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Set of polynomials where any two are orthogonal to each other
mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonal to each other
Orthogonal_polynomials
In mathematics, the multiple orthogonal polynomials (MOPs) are orthogonal polynomials in one variable that are orthogonal with respect to a finite family
Multiple orthogonal polynomials
Multiple_orthogonal_polynomials
mathematics, a sequence of discrete orthogonal polynomials is a sequence of polynomials that are pairwise orthogonal with respect to a discrete measure
Discrete orthogonal polynomials
Discrete_orthogonal_polynomials
Orthogonal symmetric polynomial family
many other families of orthogonal polynomials, such as Jack polynomials and Hall–Littlewood polynomials and Askey–Wilson polynomials, which in turn include
Macdonald_polynomials
Sequence of differential equation solutions
generalized Laguerre polynomials, as will be done here (alternatively associated Laguerre polynomials or, rarely, Sonine polynomials, after their inventor
Laguerre_polynomials
Various meanings of the terms
families of functions are used to form an orthogonal basis, such as in the contexts of orthogonal polynomials, orthogonal functions, and combinatorics. In optics
Orthogonality
Pair of polynomial sequences
The Chebyshev polynomials are two sequences of orthogonal polynomials related to the cosine and sine functions, notated as T n ( x ) {\displaystyle T_{n}(x)}
Chebyshev_polynomials
Polynomial sequence
In mathematics, the Zernike polynomials are a sequence of polynomials that are orthogonal on the unit disk. Named after optical physicist Frits Zernike
Zernike_polynomials
In mathematics, the big q-Legendre polynomials are an orthogonal family of polynomials defined in terms of Heine's basic hypergeometric series as P n
Big_q-Legendre_polynomials
Mathematical method for approximating solutions to differential and integral equations
subspace spanned by the first N vectors in some orthogonal polynomial basis, such as the Legendre polynomials. Top F1 teams began switching from Quasi-Static
Collocation_method
Polynomial with reversed root positions
self-reciprocal polynomial satisfy ai = an−i for all i. Reciprocal polynomials have several connections with their original polynomials, including: deg
Reciprocal_polynomial
Statistics concept
interval (0, 1). Although the correlation can be reduced by using orthogonal polynomials, it is generally more informative to consider the fitted regression
Polynomial_regression
Complex-valued function
formula, using the fact that the Hermite polynomials are a special case of the associated Laguerre polynomials: H 2 n ( x ) = ( − 1 ) n 2 2 n n ! L n (
Mehler_kernel
Algorithm to smooth data points
to fitting m data points by a simple polynomial in the subsidiary variable, z, is to use orthogonal polynomials. Y = b 0 P 0 ( z ) + b 1 P 1 ( z ) ⋯ +
Savitzky–Golay_filter
Geometry of the location of polynomial roots
real roots of a polynomial Root-finding of polynomials – Algorithms for finding zeros of polynomials Square-free polynomial – Polynomial with no repeated
Geometrical properties of polynomial roots
Geometrical_properties_of_polynomial_roots
Mathematical problems related to differential equations
Riemann–Hilbert problems play a central role in integrable systems, orthogonal polynomials, random matrix theory, inverse monodromy, and asymptotic analysis
Riemann–Hilbert_problem
Formula for the Legendre polynomials
it. The term is also used to describe similar formulas for other orthogonal polynomials. Askey (2005) describes the history of the Rodrigues formula in
Rodrigues'_formula
Special mathematical functions defined on the surface of a sphere
harmonic polynomials R 3 → C that are homogeneous of degree ℓ } . {\displaystyle \mathbf {A} _{\ell }=\left\{{\text{harmonic polynomials }}\mathbb
Spherical_harmonics
Physics model in statistical mechanics
Retrieved 13 November 2023. Mukhin, E.; Varchenko, A. (2007). "Multiple Orthogonal Polynomials and a Counterexample to the Gaudin Bethe Ansatz Conjecture"
Gaudin_model
Mathematical operation
goal of lattice basis reduction is to find a basis with short, nearly orthogonal vectors when given an integer lattice basis as input. This is realized
