Loosen Up! an Introduction to Frames. Bases Frames

Frames

Vectors and Vector Spaces

Orthonormal Loosen Up! An introduction to frames. Bases Frames

4 (Interrelated) Research Areas Keri A. Kornelson Applied Math Linear Algebra Geometry University of Oklahoma - Norman Operator Theory [email protected] Frames for Undergraduates Joint AMS/MAA Meetings Panel: This could be YOUR graduate research! New Orleans, LA January 7, 2011 Quick review of vector spaces, Rn: Frames

Vectors and Vector Spaces

Orthonormal Vectors in Rn are sometimes represented as columns: Bases Frames

x(1) 4 (Interrelated) x(2) Research Areas x   Applied Math = . Linear Algebra . Geometry   Operator Theory x(n)   Frames for   Undergraduates Quick review of vector spaces, Rn: Frames

n Vectors and Vector Vectors in R are sometimes represented as columns: Spaces Orthonormal x(1) Bases x(2) Frames x =  .  4 (Interrelated) . Research Areas   Applied Math x(n) Linear Algebra   Geometry   Operator Theory

Frames for and sometimes as arrows: Undergraduates Norms and Dot Products Frames 2 vectors

x(1) y(1) Vectors and Vector x(2) y(2) Spaces x   y   = . = . Orthonormal . . Bases     x(n) y(n) Frames         4 (Interrelated) Deﬁnition Research Areas Applied Math The norm or length of x is Linear Algebra Geometry 1 Operator Theory n 2 Frames for 2 kxk = x(i) . Undergraduates = ! Xi 1 A vector with norm 1 is called a unit vector . Norms and Dot Products Frames 2 vectors

n hx, yi = x(i)y(i). = Xi 1 Orthogonal Vectors Frames

Vectors and Vector Deﬁnition Spaces Orthonormal Two vectors are orthogonal if their inner (dot) product is zero, Bases i.e. if their “arrows” are perpendicular. Frames

4 (Interrelated) Research Areas Applied Math Linear Algebra Geometry Operator Theory

Frames for Undergraduates Orthogonal Vectors Frames

Deﬁnition Vectors and Vector Spaces Two vectors are orthogonal if their inner (dot) product is zero, Orthonormal i.e. if their “arrows” are perpendicular. Bases

Frames

4 (Interrelated) Research Areas Applied Math Linear Algebra Geometry Operator Theory

Frames for Undergraduates Deﬁnition Two vectors are orthonormal if they are orthogonal and are both unit vectors. Orthonormal Bases Frames

Vectors and Vector Deﬁnition Spaces n A collection of vectors {bi }i=1 is an orthonormal basis (ONB) Orthonormal for Rn if the vectors are pairwise orthonormal and form a basis. Bases Frames

Some handy facts about ONBs: 4 (Interrelated) Research Areas Applied Math Linear Algebra Geometry Operator Theory

Frames for Undergraduates Orthonormal Bases Frames

Vectors and Vector Deﬁnition Spaces n A collection of vectors {bi }i=1 is an orthonormal basis (ONB) Orthonormal for Rn if the vectors are pairwise orthonormal and form a basis. Bases Frames

Some handy facts about ONBs: 4 (Interrelated) Research Areas I Applied Math The unique expansion coefﬁcients are found by the dot Linear Algebra Geometry product. Operator Theory n n Frames for x = ci bi = hx, bi ibi Undergraduates = = Xi 1 Xi 1 Orthonormal Bases Frames

Vectors and Vector Deﬁnition Spaces n A collection of vectors {bi }i=1 is an orthonormal basis (ONB) Orthonormal for Rn if the vectors are pairwise orthonormal and form a basis. Bases Frames

I Parseval’s Identity:

n 2 2 kxk = |hx, bi i| = Xi 1 A Silly Example Frames

Vectors and Vector Spaces

Orthonormal 1 0 0 Bases Frames b1 = 0 b2 = 1 b3 = 0       4 (Interrelated) 0 0 1 Research Areas       Applied Math Linear Algebra Geometry Operator Theory

Frames for Undergraduates A Silly Example Frames

Vectors and Vector Spaces

Orthonormal 1 0 0 Bases Frames b1 = 0 b2 = 1 b3 = 0       4 (Interrelated) 0 0 1 Research Areas       Applied Math Linear Algebra Geometry 4 Operator Theory x = 5 = 4b1 + 5b2 + 6b3 Frames for 6 Undergraduates   A Silly Example Frames

