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Frames
Vectors and VectorSpaces
OrthonormalBases
Frames
4 (Interrelated)Research AreasApplied Math
Linear Algebra
Geometry
Operator Theory
Frames forUndergraduates
Loosen Up! An introduction to frames.
Keri A. Kornelson
University of Oklahoma - [email protected]
Joint AMS/MAA MeetingsPanel: This could be YOUR graduate research!
New Orleans, LAJanuary 7, 2011
Frames
Vectors and VectorSpaces
OrthonormalBases
Frames
4 (Interrelated)Research AreasApplied Math
Linear Algebra
Geometry
Operator Theory
Frames forUndergraduates
Quick review of vector spaces, Rn:
Vectors in Rn are sometimes represented as columns:
x =
x(1)x(2)
...x(n)
Frames
Vectors and VectorSpaces
OrthonormalBases
Frames
4 (Interrelated)Research AreasApplied Math
Linear Algebra
Geometry
Operator Theory
Frames forUndergraduates
Quick review of vector spaces, Rn:
Vectors in Rn are sometimes represented as columns:
x =
x(1)x(2)
...x(n)
and sometimes as arrows:
Frames
Vectors and VectorSpaces
OrthonormalBases
Frames
4 (Interrelated)Research AreasApplied Math
Linear Algebra
Geometry
Operator Theory
Frames forUndergraduates
Norms and Dot Products2 vectors
x =
x(1)x(2)
...x(n)
y =
y(1)y(2)
...y(n)
DefinitionThe norm or length of x is
‖x‖ =
(
n∑
i=1
x(i)2
)12
.
A vector with norm 1 is called a unit vector .
Frames
Vectors and VectorSpaces
OrthonormalBases
Frames
4 (Interrelated)Research AreasApplied Math
Linear Algebra
Geometry
Operator Theory
Frames forUndergraduates
Norms and Dot Products2 vectors
x =
x(1)x(2)
...x(n)
y =
y(1)y(2)
...y(n)
DefinitionThe norm or length of x is
‖x‖ =
(
n∑
i=1
x(i)2
)12
.
A vector with norm 1 is called a unit vector .
DefinitionThe dot product or inner product of x and y is
〈x , y〉 =n∑
i=1
x(i)y(i).
Frames
Vectors and VectorSpaces
OrthonormalBases
Frames
4 (Interrelated)Research AreasApplied Math
Linear Algebra
Geometry
Operator Theory
Frames forUndergraduates
Orthogonal Vectors
DefinitionTwo vectors are orthogonal if their inner (dot) product is zero,i.e. if their “arrows” are perpendicular.
Frames
Vectors and VectorSpaces
OrthonormalBases
Frames
4 (Interrelated)Research AreasApplied Math
Linear Algebra
Geometry
Operator Theory
Frames forUndergraduates
Orthogonal Vectors
DefinitionTwo vectors are orthogonal if their inner (dot) product is zero,i.e. if their “arrows” are perpendicular.
DefinitionTwo vectors are orthonormal if they are orthogonal and areboth unit vectors.
Frames
Vectors and VectorSpaces
OrthonormalBases
Frames
4 (Interrelated)Research AreasApplied Math
Linear Algebra
Geometry
Operator Theory
Frames forUndergraduates
Orthonormal Bases
DefinitionA collection of vectors {bi}
ni=1 is an orthonormal basis (ONB)
for Rn if the vectors are pairwise orthonormal and form a basis.
Some handy facts about ONBs:
Frames
Vectors and VectorSpaces
OrthonormalBases
Frames
4 (Interrelated)Research AreasApplied Math
Linear Algebra
Geometry
Operator Theory
Frames forUndergraduates
Orthonormal Bases
DefinitionA collection of vectors {bi}
ni=1 is an orthonormal basis (ONB)
for Rn if the vectors are pairwise orthonormal and form a basis.
