimensional then this subsequence has
More informationDuals of g-frames and g-frame Sequences
International Mathematical Forum, Vol. 8, 2013, no. 7, 301-310 Duals of g-frames and g-frame Sequences Mostafa Madadian Department of Mathematics, Tabriz Branch Islamic Azad University, Tabriz, Iran madadian@iaut.ac.ir
More informationOperators with Closed Range, Pseudo-Inverses, and Perturbation of Frames for a Subspace
Canad. Math. Bull. Vol. 42 (1), 1999 pp. 37 45 Operators with Closed Range, Pseudo-Inverses, and Perturbation of Frames for a Subspace Ole Christensen Abstract. Recent work of Ding and Huang shows that
More informationOn the Equality of Fusion Frames 1
International Mathematical Forum, 4, 2009, no. 22, 1059-1066 On the Equality of Fusion Frames 1 Yao Xiyan 2, Gao Guibao and Mai Ali Dept. of Appl. Math., Yuncheng University Shanxi 044000, P. R. China
More informationAtomic decompositions of square-integrable functions
Atomic decompositions of square-integrable functions Jordy van Velthoven Abstract This report serves as a survey for the discrete expansion of square-integrable functions of one real variable on an interval
More informationA short introduction to frames, Gabor systems, and wavelet systems
Downloaded from orbit.dtu.dk on: Mar 04, 2018 A short introduction to frames, Gabor systems, and wavelet systems Christensen, Ole Published in: Azerbaijan Journal of Mathematics Publication date: 2014
More informationShift Invariant Spaces and Shift Generated Dual Frames for Local Fields
Communications in Mathematics and Applications Volume 3 (2012), Number 3, pp. 205 214 RGN Publications http://www.rgnpublications.com Shift Invariant Spaces and Shift Generated Dual Frames for Local Fields
More informationG-frames in Hilbert Modules Over Pro-C*-algebras
Available online at http://ijim.srbiau.ac.ir/ Int. J. Industrial Mathematics (ISSN 2008-5621) Vol. 9, No. 4, 2017 Article ID IJIM-00744, 9 pages Research Article G-frames in Hilbert Modules Over Pro-C*-algebras
More informationFrame expansions in separable Banach spaces
Frame expansions in separable Banach spaces Pete Casazza Ole Christensen Diana T. Stoeva December 9, 2008 Abstract Banach frames are defined by straightforward generalization of (Hilbert space) frames.
More informationSPECTRAL THEOREM FOR COMPACT SELF-ADJOINT OPERATORS
SPECTRAL THEOREM FOR COMPACT SELF-ADJOINT OPERATORS G. RAMESH Contents Introduction 1 1. Bounded Operators 1 1.3. Examples 3 2. Compact Operators 5 2.1. Properties 6 3. The Spectral Theorem 9 3.3. Self-adjoint
More informationCONSTRUCTIVE PROOF OF THE CARPENTER S THEOREM
CONSTRUCTIVE PROOF OF THE CARPENTER S THEOREM MARCIN BOWNIK AND JOHN JASPER Abstract. We give a constructive proof of Carpenter s Theorem due to Kadison [14, 15]. Unlike the original proof our approach
More informationOle Christensen 3. October 20, Abstract. We point out some connections between the existing theories for
Frames and pseudo-inverses. Ole Christensen 3 October 20, 1994 Abstract We point out some connections between the existing theories for frames and pseudo-inverses. In particular, using the pseudo-inverse
More informationConstructive Proof of the Carpenter s Theorem
Canad. Math. Bull. Vol. 57 (3), 2014 pp. 463 476 http://dx.doi.org/10.4153/cmb-2013-037-x c Canadian Mathematical Society 2013 Constructive Proof of the Carpenter s Theorem Marcin Bownik and John Jasper