Lattice_reduction
Generalization of the Jack polynomial
polynomials, and is in turn generalized by the Heckman–Opdam polynomials and Macdonald polynomials. The Jack function J κ ( α ) ( x 1 , x 2 , … , x m ) {\displaystyle
Jack_function
Algorithm for the line of best fit for a two-dimensional dataset
line through the centroid is a line of best orthogonal fit. If S ≠ 0 {\displaystyle S\neq 0} , the orthogonal regression line goes through the centroid
Deming_regression
Number with an integer power equal to 1
coefficient in the nth cyclotomic polynomial. Many restrictions are known about the values that cyclotomic polynomials can assume at integer values. For
Root_of_unity
Function in discrete mathematics
periodic functions, which can often be approximated well by trigonometric polynomials. In practice, the DFT is usually computed by efficient fast Fourier transform
Discrete_Fourier_transform
Most widely known generalized inverse of a matrix
acts as a traditional inverse of A {\displaystyle A} on the subspace orthogonal to the kernel. In the following discussion, the following conventions
Moore–Penrose_inverse
Process of developing trajectory performance
by a spline of a different order. The name comes from the use of orthogonal polynomials in the state and control splines. In pseudospectral discretization
Trajectory_optimization
Concepts from linear algebra
semidefinite (PSD) matrix yields an orthogonal basis of eigenvectors, each of which has a nonnegative eigenvalue. The orthogonal decomposition of a PSD matrix
Eigenvalues_and_eigenvectors
Matrix decomposition
approximating roots of polynomials exist, such as Newton's method, but in general it is impractical to compute the characteristic polynomial and then apply these
Eigendecomposition of a matrix
Eigendecomposition_of_a_matrix
Type of vector space in math
are frequently used to study orthogonal polynomials, because different families of orthogonal polynomials are orthogonal with respect to different weighting
Hilbert_space
mean Classical orthogonal polynomials Hermite polynomials Laguerre polynomials Jacobi polynomials Gegenbauer polynomials Legendre polynomials Euclidean space
List_of_real_analysis_topics
Matrix with the same number of rows and columns
^{\mathsf {T}}A\mathbf {y} .} An orthogonal matrix is a square matrix with real entries whose columns and rows are orthogonal unit vectors (i.e., orthonormal
Square_matrix
1966 mathematics textbook by Serge Lang
the decomposition of an orthogonal operator with respect to invariant subspaces. The next three chapters discuss polynomials in an abstract-algebraic
Linear_Algebra_(book)
Theorem about polynomials
only irreducible polynomials; any real polynomial of odd degree must have an irreducible factor of odd degree, which (having no multiple roots) must have
Complex conjugate root theorem
Complex_conjugate_root_theorem
Mathematical formula expressing equality
equation is a polynomial equation (commonly called also an algebraic equation) in which the two sides are polynomials. The sides of a polynomial equation contain
Equation
Element of a basis for a function space
basis in an inner-product space Orthogonal polynomials Fourier analysis and Fourier series Harmonic analysis Orthogonal wavelet Biorthogonal wavelet Radial
Basis_function
Hermite polynomials Hermite polynomials, a sequence of polynomials orthogonal with respect to the normal distribution Continuous q-Hermite polynomials Continuous
List of things named after Charles Hermite
List_of_things_named_after_Charles_Hermite
Orthonormalization of a set of vectors
S=\{\mathbf {v} _{1},\ldots ,\mathbf {v} _{k}\}} for k ≤ n and generates an orthogonal set S ′ = { u 1 , … , u k } {\displaystyle S'=\{\mathbf {u} _{1},\ldots
Gram–Schmidt_process
Mathematical function
Selberg. It has applications in statistical mechanics, multivariable orthogonal polynomials, random matrix theory, Calogero–Moser–Sutherland model, and Knizhnik–Zamolodchikov
Selberg_integral
Theory of getting acceptably close inexact mathematical calculations
a polynomial of degree N. One can obtain polynomials very close to the optimal one by expanding the given function in terms of Chebyshev polynomials and
