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Mathematical function
mathematics, the Chebyshev function is either a scalarising function (Tchebycheff function) or one of two related functions. The first Chebyshev function ϑ(x) or
Chebyshev_function
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
Sequence of mathematical functions
mathematics, the Chebyshev rational functions are a sequence of functions which are both rational and orthogonal. They are named after Pafnuty Chebyshev. A rational
Chebyshev_rational_functions
Type of analog or digital filter
Chebyshev filters are analog or digital filters that have a steeper roll-off than Butterworth filters, and have either passband ripple (type I) or stopband
Chebyshev_filter
Type of function
in families of rational orthogonal functions called Legendre rational functions and Chebyshev rational functions. Solutions of linear differential equations
Orthogonal_functions
Function on an integer n which is log(p) if n equals p^k and zero otherwise
Re(s) > σ0. The second Chebyshev function ψ ( x ) {\displaystyle \psi (x)} is the summatory function of the von Mangoldt function: ψ ( x ) = ∑ p k ≤ x log
Von_Mangoldt_function
Function representing the number of primes less than or equal to a given number
\zeta (s)=s\int _{0}^{\infty }\Pi _{0}(x)x^{-s-1}\,\mathrm {d} x} The Chebyshev function weights primes or prime powers pn by log p: ϑ ( x ) = ∑ p ≤ x log
Prime-counting_function
Result on density of prime numbers
to 3,000,000. Chebyshev proved it in 1852 and so it is also called the Bertrand–Chebyshev theorem or Chebyshev's theorem. Chebyshev's theorem can also
Bertrand's_postulate
Bound on probability of a random variable being far from its mean
In probability theory, Chebyshev's inequality (also called the Bienaymé–Chebyshev inequality) provides an upper bound on the probability of deviation of
Chebyshev's_inequality
polynomials Chebyshev polynomials Synchrotron function Riemann zeta function: A special case of Dirichlet series. Riemann Xi function Dirichlet eta function: An
List of mathematical functions
List_of_mathematical_functions
Roots of the Chebyshev polynomials of the first kind
In numerical analysis, Chebyshev nodes (also called Chebyshev points or a Chebyshev grid) are a set of specific algebraic numbers used as nodes for polynomial
Chebyshev_nodes
Product of the first "n" prime numbers
{\displaystyle 12} is composite. Primorials are related to the first Chebyshev function ϑ ( n ) {\displaystyle \vartheta (n)} by ln ( n # ) = ϑ ( n ) .
Primorial
Function used in signal processing
is 3. Minimizes the Chebyshev norm of the side-lobes for a given main lobe width. The zero-phase Dolph–Chebyshev window function w 0 [ n ] {\displaystyle
Window_function
Topics referred to by the same term
ψ ( n ) {\displaystyle \psi (n)} the Chebyshev function ψ ( x ) {\displaystyle \psi (x)} the polygamma function ψ m ( z ) {\displaystyle \psi ^{m}(z)}
Psi_function
Mathematical metric
In mathematics, Chebyshev distance (or Tchebychev distance), maximum metric, or L∞ metric is a metric defined on a real coordinate space where the distance
Chebyshev_distance
Topics referred to by the same term
Chebyshev may refer to: Pafnuty Chebyshev: A Russian mathematician Chebyshev function: Number-theory functions Chebyshev polynomials Chebyshev filter Chebyshev's
Chebyshev_(disambiguation)
Characterization of how many integers are prime
Pafnuty Chebyshev attempted to prove the asymptotic law of distribution of prime numbers. His work is notable for the use of the zeta function ζ(s), for
Prime_number_theorem
Penultimate letter in the Greek alphabet
constant, the division polynomials, and the supergolden ratio The second Chebyshev function Water potential in movement of water between plant cells In biochemistry
