Superoscillations and Fock Spaces: A Mathematical Exploration

What is it about?

This paper uses Fock space theory to compute the action of the Segal-Bargmann transform on special wave functions obtained by multiplying superoscillating sequences with normalized Hermite functions. The transform maps superoscillating sequences onto a superposition of coherent states, leading to specific linear combinations of the normalized reproducing kernels (coherent states) of the Fock space. The paper also investigates relations between superoscillation functions and Weyl operators and Fourier transform.

Featured Image

Why is it important?

This research is important because it explores the relationship between superoscillating sequences, Segal-Bargmann transform, and Fock spaces theory. Understanding these connections helps develop new mathematical tools and representations for analyzing complex functions. This research has implications in various fields, such as quantum mechanics, signal processing, and information theory. Key Takeaways: 1. The Segal-Bargmann transform is an integral transform that maps functions in L^2(R) to the Fock space F(C). 2. The Fock kernel can be expressed as an inner product of Bargmann kernels on the Hilbert space L^2(R). 3. Normalized Hermite functions are mapped onto an orthonormal basis of the Fock space by the Segal-Bargmann transform. 4. Superoscillating sequences and Weyl operators are studied in the context of Fock spaces theory, leading to new insights and relationships.