More informationGABOR FRAMES AND OPERATOR ALGEBRAS
GABOR FRAMES AND OPERATOR ALGEBRAS J-P Gabardo a, Deguang Han a, David R Larson b a Dept of Math & Statistics, McMaster University, Hamilton, Canada b Dept of Mathematics, Texas A&M University, College
More informationLinear Algebra and its Applications
Linear Algebra and its Applications 439 (2013) 1330 1339 Contents lists available at SciVerse ScienceDirect Linear Algebra and its Applications journal homepage: www.elsevier.com/locate/laa Maximum robustness
More informationA primer on the theory of frames
A primer on the theory of frames Jordy van Velthoven Abstract This report aims to give an overview of frame theory in order to gain insight in the use of the frame framework as a unifying layer in the
More informationDensity, Overcompleteness, and Localization of Frames. I. Theory
The Journal of Fourier Analysis and Applications Volume 2, Issue 2, 2006 Density, Overcompleteness, and Localization of Frames. I. Theory Radu Balan, Peter G. Casazza, Christopher Heil, and Zeph Landau
More informationAPPROXIMATING THE INVERSE FRAME OPERATOR FROM LOCALIZED FRAMES
APPROXIMATING THE INVERSE FRAME OPERATOR FROM LOCALIZED FRAMES GUOHUI SONG AND ANNE GELB Abstract. This investigation seeks to establish the practicality of numerical frame approximations. Specifically,
More informationIntroduction to Hilbert Space Frames
to Hilbert Space Frames May 15, 2009 to Hilbert Space Frames What is a frame? Motivation Coefficient Representations The Frame Condition Bases A linearly dependent frame An infinite dimensional frame Reconstructing
More informationOn the Feichtinger conjecture
Electronic Journal of Linear Algebra Volume 26 Volume 26 (2013) Article 35 2013 On the Feichtinger conjecture Pasc Gavruta pgavruta@yahoo.com Follow this and additional works at: http://repository.uwyo.edu/ela
More informationRobustness of Fusion Frames under Erasures of Subspaces and of Local Frame Vectors
Contemporary Mathematics Robustness of Fusion Frames under Erasures of Subspaces and of Local Frame Vectors Peter G. Casazza and Gitta Kutyniok Abstract. Fusion frames were recently introduced to model
More informationSPECTRA OF FRAME OPERATORS WITH PRESCRIBED FRAME NORMS
SPECTRA OF FRAME OPERATORS WITH PRESCRIBED FRAME NORMS MARCIN BOWNIK AND JOHN JASPER Abstract. We study the set of possible finite spectra of self-adjoint operators with fixed diagonal. In the language
More informationTHE KADISON-SINGER PROBLEM IN MATHEMATICS AND ENGINEERING
THE KADISON-SINGER PROBLEM IN MATHEMATICS AND ENGINEERING PETER G. CASAZZA AND JANET CRANDELL TREMAIN Abstract. We will see that the famous intractible 1959 Kadison-Singer Problem in C -algebras is equivalent
More informationMinimizing Fusion Frame Potential
manuscript No. (will be inserted by the editor) Minimizing Fusion Frame Potential Peter G. Casazza 1, Matthew Fickus 2 1 Department of Mathematics, University of Missouri, Columbia, Missouri 65211, e-mail:
More informationNOTES ON FRAMES. Damir Bakić University of Zagreb. June 6, 2017
NOTES ON FRAMES Damir Bakić University of Zagreb June 6, 017 Contents 1 Unconditional convergence, Riesz bases, and Bessel sequences 1 1.1 Unconditional convergence of series in Banach spaces...............