Approximation_theory
Mathematical problem
combinatorics, two Latin squares of the same size (order) are said to be orthogonal if when superimposed the ordered paired entries in the positions are all
Mutually orthogonal Latin squares
Mutually_orthogonal_Latin_squares
Matrix representing a Euclidean rotation
quadratic polynomial. We can minimize it in the usual way, by finding where its derivative is zero. For a 3 × 3 matrix, the orthogonality constraint
Rotation_matrix
Algorithm used for frequency estimation and radio direction finding
{\displaystyle \{\mathbf {v} _{1},\mathbf {v} _{2},\ldots ,\mathbf {v} _{M}\}} are orthogonal to each other. If the eigenvalues of R x {\displaystyle \mathbf {R} _{x}}
MUSIC_(algorithm)
Concept in statistics
Jong Sung. "Orthogonal Polynomial Contrasts" (PDF). Retrieved 27 April 2012. Clark, James M. (2007). Intermediate Data Analysis: Multiple Regression and
Contrast_(statistics)
Matrices similar to diagonal matrices
spaces. Additional geometric visualizations of orthogonal diagonalization, including reflection and orthogonal projection matrices, are available at Wikimedia
Diagonalizable_matrix
About simultaneous modular congruences
case of Chinese remainder theorem for polynomials is Lagrange interpolation. For this, consider k monic polynomials of degree one: P i ( X ) = X − x i
Chinese_remainder_theorem
Array of numbers
+1 or −1. A special orthogonal matrix is an orthogonal matrix with determinant +1. As a linear transformation, every orthogonal matrix with determinant
Matrix_(mathematics)
dimensions Discrete Chebyshev polynomials — polynomials orthogonal with respect to a discrete measure Favard's theorem — polynomials satisfying suitable 3-term
List of numerical analysis topics
List_of_numerical_analysis_topics
Hall–Petresco identity from group theory for a workable theory of polynomials. In particular the polynomial sequence comes with a definite degree. A family of conjectures
Nilsequence
Statistical approach
attain a specific target for) the response variable(s) of interest. Orthogonality The property that allows individual effects of the k-factors to be estimated
Response_surface_methodology
248-dimensional exceptional simple Lie group
large square matrices consisting of polynomials, the Lusztig–Vogan polynomials, an analogue of Kazhdan–Lusztig polynomials introduced for reductive groups
E8_(mathematics)
Numerical methods for matrix eigenvalue calculation
eigenvalues of a normal matrix are orthogonal. The null space and the image (or column space) of a normal matrix are orthogonal to each other. For any normal
Eigenvalue_algorithm
Group of unitary matrices
this J {\displaystyle J} is orthogonal; writing all the groups as matrix groups fixes a J {\displaystyle J} (which is orthogonal) and ensures compatibility)
Unitary_group
Parameterization of a rotation into a unit vector and angle
Plugging the three eigenvalues 1 and e±iθ and their associated three orthogonal axes in a Cartesian representation into Mercer's theorem is a convenient
Axis–angle_representation
Model for approximating non-linear effects, similar to a Taylor series
identification orthogonalization, Volterra series must be rearranged in terms of orthogonal non-homogeneous G operators (Wiener series): y ( n ) = ∑ p H p x ( n )
Volterra_series
Strategy board game
"long king", which was a promoted piece capable of moving multiple squares in any orthogonal direction. Both evolutionary paths sacrificed something from
Checkers
constant coefficients a n = b n = 1 {\displaystyle a_{n}=b_{n}=1} . Orthogonal polynomials Pn all have a TTRR with respect to degree n, P n ( x ) = ( A n x
Three-term recurrence relation
Three-term_recurrence_relation
Form of a matrix indicating its eigenvalues and their algebraic multiplicities
minimal polynomial P of a square matrix A is the unique monic polynomial of least degree, m, such that P(A) = 0. Alternatively, the set of polynomials that
Jordan_normal_form
Study of mathematical knots
one should determine that the polynomial does not change under the three Reidemeister moves. Many important knot polynomials can be defined in this way.