Some handy facts about ONBs:
I The unique expansion coefficients are found by the dotproduct.
x =
n∑
i=1
cibi =
n∑
i=1
〈x , bi〉bi
Frames
Vectors and VectorSpaces
OrthonormalBases
Frames
4 (Interrelated)Research AreasApplied Math
Linear Algebra
Geometry
Operator Theory
Frames forUndergraduates
Orthonormal Bases
DefinitionA collection of vectors {bi}
ni=1 is an orthonormal basis (ONB)
for Rn if the vectors are pairwise orthonormal and form a basis.
Some handy facts about ONBs:
I The unique expansion coefficients are found by the dotproduct.
x =
n∑
i=1
cibi =
n∑
i=1
〈x , bi〉bi
I Parseval’s Identity:
‖x‖2 =
n∑
i=1
|〈x , bi〉|2
Frames
Vectors and VectorSpaces
OrthonormalBases
Frames
4 (Interrelated)Research AreasApplied Math
Linear Algebra
Geometry
Operator Theory
Frames forUndergraduates
A Silly Example
b1 =
100
b2 =
010
b3 =
001
Frames
Vectors and VectorSpaces
OrthonormalBases
Frames
4 (Interrelated)Research AreasApplied Math
Linear Algebra
Geometry
Operator Theory
Frames forUndergraduates
A Silly Example
b1 =
100
b2 =
010
b3 =
001
x =
456
= 4b1 + 5b2 + 6b3
Frames
Vectors and VectorSpaces
OrthonormalBases
Frames
4 (Interrelated)Research AreasApplied Math
Linear Algebra
Geometry
Operator Theory
Frames forUndergraduates
A Silly Example
b1 =
100
b2 =
010
b3 =
001
x =
456
= 4b1 + 5b2 + 6b3
〈x , b1〉 = 4, 〈x , b2〉 = 5, 〈x , b3〉 = 6.
Frames
Vectors and VectorSpaces
OrthonormalBases
Frames
4 (Interrelated)Research AreasApplied Math
Linear Algebra
Geometry
Operator Theory
Frames forUndergraduates
A Silly Example
b1 =
100
b2 =
010
b3 =
001
x =
456
= 4b1 + 5b2 + 6b3
〈x , b1〉 = 4, 〈x , b2〉 = 5, 〈x , b3〉 = 6.
‖x‖2 =√
42 + 52 + 62
Frames
Vectors and VectorSpaces
OrthonormalBases
Frames
4 (Interrelated)Research AreasApplied Math
Linear Algebra
Geometry
Operator Theory
Frames forUndergraduates
Orthonormal Bases, cont.
ONB’s are pretty restrictive...they all look alike somehow.
Frames
Vectors and VectorSpaces
OrthonormalBases
Frames
4 (Interrelated)Research AreasApplied Math
Linear Algebra
Geometry
Operator Theory
Frames forUndergraduates
Orthonormal Bases, cont.
ONB’s are pretty restrictive...they all look alike somehow.
I All vectors have norm 1.
Frames
Vectors and VectorSpaces
OrthonormalBases
Frames
4 (Interrelated)Research AreasApplied Math
Linear Algebra
Geometry
Operator Theory
Frames forUndergraduates
Orthonormal Bases, cont.
ONB’s are pretty restrictive...they all look alike somehow.
I All vectors have norm 1.
I The number of vectors n equals the dimension of thespace: Rn.
Frames
Vectors and VectorSpaces
OrthonormalBases
Frames
4 (Interrelated)Research AreasApplied Math
Linear Algebra
Geometry
Operator Theory
Frames forUndergraduates
Orthonormal Bases, cont.
ONB’s are pretty restrictive...they all look alike somehow.
I All vectors have norm 1.
I The number of vectors n equals the dimension of thespace: Rn.
I All the vectors are pairwise orthogonal.
Frames
Vectors and VectorSpaces
OrthonormalBases
Frames
4 (Interrelated)Research AreasApplied Math
Linear Algebra
Geometry
Operator Theory
Frames forUndergraduates
Orthonormal Bases, cont.
ONB’s are pretty restrictive...they all look alike somehow.
I All vectors have norm 1.
I The number of vectors n equals the dimension of thespace: Rn.
I All the vectors are pairwise orthogonal.