More informationPERTURBATION OF FRAMES FOR A SUBSPACE OF A HILBERT SPACE
ROCKY MOUNTIN JOURNL OF MTHEMTICS Volume 30, Number 4, Winter 2000 PERTURBTION OF FRMES FOR SUBSPCE OF HILBERT SPCE OLE CHRISTENSEN, CHRIS LENNRD ND CHRISTINE LEWIS BSTRCT. frame sequence {f i } i=1 in
More informationarxiv: v1 [math.oa] 2 Mar 2014
FRAMES AND OPERATORS IN HILBERT C -MODULES arxiv:403.0205v [math.oa] 2 Mar 204 ABBAS NAJATI, M. MOHAMMADI SAEM AND AND P. GĂVRUŢA Abstract. In this paper we introduce the concepts of atomic systems for
More informationAffine and Quasi-Affine Frames on Positive Half Line
Journal of Mathematical Extension Vol. 10, No. 3, (2016), 47-61 ISSN: 1735-8299 URL: http://www.ijmex.com Affine and Quasi-Affine Frames on Positive Half Line Abdullah Zakir Husain Delhi College-Delhi
More informationOptimal dual fusion frames for probabilistic erasures
Electronic Journal of Linear Algebra Volume 32 Volume 32 (2017) Article 16 2017 Optimal dual fusion frames for probabilistic erasures Patricia Mariela Morillas Universidad Nacional de San Luis and CONICET,
More informationAnalysis Preliminary Exam Workshop: Hilbert Spaces
Analysis Preliminary Exam Workshop: Hilbert Spaces 1. Hilbert spaces A Hilbert space H is a complete real or complex inner product space. Consider complex Hilbert spaces for definiteness. If (, ) : H H
More informationDENSITY, OVERCOMPLETENESS, AND LOCALIZATION OF FRAMES. I. THEORY
DENSITY, OVERCOMPLETENESS, AND LOCALIZATION OF FRAMES. I. THEORY RADU BALAN, PETER G. CASAZZA, CHRISTOPHER HEIL, AND ZEPH LANDAU Abstract. This work presents a quantitative framework for describing the
More informationFRAMES AND TIME-FREQUENCY ANALYSIS
FRAMES AND TIME-FREQUENCY ANALYSIS LECTURE 5: MODULATION SPACES AND APPLICATIONS Christopher Heil Georgia Tech heil@math.gatech.edu http://www.math.gatech.edu/ heil READING For background on Banach spaces,
More informationMORE ON SUMS OF HILBERT SPACE FRAMES
Bull. Korean Math. Soc. 50 (2013), No. 6, pp. 1841 1846 http://dx.doi.org/10.4134/bkms.2013.50.6.1841 MORE ON SUMS OF HILBERT SPACE FRAMES A. Najati, M. R. Abdollahpour, E. Osgooei, and M. M. Saem Abstract.
More informationOn Frame Wavelet Sets and Some Related Topics
On Frame Wavelet Sets and Some Related Topics Xingde Dai and Yuanan Diao Abstract. A special type of frame wavelets in L 2 (R) or L 2 (R d ) consists of those whose Fourier transforms are defined by set
More informationPreconditioning of Frames
Preconditioning of Frames Gitta Kutyniok a, Kasso A. Okoudjou b, and Friedrich Philipp a a Technische Universität Berlin, Institut für Mathematik, Strasse des 17. Juni 136, 10623 Berlin, Germany b University
More informationLOSS-INSENSITIVE VECTOR ENCODING WITH TWO-UNIFORM FRAMES
LOSS-INSENSITIVE VECTOR ENCODING WITH TWO-UNIFORM FRAMES BERNHARD G. BODMANN AND VERN I. PAULSEN Abstract. The central topic of this paper is the linear, redundant encoding of vectors using frames for
More informationThe Kadison-Singer and Paulsen Problems in Finite Frame Theory
Chapter 1 The Kadison-Singer and Paulsen Problems in Finite Frame Theory Peter G. Casazza Abstract We now know that some of the basic open problems in frame theory are equivalent to fundamental open problems
More informationInequalities in Hilbert Spaces
Inequalities in Hilbert Spaces Jan Wigestrand Master of Science in Mathematics Submission date: March 8 Supervisor: Eugenia Malinnikova, MATH Norwegian University of Science and Technology Department of
More informationWaveform design and quantum detection matched filtering
Waveform design and quantum detection matched filtering John J. Benedetto Norbert Wiener Center, Department of Mathematics University of Maryland, College Park http://www.norbertwiener.umd.edu Waveform
More informationLinear Independence of Finite Gabor Systems
Linear Independence of Finite Gabor Systems 1 Linear Independence of Finite Gabor Systems School of Mathematics Korea Institute for Advanced Study Linear Independence of Finite Gabor Systems 2 Short trip
More informationOn Some Properties of Generalized Fock Space F 2 (d v α ) by Frame Theory on the C n
Communications in Mathematics and Applications Volume 1, Number (010), pp. 105 111 RGN Publications http://www.rgnpublications.com On Some Properties of Generalized Fock Space F (d v α ) by Frame Theory
More informationLOCAL AND GLOBAL STABILITY OF FUSION FRAMES
LOCAL AND GLOBAL STABILITY OF FUSION FRAMES Jerry Emidih Norbert Wiener Center Department of Mathematics University of Maryland, College Park November 22 2016 OUTLINE 1 INTRO 2 3 4 5 OUTLINE 1 INTRO 2
More informationDUALITY PRINCIPLE IN g-frames
Palestine Journal of Mathematics Vol. 6(2)(2017), 403 411 Palestine Polytechnic University-PPU 2017 DUAITY PRINCIPE IN g-frames Amir Khosravi and Farkhondeh Takhteh Communicated by Akram Aldroubi MSC 2010
More informationThe Theory of Wavelets with Composite Dilations
The Theory of Wavelets with Composite Dilations Kanghui Guo 1, Demetrio Labate 2, Wang Q Lim 3, Guido Weiss 4, and Edward Wilson 5 1 Department of Mathematics, Southwest Missouri State University, Springfield,
More informationFrames and a vector-valued ambiguity function
Norbert Wiener Center Department of Mathematics University of Maryland, College Park http://www.norbertwiener.umd.edu Outline 1 Problem and goal 2 Frames 3 Multiplication problem and A 1 p 4 A d p : Z
More informationMath 350 Fall 2011 Notes about inner product spaces. In this notes we state and prove some important properties of inner product spaces.
