For other uses, see Spectrum (disambiguation).

The noun spectrum (plural: spectra) has a variety of meanings. In most modern usages, there is a unifying theme of a variety of possible cases between extremes at either end. Older usages were not necessarily on that same unifying theme, but nonetheless led to the modern ones through a sequence of events set out below. Some modern usages in mathematics evolved out of that unifying theme but may be difficult to recognize as fitting into it.

Ghosts

Originally a spectrum was what is now called a spectre, i.e., a phantom or apparition. Spectral evidence is testimony about what was done by spectres of persons not present physically, or hearsay evidence about what ghosts or apparitions of Satan said. It was used to convict a number of persons of witchcraft at Salem, Massachusetts in the late 17th century.

Modern (17th through 21st centuries) meaning in the physical sciences

High-resolution spectrum of the Sun showing thousands of elemental absorption lines (fraunhofer lines).

In the 17th century the word spectrum was introduced into optics, 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 as a function of frequency or wavelength. Max Planck later realized that frequency represents electromagnetic energy:

E = hν

where E is the energy of a photon, h is Planck's constant, and ν is the frequency of the light.

The word spectrum then took on the obvious analogous meaning in reference to other sorts of waves, such as sound wave, or other sorts of decomposition into frequency components. Thus a spectrum is a usually 2-dimensional plot, of a compound signal, depicting the components by another measure. Sometimes, the word spectrum refers to the compound signal itself, such as the "spectrum of visible light", a reference to those electromagnetic waves which are visible to the human eye.

• The electromagnetic spectrum is the power spectrum of an electromagnetic signal. It can be measured by spectroscopy.
• The optical spectrum is the electromagnetic spectrum of visible light
• The power spectrum is the distribution of the energy of a function in the frequency domain, which is actually the same as the magnitude of the frequency spectrum. See spectroscopy, spectrum of gravity waves.
• Spectroscopy is the study of spectra.
• The spectrogram is the result of calculating the frequency spectrum of windowed frames of the signal.
• In telecommunication, a spread spectrum is any of certain kinds of signal transmission.

Physical acoustics of music

• See timbre. Spectrum is one of the determinants of the timbre or quality of a sound. It is the relative strength of pitches called harmonics and partials (collectively overtones) at various frequencies usually above the fundamental frequency, which is the actual note named (eg. an A).

Intuitive approach

A spectrum in physics relates to waves: light waves, sound waves, etc. Waves can be represented by graphs that look like waves in the sea and represent some physical quantity as a function of time (or space). However, waves can also be represented as a function of a frequency. A tone without overtones, or monochromatic light can be represented as a sine function of time, but also as a single line as a function of frequency. Sound and light in general can be represented as a superposition of "pure" sine functions with different frequencies and varying intensities. A graph showing the intensities of the "pure" sine functions composing some actual time dependent phenomenon as a function of the frequency is said to show the "spectrum" of that phenomenon.

Essentially, the construction of a spectrum from some time dependent phenomenon is a purely mathematical construct. However, our eyes and ears also perform the mathematical transformation from the "time domain" to the "frequency" domain. Our eyes distinguish colors (light of different frequencies) and our ears distinguish "low" and "high" tones (sounds of different frequencies). Therefore, we are inclined to assume that spectra are real, and to forget that these phenomena always can be represented as a function of time as well.

Spectra are also apparently very realistic, very tangible in communication technology. Parts of the radio spectrum are auctioned and sold like commodities. This is based on the fact, that it is technically fairly easy to filter parts of a spectrum. There is actually just a single ratio signal in the air. However, mathematically it can be seen as a composition of signals of different frequencies. And technologically it is easy to select just the signal related to a certain range of frequencies. That may be related to a radio station. (It should be noted however that such "frequency division multipexing" of radio stations is not the only technology available. There is also "time division multiplexing" where a single channel (from the user perspective) is not related to a confined band of frequencies).

Glass and water transmit light of different frequencies at different speeds. That causes "chromatic aberration" of lenses, and, better known, rainbows. Again, a spectrum appears to be a tangible thing.

The description of a time dependent phenomenon by a spectrum, i.e. a function of a frequency involves what is called: a transformation from the time domain into the frequency domain. Such transformations are also important from an abstract mathematical perspective. Differentiating (calculating the derivative) of a function of time becomes a multiplication in the time domain. Thus such transformation can be helpful to solve differential equations (which is normally a difficult endeavor).

The transformation of a wave into a superposition of sine waves allegedly was first proposed by Jean Baptiste Joseph Fourier (1768-1830), who studied the propagation of heat waves. Today, the above transformation is still called a "fourier transform" (which can be performed from numerical data using the "fast fourier transform" algorithm, invented much later).

For theoretical mathematicians, the choice of the sine function as the basic "composing" function is just one of the possible alternatives. Other sets of "orthogonal" functions may be chosen for an "integral transformations", yielding different kinds of spectra.

Meanings of spectrum in mathematics

The various meanings of the word spectrum in mathematics are derived (some fairly directly; some less so) from the meanings in the physical sciences.

• The frequency spectrum is the result of a Fourier-related transform of a mathematical function into the frequency domain.
• The spectrum of an operator has to do with the invertibility of an operator in function spaces.
• The spectrum of a matrix is the spectrum of an operator where the matrix is considered as operator. Precisely, it is the set of the matrix's eigenvalues.
• A (strange) construction, similar to the frequency spectrum, is the cepstrum of the quefrency.
• Finding a spectrum is a method of cycles analysis.
• The spectrum of a ring is the set of prime ideals of a ring.
• The spectrum of a Boolean algebra is the Stone space of the Boolean algebra.

Other disciplines

The meanings of spectrum in some other disciplines, including pharmacology, politics, and psychology evolved by analogy with the meanings in the physical sciences: just as dispersed colored light ranged from one end of the rainbow to the other, so also other things that range from one extreme to another were called spectra.

In pharmacology

• The spectrum of activity of an antibiotic evaluates how wide a range of infections can be treated.

In politics

• There is a political spectrum which is said to go from left to right.

In psychology

• There exists the concept of a bipolar spectrum.