Vectors and Vector Spaces

kxk2 = 42 + 52 + 62 p Orthonormal Bases, cont. Frames

Vectors and Vector ONB’s are pretty restrictive...they all look alike somehow. Spaces

Orthonormal Bases

Frames

4 (Interrelated) Research Areas Applied Math Linear Algebra Geometry Operator Theory

Frames for Undergraduates Orthonormal Bases, cont. Frames

Vectors and Vector ONB’s are pretty restrictive...they all look alike somehow. Spaces

Orthonormal I All vectors have norm 1. Bases

Frames

4 (Interrelated) Research Areas Applied Math Linear Algebra Geometry Operator Theory

Frames for Undergraduates Orthonormal Bases, cont. Frames

Vectors and Vector ONB’s are pretty restrictive...they all look alike somehow. Spaces

Orthonormal I All vectors have norm 1. Bases I The number of vectors n equals the dimension of the Frames Rn 4 (Interrelated) space: . Research Areas Applied Math Linear Algebra Geometry Operator Theory

Frames for Undergraduates Orthonormal Bases, cont. Frames

Vectors and Vector ONB’s are pretty restrictive...they all look alike somehow. Spaces

Orthonormal I All vectors have norm 1. Bases I The number of vectors n equals the dimension of the Frames Rn 4 (Interrelated) space: . Research Areas Applied Math I All the vectors are pairwise orthogonal. Linear Algebra Geometry Operator Theory

Frames for Undergraduates Orthonormal Bases, cont. Frames

Vectors and Vector ONB’s are pretty restrictive...they all look alike somehow. Spaces

Frames for Undergraduates Orthonormal Bases, cont. Frames

Vectors and Vector ONB’s are pretty restrictive...they all look alike somehow. Spaces

Frames for Undergraduates

I There’s not much ﬂexibility to tailor an ONB to a particular application, and there is no resilience to losses or errors in data reconstruction. Signal transmission Frames

n If we agree on an ONB {bi } = , then I can just send you the Vectors and Vector i 1 Spaces coefﬁcients of x and you can ﬁnd x. Orthonormal In our silly example, I send you 4, 5, 6 and you can compute Bases Frames 4 4 (Interrelated) Research Areas x = 4b1 + 5b2 + 6b3 = 5 . Applied Math   Linear Algebra 6 Geometry   Operator Theory Frames for Same idea works for voice on a cell phone or pictures sent Undergraduates over the internet. Signal transmission Frames

4 4,?, 6 −→ 4b1 + ?b2 + 6b3 = ? 6   Loosen up! Frames The concept of a frame for a vector space allows for more Vectors and Vector wiggle room than ONBs. The vectors are allowed to Spaces

Orthonormal Bases

Frames

4 (Interrelated) Research Areas Applied Math Linear Algebra Geometry Operator Theory

Frames for Undergraduates Loosen up! Frames The concept of a frame for a vector space allows for more Vectors and Vector wiggle room than ONBs. The vectors are allowed to Spaces

Orthonormal Stretch out Bases

Frames

4 (Interrelated) Research Areas Applied Math Linear Algebra Geometry Operator Theory

Frames for Undergraduates Loosen up! Frames The concept of a frame for a vector space allows for more Vectors and Vector wiggle room than ONBs. The vectors are allowed to Spaces

Orthonormal Stretch out Bases

Frames

4 (Interrelated) Research Areas Applied Math Linear Algebra Geometry Move around a bit Operator Theory Frames for Undergraduates Loosen up! Frames The concept of a frame for a vector space allows for more Vectors and Vector wiggle room than ONBs. The vectors are allowed to Spaces

Orthonormal Stretch out Bases

Frames

4 (Interrelated) Research Areas Applied Math Linear Algebra Geometry Move around a bit Operator Theory Frames for Undergraduates

Even invite a few friends over! Frames Frames

n Vectors and Vector 2 2 Recall Parseval’s Identity for ONB {bi }: kxk = |hx, bi i| Spaces i=1 Orthonormal X Bases

Frames

4 (Interrelated) Research Areas Applied Math Linear Algebra Geometry Operator Theory

Frames for Undergraduates Frames Frames

n Vectors and Vector 2 2 Recall Parseval’s Identity for ONB {bi }: kxk = |hx, bi i| Spaces i=1 Orthonormal X Bases