Math 350 Fall 2011 Notes about inner product spaces In this notes we state and prove some important properties of inner product spaces. First, recall the dot product on R n : if x, y R n, say x = (x 1,...,
More informationFast Angular Synchronization for Phase Retrieval via Incomplete Information
Fast Angular Synchronization for Phase Retrieval via Incomplete Information Aditya Viswanathan a and Mark Iwen b a Department of Mathematics, Michigan State University; b Department of Mathematics & Department
More informationMULTIPLIERS OF GENERALIZED FRAMES IN HILBERT SPACES. Communicated by Heydar Radjavi. 1. Introduction
Bulletin of the Iranian Mathematical Society Vol. 37 No. 1 (2011), pp 63-80. MULTIPLIERS OF GENERALIZED FRAMES IN HILBERT SPACES A. RAHIMI Communicated by Heydar Radjavi Abstract. In this paper, we introduce
More informationSemi-orthogonal wavelet frames on positive half-line using the Walsh Fourier transform
NTMSCI 6, No., 175-183 018) 175 New Trends in Mathematical Sciences http://dx.doi.org/10.085/ntmsci.018.83 Semi-orthogonal wavelet frames on positive half-line using the Walsh Fourier transform Abdullah
More informationPROJECTIONS AND THE KADISON-SINGER PROBLEM
PROJECTIONS AND THE KADISON-SINGER PROBLEM PETE CASAZZA, DAN EDIDIN, DEEPTI KALRA, AND VERN I. PAULSEN Abstract. We prove some new equivalences of the paving conjecture and obtain some estimates on the
More informationFrame Wavelet Sets in R d
Frame Wavelet Sets in R d X. DAI, Y. DIAO Department of Mathematics University of North Carolina at Charlotte Charlotte, NC 28223 xdai@uncc.edu Q. GU Department of Mathematics Each China Normal University
More informationStability of alternate dual frames
Stability of alternate dual frames Ali Akbar Arefijamaal Abstrat. The stability of frames under perturbations, whih is important in appliations, is studied by many authors. It is worthwhile to onsider
More informationIntroduction to Bases in Banach Spaces
Introduction to Bases in Banach Spaces Matt Daws June 5, 2005 Abstract We introduce the notion of Schauder bases in Banach spaces, aiming to be able to give a statement of, and make sense of, the Gowers
More informationDual and Similar Frames in Krein Spaces
International Journal of Mathematical Analysis Vol. 10, 2016, no. 19, 939-952 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijma.2016.6469 Dual and Similar Frames in Krein Spaces Kevin Esmeral,
More informationInvariances of Frame Sequences under Perturbations
Invariances of Frame Sequences under Perturbations Shannon Bishop a,1, Christopher Heil b,1,, Yoo Young Koo c,2, Jae Kun Lim d a School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia
More informationarxiv:math.oa/ v1 22 Nov 2000
arxiv:math.oa/0011184 v1 22 Nov 2000 A module frame concept for Hilbert C*-modules Michael Frank and David R. Larson Abstract. The goal of the present paper is a short introduction to a general module
More informationFourier and Wavelet Signal Processing
Ecole Polytechnique Federale de Lausanne (EPFL) Audio-Visual Communications Laboratory (LCAV) Fourier and Wavelet Signal Processing Martin Vetterli Amina Chebira, Ali Hormati Spring 2011 2/25/2011 1 Outline
More informationOperator representations of frames: boundedness, duality, and stability.