Frames
Vectors and VectorSpaces
OrthonormalBases
Frames
4 (Interrelated)Research AreasApplied Math
Linear Algebra
Geometry
Operator Theory
Frames forUndergraduates
Orthonormal Bases, cont.
ONB’s are pretty restrictive...they all look alike somehow.
I All vectors have norm 1.
I The number of vectors n equals the dimension of thespace: Rn.
I All the vectors are pairwise orthogonal.
I There’s not much flexibility to tailor an ONB to a particularapplication, and there is no resilience to losses or errors indata reconstruction.
Frames
Vectors and VectorSpaces
OrthonormalBases
Frames
4 (Interrelated)Research AreasApplied Math
Linear Algebra
Geometry
Operator Theory
Frames forUndergraduates
Signal transmission
If we agree on an ONB {bi}ni=1, then I can just send you the
coefficients of x and you can find x .
In our silly example, I send you 4, 5, 6 and you can compute
x = 4b1 + 5b2 + 6b3 =
456
.
Same idea works for voice on a cell phone or pictures sentover the internet.
Frames
Vectors and VectorSpaces
OrthonormalBases
Frames
4 (Interrelated)Research AreasApplied Math
Linear Algebra
Geometry
Operator Theory
Frames forUndergraduates
Signal transmission
If we agree on an ONB {bi}ni=1, then I can just send you the
coefficients of x and you can find x .
In our silly example, I send you 4, 5, 6 and you can compute
x = 4b1 + 5b2 + 6b3 =
456
.
Same idea works for voice on a cell phone or pictures sentover the internet.
If one data point gets lost using an ONB, there is noinformation about what it was.
4, ?, 6 −→ 4b1 + ?b2 + 6b3 =
4?
6
Frames
Vectors and VectorSpaces
OrthonormalBases
Frames
4 (Interrelated)Research AreasApplied Math
Linear Algebra
Geometry
Operator Theory
Frames forUndergraduates
Loosen up!The concept of a frame for a vector space allows for morewiggle room than ONBs. The vectors are allowed to
Frames
Vectors and VectorSpaces
OrthonormalBases
Frames
4 (Interrelated)Research AreasApplied Math
Linear Algebra
Geometry
Operator Theory
Frames forUndergraduates
Loosen up!The concept of a frame for a vector space allows for morewiggle room than ONBs. The vectors are allowed to
Stretch out
Frames
Vectors and VectorSpaces
OrthonormalBases
Frames
4 (Interrelated)Research AreasApplied Math
Linear Algebra
Geometry
Operator Theory
Frames forUndergraduates
Loosen up!The concept of a frame for a vector space allows for morewiggle room than ONBs. The vectors are allowed to
Stretch out
Move around a bit
Frames
Vectors and VectorSpaces
OrthonormalBases
Frames
4 (Interrelated)Research AreasApplied Math
Linear Algebra
Geometry
Operator Theory
Frames forUndergraduates
Loosen up!The concept of a frame for a vector space allows for morewiggle room than ONBs. The vectors are allowed to
Stretch out
Move around a bit
Even invite a few friends over!
Frames
Vectors and VectorSpaces
OrthonormalBases
Frames
4 (Interrelated)Research AreasApplied Math
Linear Algebra
Geometry
Operator Theory
Frames forUndergraduates
Frames
Recall Parseval’s Identity for ONB {bi}: ‖x‖2 =n∑
i=1
|〈x , bi〉|2
Frames
Vectors and VectorSpaces
OrthonormalBases
Frames
4 (Interrelated)Research AreasApplied Math
Linear Algebra
Geometry
Operator Theory
Frames forUndergraduates
Frames
Recall Parseval’s Identity for ONB {bi}: ‖x‖2 =n∑
i=1
|〈x , bi〉|2
DefinitionA frame for Rn is a collection of vectors {fi}k
i=1 that satisfy alooser condition than Parseval’s identity. There are constantsA,B > 0 (called frame bounds ) such that
A‖x‖2 ≤
k∑
i=1
|〈x , fi〉|2 ≤ B‖x‖2.
