Spaces of Harmonic Functions
Since the Laplace equation is linear, the set of harmonic functions defined on a given domain is, in fact, a vector space. By defining suitable norms and/or inner products, one can exhibit sets of harmonic functions which form Hilbert or Banach spaces. In this fashion, one obtains such spaces as the Hardy space, Bloch space, and Bergman space.
Read more about this topic: Potential Theory
Famous quotes containing the words spaces of, spaces, harmonic and/or functions:
“through the spaces of the dark
Midnight shakes the memory
As a madman shakes a dead geranium.”
—T.S. (Thomas Stearns)
“Every true man is a cause, a country, and an age; requires infinite spaces and numbers and time fully to accomplish his design;—and posterity seem to follow his steps as a train of clients.”
—Ralph Waldo Emerson (1803–1882)
“For decades child development experts have erroneously directed parents to sing with one voice, a unison chorus of values, politics, disciplinary and loving styles. But duets have greater harmonic possibilities and are more interesting to listen to, so long as cacophony or dissonance remains at acceptable levels.”
—Kyle D. Pruett (20th century)
“When Western people train the mind, the focus is generally on the left hemisphere of the cortex, which is the portion of the brain that is concerned with words and numbers. We enhance the logical, bounded, linear functions of the mind. In the East, exercises of this sort are for the purpose of getting in tune with the unconscious—to get rid of boundaries, not to create them.”
—Edward T. Hall (b. 1914)