In the past 50 years, quantum physicists have discovered, and experimentally demonstrated, a phenomenon which they termed superoscillations. Aharonov and his collaborators showed that superoscillations naturally arise when dealing with weak values, a notion that provides a fundamentally different way to regard measurements in quantum physics. From a mathematical point of view, superoscillating functions are a superposition of small Fourier components with a bounded Fourier spectrum, which result, when appropriately summed, in a shift that can be arbitrarily large, and well outside the spectrum. The purpose of this work is twofold: on one hand the authors provide a self-contained survey of the existing literature, in order to offer a systematic mathematical approach to superoscillations; on the other hand, they obtain some new and unexpected results, by showing that superoscillating sequences can be seen of as solutions to a large class of convolution equations and can therefore be treated within the theory of analytically uniform spaces. In particular, the authors will also discuss the persistence of the superoscillatory behavior when superoscillating sequences are taken as initial values of the Schrödinger equation and other equations.
In the past 50 years, quantum physicists have discovered, and experimentally demonstrated, a phenomenon which they termed superoscillations. Aharonov and his collaborators showed that superoscillations naturally arise when dealing with weak values, a notion that provides a fundamentally different way to regard measurements in quantum physics. From a mathematical point of view, superoscillating functions are a superposition of small Fourier components with a bounded Fourier spectrum, which result, when appropriately summed, in a shift that can be arbitrarily large, and well outside the spectrum. The purpose of this work is twofold: on one hand the authors provide a self-contained survey of the existing literature, in order to offer a systematic mathematical approach to superoscillations; on the other hand, they obtain some new and unexpected results, by showing that superoscillating sequences can be seen of as solutions to a large class of convolution equations and can therefore be treated within the theory of analytically uniform spaces. In particular, the authors will also discuss the persistence of the superoscillatory behavior when superoscillating sequences are taken as initial values of the Schrödinger equation and other equations.
In this Ebook edition of The Land Between, author Jeff Manion uses the biblical story of the Israelites’ journey through Sinai desert as a metaphor for being in undesired, transitional space. After enduring generations of slavery in Egypt, the descendants of Jacob travel through the desert (the land between) toward their new home in Canaan. They crave the food of their former home in Egypt and despise their present environment. They are unable to go back and incapable of moving forward. The Land Between explores the way in which the Israelites’ reactions can provide insight and guidance on how to respond to God during our own seasons of difficult transition. It also provides fresh biblical insight for people traveling through undesired transitions—foreclosure, unemployment, parents in declining health, post-graduate uncertainty, business failure—who are looking for hope, guidance, and encouragement. While it is possible to move through transitions and learn little, they provide our greatest opportunity for spiritual growth. God desires to meet us in our chaos and emotional upheaval, and he intends for us to encounter his goodness and provision during these upsetting seasons.
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