The Fourier transform (FT) decomposes a function into its constituent frequencies. A special case is the expression of a musical chord in terms of the volumes and frequencies of its constituent notes. The term Fourier transform refers to both the frequency domain representation and the mathematical operation that associates the frequency domain representation to a function of time. The Fourier transform of a function of time is itself a complex-valued function of frequency, whose magnitude (modulus) represents the amount of that frequency present in the original function, and whose argument is the phase offset of the basic sinusoid in that frequency. The Fourier transform is not limited to functions of time, but the domain of the original function is commonly referred to as the time domain. There is also an inverse Fourier transform that mathematically synthesizes the original function from its frequency domain representation.
In mathematics, a Fourier series is a periodic function composed of harmonically related sinusoids, combined by a weighted summation. With appropriate weights, one cycle of the summation can be made to approximate an arbitrary function in that interval. As such, the summation is a synthesis of another function. The discrete-time Fourier transform is an example of Fourier series. The process of deriving the weights that describe a given function is a form of Fourier analysis. For functions on unbounded intervals, the analysis and synthesis analogies are Fourier transform and inverse transform.
In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular group, and also satisfying a growth condition. The theory of modular forms therefore belongs to complex analysis but the main importance of the theory has traditionally been in its connections with number theory. Modular forms appear in other areas, such as algebraic topology, sphere packing, and string theory.
In mathematics, the Poisson summation formula is an equation that relates the Fourier series coefficients of the periodic summation of a function to values of the function's continuous Fourier transform. Consequently, the periodic summation of a function is completely defined by discrete samples of the original function's Fourier transform. And conversely, the periodic summation of a function's Fourier transform is completely defined by discrete samples of the original function. The Poisson summation formula was discovered by Siméon Denis Poisson and is sometimes called Poisson resummation.
In mathematics, in particular in the theory of modular forms, a Hecke operator, studied by Hecke (1937), is a certain kind of "averaging" operator that plays a significant role in the structure of vector spaces of modular forms and more general automorphic representations.
In mathematics, even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking additive inverses. They are important in many areas of mathematical analysis, especially the theory of power series and Fourier series. They are named for the parity of the powers of the power functions which satisfy each condition: the function is an even function if n is an even integer, and it is an odd function if n is an odd integer.
In mathematics, the Ramanujan conjecture, due to Srinivasa Ramanujan (1916, p.176), states that Ramanujan's tau function given by the Fourier coefficients τ(n) of the cusp form Δ(z) of weight 12
In mathematics, the Selberg trace formula, introduced by Selberg (1956), is an expression for the character of the unitary representation of G on the space L2(G/Γ) of square-integrable functions, where G is a Lie group and Γ a cofinite discrete group. The character is given by the trace of certain functions on G.
In Fourier analysis, a multiplier operator is a type of linear operator, or transformation of functions. These operators act on a function by altering its Fourier transform. Specifically they multiply the Fourier transform of a function by a specified function known as the multiplier or symbol. Occasionally, the term multiplier operator itself is shortened simply to multiplier. In simple terms, the multiplier reshapes the frequencies involved in any function. This class of operators turns out to be broad: general theory shows that a translation-invariant operator on a group which obeys some regularity conditions can be expressed as a multiplier operator, and conversely. Many familiar operators, such as translations and differentiation, are multiplier operators, although there are many more complicated examples such as the Hilbert transform.
In mathematics, the Riemann–Lebesgue lemma, named after Bernhard Riemann and Henri Lebesgue, states that the Fourier transform or Laplace transform of an L1 function vanishes at infinity. It is of importance in harmonic analysis and asymptotic analysis.
In mathematics, a Kloosterman sum is a particular kind of exponential sum. They are named for the Dutch mathematician Hendrik Kloosterman, who introduced them in 1926 when he adapted the Hardy–Littlewood circle method to tackle a problem involving positive definite diagonal quadratic forms in four as opposed to five or more variables, which he had dealt with in his dissertation in 1924.
The Selberg zeta-function was introduced by Atle Selberg (1956). It is analogous to the famous Riemann zeta function
In mathematical analysis, Wiener's tauberian theorem is any of several related results proved by Norbert Wiener in 1932. They provide a necessary and sufficient condition under which any function in L1 or L2 can be approximated by linear combinations of translations of a given function.
In mathematics, a Schwartz–Bruhat function, named after Laurent Schwartz and François Bruhat, is a complex valued function on a locally compact abelian group, such as the adeles, that generalizes a Schwartz function on a real vector space. A tempered distribution is defined as a continuous linear functional on the space of Schwartz–Bruhat functions.
In mathematics, Siegel modular forms are a major type of automorphic form. These generalize conventional elliptic modular forms which are closely related to elliptic curves. The complex manifolds constructed in the theory of Siegel modular forms are Siegel modular varieties, which are basic models for what a moduli space for abelian varieties should be and are constructed as quotients of the Siegel upper half-space rather than the upper half-plane by discrete groups.
In mathematics, Maass forms or Maass wave forms are studied in the theory of automorphic forms. Maass forms are complex-valued smooth functions of the upper half plane, which transform in a similar way under the operation of a discrete subgroup of as modular forms. They are Eigenforms of the hyperbolic Laplace Operator defined on and satisfy certain growth conditions at the cusps of a fundamental domain of . In contrast to the modular forms the Maass forms need not be holomorphic. They were studied first by Hans Maass in 1949.
In analytic number theory, the Petersson trace formula is a kind of orthogonality relation between coefficients of a holomorphic modular form. It is a specialization of the more general Kuznetsov trace formula.
In mathematics, the Wiener algebra, named after Norbert Wiener and usually denoted by A(T), is the space of absolutely convergent Fourier series. Here T denotes the circle group.
In mathematics, a weak Maass form is a smooth function on the upper half plane, transforming like a modular form under the action of the modular group, being an eigenfunction of the corresponding hyperbolic Laplace operator, and having at most linear exponential growth at the cusps. If the eigenvalue of under the Laplacian is zero, then is called a harmonic weak Maass form, or briefly a harmonic Maass form.
In mathematics, a Voronoi formula is an equality involving Fourier coefficients of automorphic forms, with the coefficients twisted by additive characters on either side. It can be regarded as a Poisson summation formula for non-abelian groups. The Voronoi (summation) formula for GL(2) has long been a standard tool for studying analytic properties of automorphic forms and their L-functions. There have been numerous results coming out the Voronoi formula on GL(2). The concept is named after Georgy Voronoy.