Psi_(Greek)
Number of integers coprime to and less than n
Hardy & Wright 1979, thm. 326 Hardy & Wright 1979, thm. 327 In fact Chebyshev's theorem (Hardy & Wright 1979, thm.7) and Mertens' third theorem is all
Euler's_totient_function
Eighth letter of the Greek alphabet
variable in trigonometry A special function ϑ(z; τ) of several complex variables θ. The first Chebyshev function θ(x) in prime number theory The potential
Theta
Polynomial sequence
interval [−1,1] with respect to the weight function (1 − x2)α–1/2. They generalize Legendre polynomials and Chebyshev polynomials, and are special cases of
Gegenbauer_polynomials
Chebyshev function in number theory Chebyshev integral Chebyshev iteration Chebyshev method Chebyshev nodes Chebyshev polynomials and the "Chebyshev form"
List of things named after Pafnuty Chebyshev
List_of_things_named_after_Pafnuty_Chebyshev
direction between function values at a set of Chebyshev nodes and coefficients of a function in Chebyshev polynomial basis. Like the Chebyshev polynomials,
Discrete_Chebyshev_transform
Conjecture on zeros of the zeta function
{\displaystyle x\geq 73.2} , where ψ ( x ) {\displaystyle \psi (x)} is Chebyshev's second function. Adrian Dudek proved that the Riemann hypothesis implies that
Riemann_hypothesis
Function specifying the behavior of a component in an electronic or control system
a transfer function (also known as system function or network function) of a system, sub-system, or component is a mathematical function that models
Transfer_function
Summatory function of the Möbius function
A curious relation given by Mertens himself involving the second Chebyshev function is ψ ( x ) = M ( x 2 ) log 2 + M ( x 3 ) log 3 + M ( x 4 ) log
Mertens_function
Topics referred to by the same term
Melchior Islands, Antarctica Chebyshev function Dedekind psi function Digamma function Polygamma functions Stream function, in two-dimensional flows Polar
Psi
Analytic function in mathematics
Chebyshev extended the definition to Re(s) > 1. The above series is a prototypical Dirichlet series that converges absolutely to an analytic function
Riemann_zeta_function
Function whose domain is the positive integers
} The second Chebyshev function ψ(x) is the summation function of the von Mangoldt function just below. Λ(n), the von Mangoldt function, is 0 unless the
Arithmetic_function
Symbols for constants, special functions
the stream function in fluid dynamics the reciprocal Fibonacci constant the second Chebyshev function in number theory the polygamma function in mathematics
Greek letters used in mathematics, science, and engineering
Greek_letters_used_in_mathematics,_science,_and_engineering
Topics referred to by the same term
sequences Chebyshev's equioscillation theorem, on the approximation of continuous functions with polynomials The statement that if the function π ( x )
Chebyshev's_theorem
Topics referred to by the same term
first Chebyshev function ϑ ( x ) {\displaystyle \vartheta (x)} , the sum of the logarithm of all primes ≤ x {\displaystyle \leq x} Feferman's function, θ
Theta function (disambiguation)
Theta_function_(disambiguation)
Linkages of different dimensions with the same output motion
four-bar linkage coupler cognates, the Roberts–Chebyshev Theorem, after Samuel Roberts and Pafnuty Chebyshev, states that each coupler curve can be generated
Cognate_linkage
Family of probability distributions
Poisson–gamma distribution and they exhibit multifractality. The second Chebyshev function ψ(x) is given by, ψ ( x ) = ∑ p ^ k ≤ x log p ^ = ∑ n ≤ x Λ ( n
Tweedie_distribution
Smallest positive number divisible by two integers
ideals is always an ideal). Anomalous cancellation Coprime integers Chebyshev function Weisstein, Eric W. "Least Common Multiple". mathworld.wolfram.com