Knot_theory
Problem in combinatorial optimization
2021). "Improving Schroeppel and Shamir's Algorithm for Subset Sum via Orthogonal Vectors". arXiv:2010.08576 [cs.DS]. Schroeppel, Richard; Shamir, Adi (August
Knapsack_problem
Algebraic structure in linear algebra
all polynomials p ( t ) {\displaystyle p(t)} forms an algebra known as the polynomial ring: using that the sum of two polynomials is a polynomial, they
Vector_space
Relation between sides of a right triangle
identity can be extended to sums of more than two orthogonal vectors. If v1, v2, ..., vn are pairwise-orthogonal vectors in an inner-product space, then application
Pythagorean_theorem
Statistical theorem in the analysis of variance
{\displaystyle \sum _{i}r_{i}=\operatorname {rank} (A)} . But after an orthogonal transform, A = diag ( I M , 0 ) {\displaystyle A=\operatorname {diag}
Cochran's_theorem
In mathematics, invariant of square matrices
of an orthogonal basis, the magnitude of the determinant is equal to the product of the lengths of the basis vectors. For instance, an orthogonal matrix
Determinant
special case of that, since the Legendre polynomials are the special case of the ultraspherical polynomial when α = 1 / 2 {\displaystyle \alpha =1/2}
Zonal_spherical_harmonics
Infinite series of Bessel functions
}(\gamma _{n}x/b)\,f(x)\,x\,dx.} Orthogonality Generalized Fourier series Hankel transform Kapteyn series Neumann polynomial Schlömilch's series Magnus, Wilhelm;
Fourier–Bessel_series
Matrix operation which flips a matrix over its diagonal
whose transpose is equal to its inverse is called an orthogonal matrix; that is, A is orthogonal if A T = A − 1 . {\displaystyle \mathbf {A} ^{\text{T}}=\mathbf
Transpose
Pattern defining an infinite sequence of numbers
series. Special cases of these lead to recurrence relations for the orthogonal polynomials, and many special functions. For example, the solution to J n +
Recurrence_relation
Decomposition of periodic functions
\mathbf {a} _{i}} are three linearly independent but not necessarily orthogonal vectors. Let us consider some function f ( r ) {\displaystyle f(\mathbf
Fourier_series
Form of a matrix
space to the real orthogonal group O ( n ) {\displaystyle \mathrm {O} (n)} at the identity matrix; formally, the special orthogonal Lie algebra. In this
Skew-symmetric_matrix
Discrete Fourier transform algorithm
a recursive factorization of the polynomial z n − 1 {\displaystyle z^{n}-1} , here into real-coefficient polynomials of the form z m − 1 {\displaystyle
Fast_Fourier_transform
Group of 𝑛 × 𝑛 invertible matrices
and symmetries of vector spaces in general, as well as the study of polynomials. The modular group may be realised as a quotient of the special linear
General_linear_group
particular invariant rings, see invariants of a binary form, symmetric polynomials. For geometric terms used in invariant theory see the glossary of classical
Glossary_of_invariant_theory
Square root of the determinant of a skew-symmetric square matrix
first proved by Cayley (1849), who cites Jacobi for introducing these polynomials in work on Pfaffian systems of differential equations. Cayley obtains
Pfaffian
mathematician, known for Rogers–Askey–Ismail polynomials, Al-Salam–Ismail polynomials and Chihara–Ismail polynomials Peter Medawar, Lebanese-British biologist
List of modern Arab scientists and engineers
List_of_modern_Arab_scientists_and_engineers
Construct related to weighted sums and averages
See the entry on orthogonal polynomials for examples of weighted orthogonal functions. Center of mass Numerical integration Orthogonality Weighted mean Linear
Weight_function
Method of frequency estimation
eigenvector corresponding to the minimum eigenvalue. This eigenvector is orthogonal to each of the signal vectors. The frequency estimates may be determined
Pisarenko harmonic decomposition
Pisarenko_harmonic_decomposition
is an orthogonal expansion for nonlinear functionals closely related to the Volterra series and having the same relation to it as an orthogonal Hermite
Wiener_series
Concept in multilinear algebra and representation theory
f(\mathbf {Q} \mathbf {A} \mathbf {Q} ^{T})=f(\mathbf {A} )} for all orthogonal Q {\displaystyle \mathbf {Q} } . This means that a formula expressing
Invariants_of_tensors
Number with a real and an imaginary part
of all such polynomials is denoted by R [ X ] {\displaystyle \mathbb {R} [X]} . Since sums and products of polynomials are again polynomials, this set R
Complex_number
y)D_{2}} unitary (in a real field or a field of a finite characteristic, or orthogonal in the field of complex numbers), C ∗ 2 ( x , y ) {\displaystyle C^{*2}(x
