Favard's theorem is a result in functional analysis and measure theory concerning the Fourier transforms of functions in certain spaces. Specifically, it deals with the conditions under which the Fourier transform of a function in \( L^1 \) space can be represented as a limit of averages of the values of the function.
Dual \( q \)-Hahn polynomials are a class of orthogonal polynomials that arise in the context of basic hypergeometric series and \( q \)-analysis. They can be considered a $q$-analogue of classical orthogonal polynomials, such as the Hahn polynomials.
Dual Hahn polynomials are a class of orthogonal polynomials that arise in the context of approximation theory, special functions, and mathematical physics. They are part of a broader family of hypergeometric orthogonal polynomials and can be viewed as the dual version of Hahn polynomials.
Discrete \( q \)-Hermite polynomials are a family of orthogonal polynomials that arise in the context of the theory of \( q \)-special functions and quantum calculus. They represent a \( q \)-analog of the classical Hermite polynomials, which are well-known in the study of orthogonal polynomials.
Discrete orthogonal polynomials are a class of polynomials that are orthogonal with respect to a discrete measure or inner product. This means that they are specifically defined for sequences of points in a discrete set (often integers or specific values in the real line) rather than continuous intervals.
Discrete Chebyshev polynomials are a sequence of orthogonal polynomials defined on a discrete set of points, typically related to the Chebyshev polynomials of the first kind. These discrete polynomials arise in various applications, including numerical analysis, approximation theory, and computing discrete Fourier transforms. The discrete Chebyshev polynomials are defined based on the characteristic roots of the Chebyshev polynomials, which correspond to specific points on an interval.
Continuous \( q \)-Laguerre polynomials are a family of orthogonal polynomials that generalize the classical Laguerre polynomials by incorporating the concept of \( q \)-calculus, which deals with discrete analogs of calculus concepts. These polynomials arise in various areas of mathematics and physics, including approximation theory, special functions, and quantum mechanics.
Continuous q-Jacobi polynomials are a family of orthogonal polynomials that generalize the classical Jacobi polynomials in the context of q-analogs, which are important in various areas of mathematics, including combinatorics, number theory, and quantum calculus.
Continuous q-Hermite polynomials are a set of orthogonal polynomials that arise in the context of q-calculus and are related to various areas in mathematics and physics, especially in the theory of special functions and quantum groups. They are a q-analogue of the classical Hermite polynomials. ### Definition and Properties 1.
Continuous \( q \)-Hahn polynomials are a class of orthogonal polynomials that arise in the study of special functions, particularly in the context of \( q \)-series and quantum groups. They are a part of a broader family of \( q \)-analogues of classical orthogonal polynomials, which includes the \( q \)-Hahn, \( q \)-Jacobi, and others.
Continuous dual \( q \)-Hahn polynomials are a family of orthogonal polynomials that arise in the context of basic hypergeometric series and quantum group theory. They are a part of the \( q \)-Askey scheme, which organizes various families of orthogonal polynomials based on their properties and connections to special functions.
The continuous dual Hahn polynomials are a family of orthogonal polynomials that arise in the context of special functions and quantum calculus. They are part of the broader family of dual Hahn polynomials and have applications in various areas, including mathematical physics, combinatorics, and approximation theory. The continuous dual Hahn polynomials can be defined in terms of a three-parameter family of polynomials, which can be specified using recurrence relations or generating functions.
Continuous big \( q \)-Hermite polynomials are a family of orthogonal polynomials that arise in the study of special functions, particularly in the context of quantum calculus or \( q \)-analysis. They are part of the wider family of \( q \)-orthogonal polynomials, which generalize classical orthogonal polynomials by introducing a parameter \( q \).
Continuous Hahn polynomials are a family of orthogonal polynomials that arise in the context of approximation theory and quantum physics. They are part of the broader family of hypergeometric orthogonal polynomials and are linked to various mathematical fields, including special functions, approximation theory, and the theory of orthogonal polynomials.
Classical orthogonal polynomials are a set of orthogonal polynomials that arise in various areas of mathematics, especially in the context of approximation theory, numerical analysis, and mathematical physics. These polynomials are defined on specific intervals and with respect to certain weight functions, leading to their orthogonality properties.
The Christoffel–Darboux formula is a significant result in the theory of orthogonal polynomials. It provides a way to express sums of products of orthogonal polynomials in a concise form. Typically, the formula relates the orthogonal polynomials defined on a specific interval with respect to a weight function.
Biorthogonal polynomials are a generalization of orthogonal polynomials where two different systems of polynomials are orthogonal with respect to two different measures.
Big \( q \)-Laguerre polynomials are a specific family of orthogonal polynomials that arise in the context of \( q \)-analysis, a generalization of classical analysis that incorporates the parameter \( q \). These polynomials are particularly useful in various areas of mathematics and mathematical physics, including quantum calculus, combinatorics, and orthogonal polynomial theory.
Bessel polynomials are a series of orthogonal polynomials that are related to Bessel functions, which are solutions to Bessel's differential equation. The Bessel polynomials, denoted usually by \( P_n(x) \), are defined using the formula: \[ P_n(x) = \sum_{k=0}^{n} \binom{n}{k} \frac{(-1)^k}{k!} (x/2)^k.
Bateman polynomials, named after the mathematician Harry Bateman, are a family of orthogonal polynomials that arise in various contexts in mathematics, particularly in the theory of special functions and approximation theory. They are often denoted by \( B_n(x) \) and defined using a specific recurrence relation or via their generating functions.