An Etymological Dictionary of Astronomy and Astrophysics
English-French-Persian

The process of decomposing any function of time or space into a sum of sinusoidal functions using the → Fourier series and → Fourier transforms. In other words, any data analysis procedure that describes or measures the fluctuations in a time series by comparing them with sinusoids. Fourier analysis is an essential component of much of modern applied and pure mathematics. It forms an exceptionally powerful analytical tool for solving various problems in many areas of mathematics, physics, engineering, biology, finance, etc. and has opened up new realms of knowledge.

After the French mathematician Baron Jean Baptiste Joseph Fourier (1768-1830), whose work had a tremendous impact on the physical applications of mathematics; → analysis.

One of the coefficients a_n or b_n of cos (nx) and sin (nx) respectively in the → Fourier series representation of a function. They are expressed by:
a_n = (1/π) ∫f(x) cos nx dx, for n≥ 0, summed over 0 to 2π
b_n = (1/π) ∫f(x) sin nx dx, for n≥ 1, summed over 0 to 2π.

→ Fourier analysis; → series.

An integral used in the → Fourier transform.

→ Fourier analysis; → integral.

A mathematical tool used for decomposing a → periodic function into an infinite sum of sine and cosine functions. The general form of the Fourier series for a function f(x) with period 2π is:
(1/2) a₀ + Σ (a_n cos (nx) + b_n sin (nx), summed from n = 1 to ∞,
where a_n and b_n are the → Fourier coefficients, measuring the strength of contribution from each harmonic. The functions cos (nx) and sin (nx) can be used in this way because they satisfy the → orthogonality conditions. For the problem of convergence of the Fourier series see → Dirichlet conditions. The Fourier series play a very important role in the study of periodic phenomena, because they allow one to decompose a large number of complex problems into simpler ones. The generalization of this method, called the → Fourier transform, makes it possible to also decompose non-periodic functions into harmonic components. See also → complex Fourier series, → Parseval's theorem.

→ Fourier analysis; → series.

Any finite periodic motion may be analyzed into components, each of which is a simple harmonic motion of definite and determinable amplitudes and phase.

→ Fourier analysis; → theorem.

A powerful mathematical tool which is the generalization of the → Fourier series for the analysis of non-periodic functions. The Fourier transform transforms a function defined on physical space into a function defined on the space of frequencies, whose values quantify the "amount" of each periodic frequency contained in the original function. The inverse Fourier transform then reconstructs the original function from its transformed frequency components. The integral F(α) = ∫ f(u)e^-iαudu is called the Fourier transform of F(x) = (1/2π)∫ f(α)e^iαxdx, both integrals from -∞ to + ∞.

→ Fourier analysis; → transform.

The end of a solar eclipse marked by the disk of the Moon completely passing away from the disk of the Sun.

From M.E. fourthe, O.E. féowertha, from four, from O.E. feower, from P.Gmc. *petwor- (cf. Du. and Ger. vier, O.N. fjorir, Dan. fire, Sw. fyra), from PIE *qwetwor (cf. Mod.Pers. cahâr, Av. caθwar-, catur-, Skt. catvarah, Gk. tessares, L. quattuor) + -th a suffix used in the formation of ordinal numbers, from M.E. -the, -te, O.E. -tha, -the; cf. O.N. -thi, -di; L. -tus; Gk -tos; → contact.

Parmâs, → contact; cahârom cardinal form from cahâr "four," cognate with E. four, as above.

A small depression, approximately 1.5 millimeters in diameter, at the back of the → retina. It forms the area of the most acute vision.

Contraction of fovea centralis, from L. fovea "small pit," of unknown origin.

Lake-ye zard, literally "yellow spot," from laké, → spot, zard, → yellow.

A geometrical or physical structure that repeats itself or nearly repeats itself on many different scales of magnification.

From Fr. fractale, term coined by Benoit Mandelbrot (1975), from frac(tus) "broken, uneven", → fraction, + -ale→ -al.

Barxâl, from barx, → fraction, + -âl→ -al.

The postulate that the concentrations of matter in the Universe follow a → fractal structure over a wide range of scales.

→ fractal; → cosmology.

A → hierarchial structure that can be likened to fractals.

→ fractal; → structure

A rational number of the form a/b where a is called the numerator and b is called the denominator.

From L.L. fractionem (nom. fractio) "a breaking in pieces," from frangere "to break," from PIE base *bhreg- "to break" (cf. Goth. brikan, O.E. brecan "to break;" Lith. brasketi "crash, crack").

Barxé, from barx "lot, portion," variant bahr, from Mid.Pers. bahr "lot, share, portion," Av. baxəδra- "portion."

1) Math.: Pertaining to fractions; constituting a fraction.
2) Chemistry: Of or relating to any process by which parts of a mixture are separated by exploiting differences in their physical properties, such as their boiling points, solubility, or other characteristics.

→ fraction; → -al.

The portion of the 4π → steradians of the sky that a radiotelescope can observe from a given location on Earth over a 24-hour time interval.

→ fractional; → sky; → coverage.

1) To break something up into smaller parts.
2) To separate a mixture into ingredients or portions having different properties, as by distillation or otherwise.

From → fraction + -ate a suffix forming verbs or nouns, from L. -atus, -ata, -atum.

Barxândan, from barx, barxé, → fraction, + -ândan suffix of transitive verbs.

1) Any of various methods of separating the components of a mixture into fractions of different properties.
2) → isotope fractionation

Verbal noun from → fractionate.

1) (n.) A part broken off or detached.
2) (v.intr.) To collapse or break into pieces.
3) (v.tr.) To break something into pieces.

From L. fragmentum, from frangere "to break."

1) Latté, lat, variant laxt, laxté "piece, part;" pâré "piece, part, portion, fragment;" Mid.Pers. pârag "piece, part, portion; gift, offering, bribe;" Av. pāra- "debt," from par- "to remunerate, equalize; to condemn;" PIE *per- "to sell, hand over, distribute; to assigne;" cf. L. pars "part, piece, side, share," portio "share, portion;" Gk. peprotai "it has been granted;" Skt. purti- "reward;" Hitt. pars-, parsiya- "to break, crumble."
2) and 3) Verbal forms.

Generally, the process of breaking up into smaller parts. In particular, the splitting of a large molecular cloud into smaller, denser clumps. → cloud fragmentation.

From → fragmenta + -ation, a combination of -ate and -ion, used to form nouns from stems in -ate.

Latpâreš, verbal noun from latpâridan, → fragment.

The succession of physical events that results in the breaking of a → molecular cloud into several → fragments.

→ fragmentation; → process.

1) A border or case for enclosing a picture, mirror, etc.; a structure for admitting or enclosing something.
2) One of the successive pictures on a roll of movie film or videotape.
3) In computers, the information or image on a screen or monitor at any one time.

Frame, from M.E. verb framen "to prepare (timber)," from O.E. framian "to avail, profit."; cf. O.H.G. (gi)framon "to do."

1) Cârcub "frame," from câr, contraction of cahâr "four" (→ four) + cub "stick, satff, beam," Mid.Pers. côp "wood, stick." 2) Tasvirak from Ar. tasvir "image" + -ak suffix of relation and similarity (as in poštak, dastak, nâxonak), → fibril.