Frames
Vectors and VectorSpaces
OrthonormalBases
Frames
4 (Interrelated)Research AreasApplied Math
Linear Algebra
Geometry
Operator Theory
Frames forUndergraduates
Frames
Recall Parseval’s Identity for ONB {bi}: ‖x‖2 =n∑
i=1
|〈x , bi〉|2
DefinitionA frame for Rn is a collection of vectors {fi}k
i=1 that satisfy alooser condition than Parseval’s identity. There are constantsA,B > 0 (called frame bounds ) such that
A‖x‖2 ≤
k∑
i=1
|〈x , fi〉|2 ≤ B‖x‖2.
Frames
Vectors and VectorSpaces
OrthonormalBases
Frames
4 (Interrelated)Research AreasApplied Math
Linear Algebra
Geometry
Operator Theory
Frames forUndergraduates
Frames
Recall Parseval’s Identity for ONB {bi}: ‖x‖2 =n∑
i=1
|〈x , bi〉|2
DefinitionA frame for Rn is a collection of vectors {fi}k
i=1 that satisfy alooser condition than Parseval’s identity. There are constantsA,B > 0 (called frame bounds ) such that
A‖x‖2 ≤
k∑
i=1
|〈x , fi〉|2 ≤ B‖x‖2.
A frame is tight if A = B and Parseval if A = B = 1.
‖x‖2 =
k∑
i=1
|〈x , fi〉|2 (look familiar?)
Frames
Vectors and VectorSpaces
OrthonormalBases
Frames
4 (Interrelated)Research AreasApplied Math
Linear Algebra
Geometry
Operator Theory
Frames forUndergraduates
Handy facts about frames:
I In finite-dimensional spaces, the frames are exactly thespanning sets for the space.
Frames
Vectors and VectorSpaces
OrthonormalBases
Frames
4 (Interrelated)Research AreasApplied Math
Linear Algebra
Geometry
Operator Theory
Frames forUndergraduates
Handy facts about frames:
I In finite-dimensional spaces, the frames are exactly thespanning sets for the space.
I Every ONB is a Parseval frame, but there are more!
Frames
Vectors and VectorSpaces
OrthonormalBases
Frames
4 (Interrelated)Research AreasApplied Math
Linear Algebra
Geometry
Operator Theory
Frames forUndergraduates
Handy facts about frames:
I In finite-dimensional spaces, the frames are exactly thespanning sets for the space.
I Every ONB is a Parseval frame, but there are more!
I Parseval frames also satisfy the reconstruction property ofONBs:
x =
k∑
i=1
ci fi =k∑
i=1
〈x , fi〉fi
Frames
Vectors and VectorSpaces
OrthonormalBases
Frames
4 (Interrelated)Research AreasApplied Math
Linear Algebra
Geometry
Operator Theory
Frames forUndergraduates
ProjectionsTheoremEvery frame is the projection of a basis for a larger space.Every Parseval frame is the projection of a orthonormal basisfor a larger space.
Frames
Vectors and VectorSpaces
OrthonormalBases
Frames
4 (Interrelated)Research AreasApplied Math
Linear Algebra
Geometry
Operator Theory
Frames forUndergraduates
ProjectionsTheoremEvery frame is the projection of a basis for a larger space.Every Parseval frame is the projection of a orthonormal basisfor a larger space.
ExampleR
3 orthonormal basis projected onto the plane.
yields the Parseval frame with 3 equal-norm vectors for R2.
Frames
Vectors and VectorSpaces
OrthonormalBases
Frames
4 (Interrelated)Research AreasApplied Math
Linear Algebra
Geometry
Operator Theory
Frames forUndergraduates
4 frame-related research areas
1. Applied Math
2. Linear Algebra
3. Geometry
4. Operator Theory
Frames
Vectors and VectorSpaces
OrthonormalBases
Frames
4 (Interrelated)Research AreasApplied Math
Linear Algebra
Geometry
Operator Theory
Frames forUndergraduates
Sample Applied Math Research Problems
I Which frames have the best resilience to 1, 2, or moreerasures?