Deﬁnition Frames Rn k A frame for is a collection of vectors {fi }i=1 that satisfy a 4 (Interrelated) looser condition than Parseval’s identity. There are constants Research Areas Applied Math A, B > 0 (called frame bounds ) such that Linear Algebra Geometry Operator Theory k 2 2 2 Frames for Akxk ≤ |hx, fi i| ≤ Bkxk . Undergraduates = Xi 1 Frames Frames

n Vectors and Vector 2 2 Recall Parseval’s Identity for ONB {bi }: kxk = |hx, bi i| Spaces i=1 Orthonormal X Bases

A frame is tight if A = B and Parseval if A = B = 1.

k 2 2 kxk = |hx, fi i| (look familiar?) = Xi 1 Handy facts about frames: Frames

Vectors and Vector Spaces I In ﬁnite-dimensional spaces, the frames are exactly the Orthonormal spanning sets for the space. Bases Frames

4 (Interrelated) Research Areas Applied Math Linear Algebra Geometry Operator Theory

Frames for Undergraduates Handy facts about frames: Frames

Vectors and Vector Spaces I In ﬁnite-dimensional spaces, the frames are exactly the Orthonormal spanning sets for the space. Bases Frames I Every ONB is a Parseval frame, but there are more! 4 (Interrelated) Research Areas Applied Math Linear Algebra Geometry Operator Theory

Frames for Undergraduates Handy facts about frames: Frames

Frames for Undergraduates

I Parseval frames also satisfy the reconstruction property of ONBs: k k

x = ci fi = hx, fi ifi = = Xi 1 Xi 1 Projections Frames Theorem Every frame is the projection of a basis for a larger space. Vectors and Vector Spaces Every Parseval frame is the projection of a orthonormal basis Orthonormal for a larger space. Bases Frames

4 (Interrelated) Research Areas Applied Math Linear Algebra Geometry Operator Theory

Frames for Undergraduates Projections Frames Theorem Every frame is the projection of a basis for a larger space. Vectors and Vector Spaces Every Parseval frame is the projection of a orthonormal basis Orthonormal for a larger space. Bases Example Frames 4 (Interrelated) 3 R orthonormal basis projected onto the plane. Research Areas Applied Math Linear Algebra Geometry Operator Theory

Frames for Undergraduates

yields the Parseval frame with 3 equal-norm vectors for R2. 4 frame-related research areas Frames

Vectors and Vector Spaces

Orthonormal Bases

Frames

4 (Interrelated) 1. Applied Math Research Areas Applied Math Linear Algebra 2. Linear Algebra Geometry Operator Theory

3. Geometry Frames for Undergraduates 4. Operator Theory Sample Applied Math Research Problems Frames

Vectors and Vector Spaces I Orthonormal Which frames have the best resilience to 1, 2, or more Bases

erasures? Frames

4 (Interrelated) Research Areas Applied Math Linear Algebra Geometry Operator Theory

Frames for Undergraduates Sample Applied Math Research Problems Frames

Vectors and Vector Spaces I Orthonormal Which frames have the best resilience to 1, 2, or more Bases

erasures? Frames I Build tight frames which are tailored to a particular 4 (Interrelated) Research Areas application. Applied Math Linear Algebra Geometry Operator Theory

Frames for Undergraduates Sample Applied Math Research Problems Frames

Vectors and Vector Spaces I Orthonormal Which frames have the best resilience to 1, 2, or more Bases

erasures? Frames I Build tight frames which are tailored to a particular 4 (Interrelated) Research Areas application. Applied Math Linear Algebra I Geometry Build the sparsest possible tight frame of given Operator Theory

size/redundancy. Frames for Undergraduates Sample Applied Math Research Problems Frames

Vectors and Vector Spaces I Orthonormal Which frames have the best resilience to 1, 2, or more Bases

size/redundancy. Frames for Undergraduates I Find an algorithm like Gram-Schmidt that generates tight frames from a given frame sequence. Sample Applied Math Research Problems Frames

Vectors and Vector Spaces I Orthonormal Which frames have the best resilience to 1, 2, or more Bases

size/redundancy. Frames for Undergraduates I Find an algorithm like Gram-Schmidt that generates tight frames from a given frame sequence. I Find an algorithm that numerically converges to a tight frame under given constraints (same norms, for example). Sample Applied Math Research Problems Frames

Vectors and Vector Spaces I Orthonormal Which frames have the best resilience to 1, 2, or more Bases

Vectors and Vector Spaces Let f k be a frame for Rn. We can create the n k matrix S Orthonormal { i }i=1 × Bases which has the frame vectors as columns. Frames

4 (Interrelated) ↑↑ ↑ Research Areas f1 f2 ··· fk Applied Math   Linear Algebra ↓↓ ↓ Geometry Operator Theory   Frames for Theorem Undergraduates Synthesis operator, frame potential Frames