Least_common_multiple
dx=B(x;1+p,1+q),} where B ( x ; a , b ) {\displaystyle B(x;a,b)} is an incomplete beta function. Weisstein, Eric W. "Chebyshev Integral". MathWorld. v t e
Chebyshev_integral
Mathematical concept
in 1895: it started with a proof of the following formula for the Chebyshev's function ψ ψ 0 ( x ) = 1 2 π i ∫ σ − i ∞ σ + i ∞ ( − ζ ′ ( s ) ζ ( s ) )
Explicit formulae for L-functions
Explicit_formulae_for_L-functions
Mathematical function
function naturally appears when the derivative of the log-likelihood is taken for finding the maxima. Polygamma function Trigamma function Chebyshev expansions
Digamma_function
Mathematical description of quantum state
space. These include the Legendre and Laguerre polynomials as well as Chebyshev polynomials, Jacobi polynomials and Hermite polynomials. All of these
Wave_function
Extension of the factorial function
1093/IMANUM/12.4.519. Werner, Helmut; Collinge, Robert (1961). "Chebyshev approximations to the Gamma Function". Math. Comput. 15 (74): 195–197. doi:10.1090/S0025-5718-61-99220-1
Gamma_function
Concept in probability theory
(sometimes, calling it the first Chebyshev inequality, while referring to Chebyshev's inequality as the second Chebyshev inequality) or Bienaymé's inequality
Markov's_inequality
Theory of getting acceptably close inexact mathematical calculations
the function, using the Chebyshev polynomials instead of the usual trigonometric functions. If one calculates the coefficients in the Chebyshev expansion
Approximation_theory
M is called a Chebyshev set. As an example, if (X,d) is a Euclidean space (Rn with the Euclidean distance), then a set M is a Chebyshev set if and only
Metric_projection
Theorem
continuous functions using polynomials when the merit function is the maximum difference (uniform norm). Its discovery is attributed to Chebyshev. Let f {\displaystyle
Equioscillation_theorem
Special mathematical function defined as sin(x)/x
W. "Sinc Function". mathworld.wolfram.com. Retrieved 2023-06-07. Merca, Mircea (2016-03-01). "The cardinal sine function and the Chebyshev–Stirling numbers"
Sinc_function
Mathematical mentod
In numerical analysis Chebyshev–Gauss quadrature is an extension of Gaussian quadrature method for approximating the value of integrals of the following
Chebyshev–Gauss_quadrature
Formal power series
polynomial sequences generated by more complex generating functions include: Appell polynomials Chebyshev polynomials Difference polynomials Generalized Appell
Generating_function
Transcendental single-variable function
Clausen, Glaisher, and L-functions" (PDF). Archived from the original (PDF) on 1 December 2025. Kölbig, Kurt Siegfried (1995). "Chebyshev coefficients for the
Clausen_function
Types of special mathematical functions
Richard (1961). "Evaluation of the Incomplete Gamma Function of Imaginary Argument by Chebyshev Polynomials". Math. Comp. 15 (73): 7–11. doi:10
Incomplete_gamma_function
Measure of the shape of a function
intervals (Hamburger moment problem). In the mid-nineteenth century, Pafnuty Chebyshev became the first person to think systematically in terms of the moments
Moment_(mathematics)
Chebyshev: Chebyshev polynomials Elwin Bruno Christoffel, Darboux: Christoffel–Darboux relation Cyclotomic polynomials H. G. Dawson: Dawson function Richard
List of eponyms of special functions
List_of_eponyms_of_special_functions
Increasing sequence of reduced fractions
1/2}2\sin(\pi r)\right)^{2}} where ψ(N) is the second Chebyshev function. Since the Euler's totient function is directly connected to the gcd so is the number
Farey_sequence
Failure of convergence in interpolation
[citation needed] Chebyshev interpolation (i.e., on Chebyshev nodes) converges uniformly for every absolutely continuous function. Chebyshev nodes Compare