Frames
Vectors and VectorSpaces
OrthonormalBases
Frames
4 (Interrelated)Research AreasApplied Math
Linear Algebra
Geometry
Operator Theory
Frames forUndergraduates
Sample Applied Math Research Problems
I Which frames have the best resilience to 1, 2, or moreerasures?
I Build tight frames which are tailored to a particularapplication.
Frames
Vectors and VectorSpaces
OrthonormalBases
Frames
4 (Interrelated)Research AreasApplied Math
Linear Algebra
Geometry
Operator Theory
Frames forUndergraduates
Sample Applied Math Research Problems
I Which frames have the best resilience to 1, 2, or moreerasures?
I Build tight frames which are tailored to a particularapplication.
I Build the sparsest possible tight frame of givensize/redundancy.
Frames
Vectors and VectorSpaces
OrthonormalBases
Frames
4 (Interrelated)Research AreasApplied Math
Linear Algebra
Geometry
Operator Theory
Frames forUndergraduates
Sample Applied Math Research Problems
I Which frames have the best resilience to 1, 2, or moreerasures?
I Build tight frames which are tailored to a particularapplication.
I Build the sparsest possible tight frame of givensize/redundancy.
I Find an algorithm like Gram-Schmidt that generates tightframes from a given frame sequence.
Frames
Vectors and VectorSpaces
OrthonormalBases
Frames
4 (Interrelated)Research AreasApplied Math
Linear Algebra
Geometry
Operator Theory
Frames forUndergraduates
Sample Applied Math Research Problems
I Which frames have the best resilience to 1, 2, or moreerasures?
I Build tight frames which are tailored to a particularapplication.
I Build the sparsest possible tight frame of givensize/redundancy.
I Find an algorithm like Gram-Schmidt that generates tightframes from a given frame sequence.
I Find an algorithm that numerically converges to a tightframe under given constraints (same norms, for example).
Frames
Vectors and VectorSpaces
OrthonormalBases
Frames
4 (Interrelated)Research AreasApplied Math
Linear Algebra
Geometry
Operator Theory
Frames forUndergraduates
Sample Applied Math Research Problems
I Which frames have the best resilience to 1, 2, or moreerasures?
I Build tight frames which are tailored to a particularapplication.
I Build the sparsest possible tight frame of givensize/redundancy.
I Find an algorithm like Gram-Schmidt that generates tightframes from a given frame sequence.
I Find an algorithm that numerically converges to a tightframe under given constraints (same norms, for example).
I Wavelet frames.
Frames
Vectors and VectorSpaces
OrthonormalBases
Frames
4 (Interrelated)Research AreasApplied Math
Linear Algebra
Geometry
Operator Theory
Frames forUndergraduates
Synthesis operator, frame potential
Let {fi}ki=1 be a frame for Rn. We can create the n × k matrix S
which has the frame vectors as columns.
↑ ↑ ↑f1 f2 · · · fk↓ ↓ ↓
Theorem
Frames
Vectors and VectorSpaces
OrthonormalBases
Frames
4 (Interrelated)Research AreasApplied Math
Linear Algebra
Geometry
Operator Theory
Frames forUndergraduates
Synthesis operator, frame potential
Let {fi}ki=1 be a frame for Rn. We can create the n × k matrix S
which has the frame vectors as columns.
↑ ↑ ↑f1 f2 · · · fk↓ ↓ ↓
Theorem
I The frame {fi}ki=1 is Parseval if and only if the rows of S
are orthonormal.
Frames
Vectors and VectorSpaces
OrthonormalBases
Frames
4 (Interrelated)Research AreasApplied Math
Linear Algebra
Geometry
Operator Theory
Frames forUndergraduates
Synthesis operator, frame potential
Let {fi}ki=1 be a frame for Rn. We can create the n × k matrix S
which has the frame vectors as columns.
↑ ↑ ↑f1 f2 · · · fk↓ ↓ ↓
Theorem
I The frame {fi}ki=1 is Parseval if and only if the rows of S
are orthonormal.
I {fi}ki=1 is Parseval iff SS∗ is the identity matrix on R
n.