Hamiltonian_cycle_polynomial
Field of combinatorics using complex analysis
Asymptotics" (PDF). Retrieved 4 November 2023. Szegő, Gabor (1975). Orthogonal Polynomials (4th ed.). American Mathematical Society. Wilf, Herbert S. (2006)
Analytic_combinatorics
R n {\displaystyle \mathbb {R} ^{n}} it yields a lattice basis with orthogonality defect at most n n {\displaystyle n^{n}} , unlike the 2 n 2 / 2 {\displaystyle
Korkine–Zolotarev lattice basis reduction algorithm
Korkine–Zolotarev_lattice_basis_reduction_algorithm
Least squares approximation of linear functions to data
_{3}x^{2}} . Cubic, quartic and higher polynomials. For regression with high-order polynomials, the use of orthogonal polynomials is recommended. Numerical smoothing
Linear_least_squares
Branch of mathematics
one recover the set of polynomials which generate it? If U is any subset of An, define I(U) to be the set of all polynomials whose vanishing set contains
Algebraic_geometry
formula that gives the characters of representations in terms of Schur polynomials, χ λ ( g ) = s λ ( x 1 , … , x n ) {\displaystyle \chi _{\lambda }(g)=s_{\lambda
Representations of classical Lie groups
Representations_of_classical_Lie_groups
Commutative ring with no zero divisors other than zero
2^{n}\mathbb {Z} \supset 2^{n+1}\mathbb {Z} \supset \cdots } Rings of polynomials are integral domains if the coefficients come from an integral domain
Integral_domain
Any of a set of standard configurations of Redundant Arrays of Independent Disks
methods, including dual check data computations (parity and Reed–Solomon), orthogonal dual parity check data and diagonal parity, have been used to implement
Standard_RAID_levels
Mathematical optimization algorithm
\mathbf {v} \rangle .} Two vectors are conjugate if and only if they are orthogonal with respect to this inner product. Being conjugate is a symmetric relation:
Conjugate_gradient_method
Overview of GPS conversion formulas
datums for good statistics, multiple regression methods are used to fit the parameters of these polynomials. The polynomials, along with the fitted coefficients
Geographic coordinate conversion
Geographic_coordinate_conversion
Branch of discrete mathematics
partitions, and is closely related to q-series, special functions and orthogonal polynomials. Originally a part of number theory and analysis, it is now considered
Combinatorics
Statistical methods to improve the quality of manufactured goods
proposed extending each experiment with an "outer array" (possibly an orthogonal array); the "outer array" should simulate the random environment in which
Taguchi_methods
Smallest convex set containing a given set
univariate polynomials and Newton polytopes of multivariate polynomials are convex hulls of points derived from the exponents of the terms in the polynomial, and
Convex_hull
entries are the squares of the absolute values of the entries of some orthogonal matrix Precision matrix — a symmetric n×n matrix, formed by inverting
List_of_named_matrices
Matrix-valued random variable
McLaughlin, K.T.-R.; Venakides, S.; Zhou, X. (1997). "Asymptotics for polynomials orthogonal with respect to varying exponential weights". International Mathematics
Random_matrix
Movement with a fixed point is rotation
n rotations in orthogonal planes of rotation, though these planes need not be uniquely determined, and a rigid motion may fix multiple axes. Also, any
Euler's_rotation_theorem
Mathematical algorithm
signal in the time domain into a sequence of coefficients based on an orthogonal basis of small finite waves, or wavelets. The transform can be easily
Fast_wavelet_transform
Method for dividing a secret among multiple parties
Homomorphic secret sharing – A simplistic decentralized voting protocol. Orthogonal array – Used to construct some threshold schemes. Publicly verifiable
Secret_sharing
Quantum search algorithm
operator U ω {\displaystyle U_{\omega }} is a reflection at the hyperplane orthogonal to | ω ⟩ {\displaystyle |\omega \rangle } for vectors in the plane spanned
Grover's_algorithm
Design of tasks
optimal design for polynomial regression was suggested by Gergonne in 1815. In 1918, Kirstine Smith published optimal designs for polynomials of degree six
Design_of_experiments
Study of triangles in other spaces than the Euclidean plane
S2CID 7837108 Yamaleev, Robert M. (2005), "Complex algebras on n-order polynomials and generalizations of trigonometry, oscillator model and Hamilton dynamics"
Generalized_trigonometry
MULTIPLE ORTHOGONAL-POLYNOMIALS
MULTIPLE ORTHOGONAL-POLYNOMIALS
Boy/Male
Hindu, Indian, Tamil
Multiple
Boy/Male
Hebrew
God will multiply.