Runge's_phenomenon
statistics and number theory, author of the Chebyshev's inequality, Chebyshev distance, Chebyshev function, Chebyshev equation etc. Sergei Chernikov, significant
List of Russian mathematicians
List_of_Russian_mathematicians
electronic filters. (These functions are sometimes called Chebyshev rational functions, not to be confused with certain other functions of the same name). Rational
Elliptic_rational_functions
The fractional Chebyshev collocation (FCC) method is an efficient spectral method for solving a system of linear fractional-order differential equations
Fractional Chebyshev collocation method
Fractional_Chebyshev_collocation_method
Product of numbers from 1 to n
(1932). "Beweis eines Satzes von Tschebyschef" [Proof of a theorem of Chebyshev] (PDF). Acta Litt. Sci. Szeged (in German). 5: 194–198. Zbl 0004.10103
Factorial
Algorithm to approximate functions
algorithm used to find simple approximations to functions, specifically, approximations by functions in a Chebyshev space that are the best in the uniform norm
Remez_algorithm
Baltic German mathematician
some day you will be recompensed but I am still grateful to Hitler". Chebyshev function Isoperimetric inequality Low-rank approximation List of Baltic German
Erhard_Schmidt
Function defined by a hypergeometric series
Legendre polynomials, Chebyshev polynomials, Gegenbauer polynomials, Zernike polynomials can be written in terms of hypergeometric functions using 2 F 1 ( −
Hypergeometric_function
Probability distribution
the standard normal cumulative distribution function using Hart's algorithms and approximations with Chebyshev polynomials. Dia (2023) proposes the following
Normal_distribution
Signal processing filter
\infty } the elliptic rational function becomes a Chebyshev polynomial, and therefore the filter becomes a Chebyshev type I filter, with ripple factor
Elliptic_filter
In numerical linear algebra, the Chebyshev iteration is an iterative method for determining the solutions of a system of linear equations. The method
Chebyshev_iteration
Method for estimating new data within known data points
interpolation or restricting attention to Chebyshev polynomials. Linear interpolation uses a linear function for each of intervals [xk,xk+1]. Spline interpolation
Interpolation
Type of discrete orthogonal polynomials
In mathematics, discrete Chebyshev polynomials, or Gram polynomials, are a type of discrete orthogonal polynomials used in approximation theory, introduced
Discrete Chebyshev polynomials
Discrete_Chebyshev_polynomials
Polynomial sequence
scarcely recognizable form, and studied in detail by Pafnuty Chebyshev in 1859. Chebyshev's work was overlooked, and they were named later after Charles
Hermite_polynomials
statistics and number theory, author of the Chebyshev's inequality, Chebyshev distance, Chebyshev function, Chebyshev equation Boris Delaunay, inventor of Delaunay
List_of_Russian_scientists
Signal processing method
Thomas Parks in 1972, is an iterative algorithm for finding the optimal Chebyshev finite impulse response (FIR) filter. The Parks–McClellan algorithm is
Parks–McClellan filter design algorithm
Parks–McClellan_filter_design_algorithm
Type of metric geometry
for the Chebyshev distance (L∞ metric) on a plane is also a square with side length 2r parallel to the coordinate axes, so planar Chebyshev distance
Taxicab_geometry
statistics and number theory, author of the Chebyshev's inequality, Chebyshev distance, Chebyshev function, Chebyshev equation Boris Delaunay, inventor of Delaunay
List_of_Russian_people
Operation on formal power series
sequence's generating function provides a method of converting the generating function for one sequence into a generating function enumerating another.