Frames
Vectors and VectorSpaces
OrthonormalBases
Frames
4 (Interrelated)Research AreasApplied Math
Linear Algebra
Geometry
Operator Theory
Frames forUndergraduates
Sample Linear Algebra Research Problems
I Find a tight frame with k vectors for Rn, where the vectorsare all the same length.
Frames
Vectors and VectorSpaces
OrthonormalBases
Frames
4 (Interrelated)Research AreasApplied Math
Linear Algebra
Geometry
Operator Theory
Frames forUndergraduates
Sample Linear Algebra Research Problems
I Find a tight frame with k vectors for Rn, where the vectorsare all the same length.
I Find a tight frame with k vectors for Rn, where the vectorsare all lie on an ellipsoid.
Frames
Vectors and VectorSpaces
OrthonormalBases
Frames
4 (Interrelated)Research AreasApplied Math
Linear Algebra
Geometry
Operator Theory
Frames forUndergraduates
Sample Linear Algebra Research Problems
I Find a tight frame with k vectors for Rn, where the vectorsare all the same length.
I Find a tight frame with k vectors for Rn, where the vectorsare all lie on an ellipsoid.
I Find a tight frame with k vectors for Rn, where the vectorshave a given sequence of norms.
Frames
Vectors and VectorSpaces
OrthonormalBases
Frames
4 (Interrelated)Research AreasApplied Math
Linear Algebra
Geometry
Operator Theory
Frames forUndergraduates
Sample Linear Algebra Research Problems
I Find a tight frame with k vectors for Rn, where the vectorsare all the same length.
I Find a tight frame with k vectors for Rn, where the vectorsare all lie on an ellipsoid.
I Find a tight frame with k vectors for Rn, where the vectorshave a given sequence of norms.
I These all relate to classical problems about writingoperators as sums of projections!
Frames
Vectors and VectorSpaces
OrthonormalBases
Frames
4 (Interrelated)Research AreasApplied Math
Linear Algebra
Geometry
Operator Theory
Frames forUndergraduates
Sample Linear Algebra Research Problems
I Find a tight frame with k vectors for Rn, where the vectorsare all the same length.
I Find a tight frame with k vectors for Rn, where the vectorsare all lie on an ellipsoid.
I Find a tight frame with k vectors for Rn, where the vectorshave a given sequence of norms.
I These all relate to classical problems about writingoperators as sums of projections!
I Frame potential - a real quantity that is minimized at tightframes, simulating electromagnetic potential.
Frames
Vectors and VectorSpaces
OrthonormalBases
Frames
4 (Interrelated)Research AreasApplied Math
Linear Algebra
Geometry
Operator Theory
Frames forUndergraduates
R2 frames
Another way to think about a vector in R2:[
a cos θa sin θ
]
Theorem
A frame{[
ai cos θi
ai sin θi
]}k
i=1is a tight frame for R2 if and only if
k∑
i=1
[
a2i cos 2θi
a2i sin 2θi
]
=
[
00
]
.
Frames
Vectors and VectorSpaces
OrthonormalBases
Frames
4 (Interrelated)Research AreasApplied Math
Linear Algebra
Geometry
Operator Theory
Frames forUndergraduates
R2 frames
Another way to think about a vector in R2:[
a cos θa sin θ
]
Theorem
A frame{[
ai cos θi
ai sin θi
]}k
i=1is a tight frame for R2 if and only if
k∑
i=1
[
a2i cos 2θi
a2i sin 2θi
]
=
[
00
]
.
Proof.Recall S is the 2 × k matrix with the frame vectors as columns,and the frame is tight iff SS∗ is a scalar multiple of the identity.Computing S and using some trigonometric identities gives theresult.
Frames
Vectors and VectorSpaces
OrthonormalBases
Frames
4 (Interrelated)Research AreasApplied Math
Linear Algebra
Geometry
Operator Theory
Frames forUndergraduates
Sample Geometric Research Problems
I The theorem yields lots of facts about R2 tight frames —for example All 4-vector unit frames consist of 2 ONBs.