Boy/Male
Hebrew
God will multiply.
Boy/Male
Hebrew Gaelic
God will multiply.
Boy/Male
Hebrew
God shall multiply.
Boy/Male
Hebrew American Latin
God will multiply.
Boy/Male
Australian, Vietnamese
Many; Multiple
Boy/Male
Hebrew
God will multiply.
Boy/Male
Muslim
Multiple lights. Luster.
Boy/Male
Hebrew
God will multiply.
Boy/Male
Hebrew Spanish
God will multiply.
Girl/Female
Hebrew
God will multiply.
Girl/Female
Hebrew
God will multiply.
Boy/Male
Hindu, Indian
Un Countable; Multiple; Countless
Girl/Female
Hebrew
God will multiply.
Girl/Female
Hebrew
God will multiply.
Boy/Male
Hebrew
God will multiply.
Boy/Male
Hebrew Spanish
God will multiply.
Boy/Male
Hebrew
God will multiply.
Boy/Male
Dutch, German, Hebrew
God will Multiply
MULTIPLE ORTHOGONAL-POLYNOMIALS
MULTIPLE ORTHOGONAL-POLYNOMIALS
Girl/Female
Muslim/Islamic
Light of the moon
Boy/Male
Christian & English(British/American/Australian)
Bright as the Sun
Boy/Male
Hebrew
Comfort.
Girl/Female
Hindu, Indian, Marathi
Attractive
Girl/Female
Bihari, Gujarati, Hindu, Indian, Kannada, Marathi, Punjabi, Sikh
Delight; Bright
Boy/Male
Hindu, Indian
Covered with Hides; Protected; Sheltered
Boy/Male
Arabic, Muslim
Sun of Allah
Girl/Female
Arabic, Muslim
Star
Girl/Female
Arabic, Muslim
Adorning the Assembly
Girl/Female
British, English
Elf; Power
MULTIPLE ORTHOGONAL-POLYNOMIALS
MULTIPLE ORTHOGONAL-POLYNOMIALS
MULTIPLE ORTHOGONAL-POLYNOMIALS
MULTIPLE ORTHOGONAL-POLYNOMIALS
MULTIPLE ORTHOGONAL-POLYNOMIALS
n.
A quantity containing another quantity a number of times without a remainder.
a.
Containing more than once, or more than one; consisting of more than one; manifold; repeated many times; having several, or many, parts.
adv.
Perpendicularly; at right angles; as, a curve cuts a set of curves orthogonally.
v. t.
To multiply; to make manifold.
n.
The number which is to be multiplied by another number called the multiplier. See Note under Multiplication.
n.
One who, or that which, multiplies or increases number.
v. t.
To multiply; to increase.
a.
Having many flues; as, a multiflue boiler. See Boiler.
a.
Pertaining to, or evincing, orthodoxy; orthodox.
p. pr. & vb. n.
of Multiply
n.
The number by which another number is multiplied. See the Note under Multiplication.
n.
The multiplier.
adv.
So as to multiply.
a.
Right-angled; rectangular; as, an orthogonal intersection of one curve with another.
v. t.
To add (any given number or quantity) to itself a certain number of times; to find the product of by multiplication; thus 7 multiplied by 8 produces the number 56; to multiply two numbers. See the Note under Multiplication.
a.
Manifold; multiple.
n.
Multiplied diversity.
n.
The number by which another number is multiplied; a multiplier.
a.
Tending to multiply; having the power to multiply, or incease numbers.
imp. & p. p.
of Multiply