A frame in a finite-dimensional inner-product space is a collection of vectors spanning this space. In this sense, frames are generalizations of orthonormal bases, which can be used in many cases where orthonormal bases cannot be tailored to fit naturally arising applications. The wide array of possibilities introduced by working with frames rather than orthonormal bases has revolutionized mathematical areas such as wavelets and harmonic analysis. The adaptability to existing conditions allows frames to be used in applied settings including signal processing, imaging, sampling, and cryptography. The study of frames, particularly in finite dimensions, begins with exactly the topics from an undergraduate linear algebra course. This makes the topics particularly accessible to undergraduate students, yet the theory contains deep unsolved problems. This book can be used as a resource for an REU or for a topics course about frames. It is also a suitable textbook for a second linear algebra course, using frames as a thematic example to demonstrate and explore the new material. The theory of frames is increasingly broad with widespread applications. "Frames for Undergraduates" introduces students to this vibrant and important area of mathematics. Readership Undergraduate and graduate students interested in linear algebra and applications, and the theory of frames. Reviews "This incredibly readable book is an enticing introduction to frames. It contains the perfect combination of fundamental ideas from linear algebra and operator theory, and foundational examples and applications of finite frames. My students and I have found it to be an all-around pleasure to read." *-- Michael Orrison, Harvey Mudd College * "I used this book in an undergraduate special topics course on frame theory. With the prerequisite being a standard linear algebra course, I appreciated the comprehensive yet economical background on the necessary ideas from linear algebra and finite dimensional operator theory. The chapters on frame theory cover a nice range of topics, from sampling theory and image reconstruction to frames arising from unitary representations of a group to more recent developments like frame potential, which has interesting connections to physics and platonic solids. My students were inspired not only by these topics but also by the final chapter on anecdotes; many of them had never entertained the possibility that they could do original mathematical research as undergraduates." *-- Fumiko Futamura, Ph.D., Southwestern University * "The result (of this book) is a concise and accessible treatment that serves well as a semester's text." *-- Mathematical Horizon* "This book would be a good candidate for a topics course or for a second course in linear algebra." *-- MAA Reviews* "the clean conceptual layout of this little text makes it appropriate for a second semester special topics course in linear algebra for mathematics majors and certainly for directed reading for a junior level student who might be contemplating graduate study in mathematics. Exercises are included." *-- Zentralblatt MATH* |