Frames
Vectors and VectorSpaces
OrthonormalBases
Frames
4 (Interrelated)Research AreasApplied Math
Linear Algebra
Geometry
Operator Theory
Frames forUndergraduates
Sample Geometric Research Problems
I The theorem yields lots of facts about R2 tight frames —for example All 4-vector unit frames consist of 2 ONBs.
I Is there a similar kind of characterization for tight framesin 3 or 4 dimensions?
Frames
Vectors and VectorSpaces
OrthonormalBases
Frames
4 (Interrelated)Research AreasApplied Math
Linear Algebra
Geometry
Operator Theory
Frames forUndergraduates
Sample Geometric Research Problems
I The theorem yields lots of facts about R2 tight frames —for example All 4-vector unit frames consist of 2 ONBs.
I Is there a similar kind of characterization for tight framesin 3 or 4 dimensions?
I Find/characterize equiangular equal-norm tight frames(related to packing problems).
Frames
Vectors and VectorSpaces
OrthonormalBases
Frames
4 (Interrelated)Research AreasApplied Math
Linear Algebra
Geometry
Operator Theory
Frames forUndergraduates
Kadison-Singer Problem (1951)
The Kadison-Singer problem in operator theory has been opensince 1951.
It has recently been shown equivalent to a variety of problemshaving to do with finite frames and finite matrices.
ExampleDoes there exist an ε > 0 and a natural number r such that forall equal-norm Parseval frames {fi}2n
i=1 for Rn , there is apartition {Aj}
rj=1 of {1, 2, . . . , 2n} such that {fi}i∈Aj has Bessel
bound ≤ 1 − ε for all j = 1, 2, . . . , r .
Frames
Vectors and VectorSpaces
OrthonormalBases
Frames
4 (Interrelated)Research AreasApplied Math
Linear Algebra
Geometry
Operator Theory
Frames forUndergraduates
For further reading...
Frames for UndergraduatesDeguang Han, University of Central Florida, Orlando, FL, Keri Kornelson, Grinnell College, IA, David Larson, Texas A&M University, College Station, TX, and Eric Weber, Iowa State University, Ames, IA
Student Mathematical Library
2007; 295 pp; softcover
Volume: 40
ISBN: 978-0-8218-4212-6
List Price: US$49
Member Price: US$39
Order Code: STML/40
Frames are a generalization of bases.
�eir study has a powerful impact in both
abstract and applied settings. �is book
provides an undergraduate-level introduc-
tion to the theory of frames, primarily in
finite-dimensional Hilbert spaces.
Instructional Venues:
• A special topics course about
frames and bases.
• A second linear algebra course.
• A resource for an undergraduate
research activity.
AMERICAN MATHEMATICAL SOCIETY
For many more publications of interest,
visit the AMS Bookstore
www.ams.org/bookstore
1-800-321-4AMS (4267), in the U. S. and Canada, or 1-401-455-4000 (worldwide); fax:1-401-455-4046; email:
[email protected]. American Mathematical Society, 201 Charles Street, Providence, RI 02904-2294 USA
Frames
Vectors and VectorSpaces
OrthonormalBases
Frames
4 (Interrelated)Research AreasApplied Math
Linear Algebra
Geometry
Operator Theory
Frames forUndergraduates
Where you might go to study frames.... . . in no particular order:
I University of Oklahoma
Frames
Vectors and VectorSpaces
OrthonormalBases
Frames
4 (Interrelated)Research AreasApplied Math
Linear Algebra
Geometry
Operator Theory
Frames forUndergraduates
Where you might go to study frames.... . . in no particular order:
I University of Oklahoma
I University of Maryland - also check out the NorbertWeiner Center
Frames
Vectors and VectorSpaces
OrthonormalBases
Frames
4 (Interrelated)Research AreasApplied Math
Linear Algebra
Geometry
Operator Theory
Frames forUndergraduates
Where you might go to study frames.... . . in no particular order:
I University of Oklahoma
I University of Maryland - also check out the NorbertWeiner Center
I Texas A& M University
Frames
Vectors and VectorSpaces
OrthonormalBases
Frames
4 (Interrelated)Research AreasApplied Math
Linear Algebra
Geometry
Operator Theory
Frames forUndergraduates
Where you might go to study frames.... . . in no particular order:
I University of Oklahoma