arxiv:1704.08918v1 [math.fa] 28 Apr 2017 Operator representations of frames: boundedness, duality, and stability. Ole Christensen, Marzieh Hasannasab May 1, 2017 Abstract The purpose of the paper is to
More informationPAVING AND THE KADISON-SINGER PROBLEM
PAVING AND THE KADISON-SINGER PROBLEM PETE CASAZZA, VERN PAULSEN, AND GARY WEISS Abstract. This is an introduction to problems surrounding the Paving Conjecture.. Paving Parameters and Notation Notation..
More informationFinite-dimensional spaces. C n is the space of n-tuples x = (x 1,..., x n ) of complex numbers. It is a Hilbert space with the inner product
Chapter 4 Hilbert Spaces 4.1 Inner Product Spaces Inner Product Space. A complex vector space E is called an inner product space (or a pre-hilbert space, or a unitary space) if there is a mapping (, )
More informationParseval Frame Construction
LSU, LSU, USA July 6, 2012 1 Introduction 1 Introduction 2 1 Introduction 2 3 1 Introduction 2 3 4 1 Introduction 2 3 4 5 Introduction A vector space, V, is a nonempty set with two operations: addition
More informationContents. 0.1 Notation... 3
Contents 0.1 Notation........................................ 3 1 A Short Course on Frame Theory 4 1.1 Examples of Signal Expansions............................ 4 1.2 Signal Expansions in Finite-Dimensional
More informationOperator Theory and Modulation Spaces
To appear in: Frames and Operator Theory in Analysis and Signal Processing (San Antonio, 2006), Comtemp. Math., Amer. Math. Soc. Operator Theory and Modulation Spaces Christopher Heil and David Larson
More information5 Compact linear operators
5 Compact linear operators One of the most important results of Linear Algebra is that for every selfadjoint linear map A on a finite-dimensional space, there exists a basis consisting of eigenvectors.
More informationUNCERTAINTY PRINCIPLES FOR THE FOCK SPACE
UNCERTAINTY PRINCIPLES FOR THE FOCK SPACE KEHE ZHU ABSTRACT. We prove several versions of the uncertainty principle for the Fock space F 2 in the complex plane. In particular, for any unit vector f in
More information446 SCIENCE IN CHINA (Series F) Vol. 46 introduced in refs. [6, ]. Based on this inequality, we add normalization condition, symmetric conditions and
Vol. 46 No. 6 SCIENCE IN CHINA (Series F) December 003 Construction for a class of smooth wavelet tight frames PENG Lizhong (Λ Π) & WANG Haihui (Ξ ) LMAM, School of Mathematical Sciences, Peking University,
More informationLinear Independence of Gabor Systems in Finite Dimensional Vector Spaces
The Journal of Fourier Analysis and Applications Volume 11, Issue 6, 2005 Linear Independence of Gabor Systems in Finite Dimensional Vector Spaces Jim Lawrence, Götz E. Pfander, and David Walnut Communicated
More informationUniversity of Missouri. In Partial Fulllment LINDSEY M. WOODLAND MAY 2015
Frames and applications: Distribution of frame coecients, integer frames and phase retrieval A Dissertation presented to the Faculty of the Graduate School University of Missouri In Partial Fulllment of
More informationFrames in Hilbert C -modules. Wu Jing
Frames in Hilbert C -modules by Wu Jing B.S. Ludong University, 1991 M.S. Qufu Normal University, 1994 A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy
More informationFrame expansions of test functions, tempered distributions, and ultradistributions
arxiv:1712.06739v1 [math.fa] 19 Dec 2017 Frame expansions of test functions, tempered distributions, and ultradistributions Stevan Pilipović a and Diana T. Stoeva b a Department of Mathematics and Informatics,
More information