Noncommutative harmonic analysisIn mathematics, noncommutative harmonic analysis is the field in which results from Fourier analysis are extended to topological groups that are not commutative. Since locally compact abelian groups have a well-understood theory, Pontryagin duality, which includes the basic structures of Fourier series and Fourier transforms, the major business of non-commutative harmonic analysis is usually taken to be the extension of the theory to all groups G that are locally compact.
Riesz projectorIn mathematics, or more specifically in spectral theory, the Riesz projector is the projector onto the eigenspace corresponding to a particular eigenvalue of an operator (or, more generally, a projector onto an invariant subspace corresponding to an isolated part of the spectrum). It was introduced by Frigyes Riesz in 1912. Let be a closed linear operator in the Banach space . Let be a simple or composite rectifiable contour, which encloses some region and lies entirely within the resolvent set () of the operator .
Spectrum (physical sciences)In the physical sciences, the term spectrum was introduced first into optics by Isaac Newton in the 17th century, referring to the range of colors observed when white light was dispersed through a prism. Soon the term referred to a plot of light intensity or power as a function of frequency or wavelength, also known as a spectral density plot. Later it expanded to apply to other waves, such as sound waves and sea waves that could also be measured as a function of frequency (e.g., noise spectrum, sea wave spectrum).
Spectral theory of ordinary differential equationsIn mathematics, the spectral theory of ordinary differential equations is the part of spectral theory concerned with the determination of the spectrum and eigenfunction expansion associated with a linear ordinary differential equation. In his dissertation, Hermann Weyl generalized the classical Sturm–Liouville theory on a finite closed interval to second order differential operators with singularities at the endpoints of the interval, possibly semi-infinite or infinite.
Resolvent setIn linear algebra and operator theory, the resolvent set of a linear operator is a set of complex numbers for which the operator is in some sense "well-behaved". The resolvent set plays an important role in the resolvent formalism. Let X be a Banach space and let be a linear operator with domain . Let id denote the identity operator on X. For any , let A complex number is said to be a regular value if the following three statements are true: is injective, that is, the corestriction of to its image has an inverse ; is a bounded linear operator; is defined on a dense subspace of X, that is, has dense range.