I University of Maryland - also check out the NorbertWeiner Center
I Texas A& M University
I University of Missouri
Frames
Vectors and VectorSpaces
OrthonormalBases
Frames
4 (Interrelated)Research AreasApplied Math
Linear Algebra
Geometry
Operator Theory
Frames forUndergraduates
Where you might go to study frames.... . . in no particular order:
I University of Oklahoma
I University of Maryland - also check out the NorbertWeiner Center
I Texas A& M University
I University of Missouri
I University of Cincinnati
Frames
Vectors and VectorSpaces
OrthonormalBases
Frames
4 (Interrelated)Research AreasApplied Math
Linear Algebra
Geometry
Operator Theory
Frames forUndergraduates
Where you might go to study frames.... . . in no particular order:
I University of Oklahoma
I University of Maryland - also check out the NorbertWeiner Center
I Texas A& M University
I University of Missouri
I University of Cincinnati
I University of Colorado
Frames
Vectors and VectorSpaces
OrthonormalBases
Frames
4 (Interrelated)Research AreasApplied Math
Linear Algebra
Geometry
Operator Theory
Frames forUndergraduates
Where you might go to study frames.... . . in no particular order:
I University of Oklahoma
I University of Maryland - also check out the NorbertWeiner Center
I Texas A& M University
I University of Missouri
I University of Cincinnati
I University of Colorado
I University of Iowa
Frames
Vectors and VectorSpaces
OrthonormalBases
Frames
4 (Interrelated)Research AreasApplied Math
Linear Algebra
Geometry
Operator Theory
Frames forUndergraduates
Where you might go to study frames.... . . in no particular order:
I University of Oklahoma
I University of Maryland - also check out the NorbertWeiner Center
I Texas A& M University
I University of Missouri
I University of Cincinnati
I University of Colorado
I University of Iowa
I Iowa State University
Frames
Vectors and VectorSpaces
OrthonormalBases
Frames
4 (Interrelated)Research AreasApplied Math
Linear Algebra
Geometry
Operator Theory
Frames forUndergraduates
Where you might go to study frames.... . . in no particular order:
I University of Oklahoma
I University of Maryland - also check out the NorbertWeiner Center
I Texas A& M University
I University of Missouri
I University of Cincinnati
I University of Colorado
I University of Iowa
I Iowa State University
I University of Houston
Frames
Vectors and VectorSpaces
OrthonormalBases
Frames
4 (Interrelated)Research AreasApplied Math
Linear Algebra
Geometry
Operator Theory
Frames forUndergraduates
Where you might go to study frames.... . . in no particular order:
I University of Oklahoma
I University of Maryland - also check out the NorbertWeiner Center
I Texas A& M University
I University of Missouri
I University of Cincinnati
I University of Colorado
I University of Iowa
I Iowa State University
I University of Houston
I University of Oregon
Frames
Vectors and VectorSpaces
OrthonormalBases
Frames
4 (Interrelated)Research AreasApplied Math
Linear Algebra
Geometry
Operator Theory
Frames forUndergraduates
Where you might go to study frames.... . . in no particular order:
I University of Oklahoma
I University of Maryland - also check out the NorbertWeiner Center
I Texas A& M University
I University of Missouri
I University of Cincinnati
I University of Colorado
I University of Iowa
I Iowa State University
I University of Houston
I University of Oregon
I Georgia Institute of Technology
Frames
Vectors and VectorSpaces
OrthonormalBases
Frames
4 (Interrelated)Research AreasApplied Math
Linear Algebra
Geometry
Operator Theory
Frames forUndergraduates
Where you might go to study frames.... . . in no particular order:
I University of Oklahoma
I University of Maryland - also check out the NorbertWeiner Center
I Texas A& M University
I University of Missouri
I University of Cincinnati
I University of Colorado
I University of Iowa
I Iowa State University
I University of Houston
I University of Oregon
I Georgia Institute of Technology
I Vanderbilt University