Vectors and Vector Spaces Let f k be a frame for Rn. We can create the n k matrix S Orthonormal { i }i=1 × Bases which has the frame vectors as columns. Frames

4 (Interrelated) ↑↑ ↑ Research Areas f1 f2 ··· fk Applied Math   Linear Algebra ↓↓ ↓ Geometry Operator Theory   Frames for Theorem Undergraduates

I k The frame {fi }i=1 is Parseval if and only if the rows of S are orthonormal. Synthesis operator, frame potential Frames

Vectors and Vector Spaces Let f k be a frame for Rn. We can create the n k matrix S Orthonormal { i }i=1 × Bases which has the frame vectors as columns. Frames

4 (Interrelated) ↑↑ ↑ Research Areas f1 f2 ··· fk Applied Math   Linear Algebra ↓↓ ↓ Geometry Operator Theory   Frames for Theorem Undergraduates

I k The frame {fi }i=1 is Parseval if and only if the rows of S are orthonormal. I k ∗ Rn {fi }i=1 is Parseval iff SS is the identity matrix on . Sample Linear Algebra Research Problems Frames

Vectors and Vector Spaces

Orthonormal I Find a tight frame with k vectors for Rn, where the vectors Bases are all the same length. Frames 4 (Interrelated) Research Areas Applied Math Linear Algebra Geometry Operator Theory

Frames for Undergraduates Sample Linear Algebra Research Problems Frames

Vectors and Vector Spaces

Frames for Undergraduates Sample Linear Algebra Research Problems Frames

Vectors and Vector Spaces

Orthonormal I Find a tight frame with k vectors for Rn, where the vectors Bases are all the same length. Frames 4 (Interrelated) I Find a tight frame with k vectors for Rn, where the vectors Research Areas Applied Math are all lie on an ellipsoid. Linear Algebra Geometry I Find a tight frame with k vectors for Rn, where the vectors Operator Theory Frames for have a given sequence of norms. Undergraduates Sample Linear Algebra Research Problems Frames

Vectors and Vector Spaces

Vectors and Vector a cos θ Spaces Another way to think about a vector in R2: a sin θ Orthonormal Bases Theorem Frames k ai cos θi R2 4 (Interrelated) A frame is a tight frame for if and only if Research Areas ai sin θi = i 1 Applied Math Linear Algebra k Geometry a2 cos 2θ 0 Operator Theory i i = . a2 sin 2θ 0 Frames for = i i Undergraduates Xi 1 R2 frames Frames

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 the result. Sample Geometric Research Problems Frames

Vectors and Vector Spaces

Orthonormal Bases

Frames I The theorem yields lots of facts about R2 tight frames — 4 (Interrelated) Research Areas for example All 4-vector unit frames consist of 2 ONBs. Applied Math Linear Algebra Geometry Operator Theory

Frames for Undergraduates Sample Geometric Research Problems Frames

Vectors and Vector Spaces

Orthonormal Bases

in 3 or 4 dimensions? Frames for Undergraduates Sample Geometric Research Problems Frames

Vectors and Vector Spaces

Orthonormal Bases

in 3 or 4 dimensions? Frames for Undergraduates I Find/characterize equiangular equal-norm tight frames (related to packing problems). Kadison-Singer Problem (1951) Frames

Vectors and Vector Spaces

Orthonormal Bases The Kadison-Singer problem in operator theory has been open Frames since 1951. 4 (Interrelated) Research Areas It has recently been shown equivalent to a variety of problems Applied Math having to do with ﬁnite frames and ﬁnite matrices. Linear Algebra Geometry Operator Theory Example Frames for Undergraduates Does there exist an > 0 and a natural number r such that for 2n Rn all equal-norm Parseval frames {fi }i=1 for , there is a r partition {Aj }j=1 of {1, 2,..., 2n} such that {fi }i∈Aj has Bessel bound ≤ 1 − for all j = 1, 2,..., r. For further reading... Frames

AMERICAN MATHEMATICAL SOCIETY Vectors and Vector Spaces

Frames for Orthonormal Undergraduates Bases Frames Deguang Han, University of Central Florida, Orlando, FL , Keri Kornelson, Grinnell 4 (Interrelated) College, IA , David Larson, Texas A&M University, College Station, TX , and Eric Research Areas Weber, Iowa State University, Ames, IA Applied Math Linear Algebra Frames are a generalization of bases. Geometry eir study has a powerful impact in both Operator Theory abstract and applied settings. is book provides an undergraduate-level introduction to the theory of frames, primarily in Frames for finite-dimensional Hilbert spaces. Undergraduates Instructional Venues: Student Mathematical Library • A special topics course about 2007; 295 pp; softcover Volume: 40 frames and bases. ISBN: 978-0-8218-4212-6 List Price: US$49 • A second linear algebra course. Member Price: US$39 Order Code: STML/40 • A resource for an undergraduate research activity.

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 Where you might go to study frames... Frames . . . in no particular order:

Vectors and Vector I University of Oklahoma Spaces Orthonormal Bases

Frames

4 (Interrelated) Research Areas Applied Math Linear Algebra Geometry Operator Theory

Frames for Undergraduates Where you might go to study frames... Frames . . . in no particular order:

Vectors and Vector I University of Oklahoma Spaces Orthonormal I University of Maryland - also check out the Norbert Bases Weiner Center Frames 4 (Interrelated) Research Areas Applied Math Linear Algebra Geometry Operator Theory

Frames for Undergraduates Where you might go to study frames... Frames . . . in no particular order:

Vectors and Vector I University of Oklahoma Spaces Orthonormal I University of Maryland - also check out the Norbert Bases Weiner Center Frames 4 (Interrelated) I Texas A& M University Research Areas Applied Math I University of Missouri Linear Algebra Geometry Operator Theory I University of Cincinnati Frames for Undergraduates Where you might go to study frames... Frames . . . in no particular order:

Vectors and Vector I University of Oklahoma Spaces Orthonormal I University of Maryland - also check out the Norbert Bases Weiner Center Frames 4 (Interrelated) I Texas A& M University Research Areas Applied Math I University of Missouri Linear Algebra Geometry Operator Theory I University of Cincinnati Frames for I University of Colorado Undergraduates Where you might go to study frames... Frames . . . in no particular order:

Vectors and Vector I University of Oklahoma Spaces Orthonormal I University of Maryland - also check out the Norbert Bases Weiner Center Frames 4 (Interrelated) I Texas A& M University Research Areas Applied Math I University of Missouri Linear Algebra Geometry Operator Theory I University of Cincinnati Frames for I University of Colorado Undergraduates I University of Iowa Where you might go to study frames... Frames . . . in no particular order:

Vectors and Vector I University of Oklahoma Spaces Orthonormal I University of Maryland - also check out the Norbert Bases Weiner Center Frames 4 (Interrelated) I Texas A& M University Research Areas Applied Math I University of Missouri Linear Algebra Geometry Operator Theory I University of Cincinnati Frames for I University of Colorado Undergraduates I University of Iowa I Iowa State University Where you might go to study frames... Frames . . . in no particular order:

Vectors and Vector I University of Oklahoma Spaces Orthonormal I University of Maryland - also check out the Norbert Bases Weiner Center Frames 4 (Interrelated) I Texas A& M University Research Areas Applied Math I University of Missouri Linear Algebra Geometry Operator Theory I University of Cincinnati Frames for I University of Colorado Undergraduates I University of Iowa I Iowa State University I University of Houston Where you might go to study frames... Frames . . . in no particular order:

Vectors and Vector I University of Oklahoma Spaces Orthonormal I University of Maryland - also check out the Norbert Bases Weiner Center Frames 4 (Interrelated) I Texas A& M University Research Areas Applied Math I University of Missouri Linear Algebra Geometry Operator Theory I University of Cincinnati Frames for I University of Colorado Undergraduates I University of Iowa I Iowa State University I University of Houston I University of Oregon Where you might go to study frames... Frames . . . in no particular order:

Vectors and Vector I University of Oklahoma Spaces Orthonormal I University of Maryland - also check out the Norbert Bases Weiner Center Frames 4 (Interrelated) I Texas A& M University Research Areas Applied Math I University of Missouri Linear Algebra Geometry Operator Theory I University of Cincinnati Frames for I University of Colorado Undergraduates I University of Iowa I Iowa State University I University of Houston I University of Oregon I Georgia Institute of Technology Where you might go to study frames... Frames . . . in no particular order:

Vectors and Vector I University of Oklahoma Spaces Orthonormal I University of Maryland - also check out the Norbert Bases Weiner Center Frames 4 (Interrelated) I Texas A& M University Research Areas Applied Math I University of Missouri Linear Algebra Geometry Operator Theory I University of Cincinnati Frames for I University of Colorado Undergraduates I University of Iowa I Iowa State University I University of Houston I University of Oregon I Georgia Institute of Technology I Vanderbilt University