Generating function transformation
Generating_function_transformation
Number theory related to prime numbers
In number theory, Chebyshev's bias is the phenomenon that most of the time, there are more primes of the form 4k + 3 than of the form 4k + 1, up to the
Chebyshev's_bias
Vector distance function
limiting case of p {\displaystyle p} approaching infinity, we obtain the Chebyshev distance: lim p → ∞ ( ∑ i = 1 n | x i − y i | p ) 1 p = max i = 1 n |
Minkowski_distance
Space of bounded sequences
by L ∞ . {\displaystyle L^{\infty }.} Uniform norm – Function in mathematical analysis Chebyshev distance – Mathematical metric "Elementary set theory
L-infinity
Type of signal processing filter
phase response in the passband than Chebyshev Type I/Type II and elliptic filters can achieve. A transfer function of a third-order low-pass Butterworth
Butterworth_filter
Result of repeatedly applying a mathematical function
equation. On a logarithmic scale, this reduces to the nesting property of Chebyshev polynomials, Tm(Tn(x)) = Tm n(x), since Tn(x) = cos(n arccos(x)). The
Iterated_function
Second-order linear differential equation
Chebyshev's equation is the second order linear differential equation ( 1 − x 2 ) d 2 y d x 2 − x d y d x + p 2 y = 0 , {\displaystyle (1-x^{2}){d^{2}y
Chebyshev_equation
Numerical integration method
Briefly, the function f ( x ) {\displaystyle f(x)} to be integrated is evaluated at the N {\displaystyle N} extrema or roots of a Chebyshev polynomial and
Clenshaw–Curtis_quadrature
Family of orthogonal polynomials
hypergeometric orthogonal polynomials, introduced by Pafnuty Chebyshev in 1875 (Chebyshev 1907) and rediscovered by Wolfgang Hahn (Hahn 1949). The Hahn
Hahn_polynomials
Method in numerical analysis
monomials. It generalizes to more than just Chebyshev polynomials; it applies to any class of functions that can be defined by a three-term recurrence
Clenshaw_algorithm
Property of having a unique mode or maximum value
the Vysochanskij–Petunin inequality, a refinement of the Chebyshev inequality. The Chebyshev inequality guarantees that in any probability distribution
Unimodality
Function in analytic number theory
Borwein used approximations involving Chebyshev polynomials to produce a method for efficient evaluation of the eta function. If d k = n ∑ ℓ = 0 k ( n + ℓ −
Dirichlet_eta_function
Function in mathematical analysis
|f(s)|:s\in S\,\right\}.} This norm is also called the supremum norm, the Chebyshev norm, the infinity norm, or, when the supremum is in fact the maximum
Uniform_norm
Dimensionless parameter
Frequency response of the low-pass prototype filters is characterized by Chebyshev function of the first kind. The formulas were first published in. They have
Coupling coefficient of resonators
Coupling_coefficient_of_resonators
where Tn is the Chebyshev polynomial.[citation needed] The following relationship holds for the sine function ∏ k = 1 n − 1 sin ( k π n
List of trigonometric identities
List_of_trigonometric_identities
Special function defined by an integral
380–385. doi:10.1007/BF02162153. S2CID 121794086. Cody, William J. (1968). "Chebyshev approximations for the Fresnel integrals" (PDF). Math. Comp. 22 (102):
Fresnel_integral
Type of orthogonal polynomials
polynomials (including as a special case the Gegenbauer polynomials, Chebyshev polynomials, and Legendre polynomials). They have many important applications
Classical orthogonal polynomials
Classical_orthogonal_polynomials
Russian mathematician (1857–1918)
the university. Among the Saint Petersburg mathematics professors were Chebyshev and his students Aleksandr Nikolaevich Korkin and Yegor Ivanovich Zolotarev
Aleksandr_Lyapunov
Family of power series in mathematics
polynomials and Chebyshev polynomials. A wide range of integrals of elementary functions can be expressed using the hypergeometric function, e.g.: ∫ 0 x
Generalized hypergeometric function
Generalized_hypergeometric_function
Type of artificial neural network architecture
"Kolmogorov-Arnold Networks are Radial Basis Function Networks". arXiv:2405.06721 [cs.LG]. SS, S.; AR, K.; KP, A. (2024). "Chebyshev Polynomial-based Kolmogorov-Arnold
Kolmogorov–Arnold_Networks
American mathematician
{{\sqrt {x}}\,\ln ^{2}x}{8\pi }}} for all x ≥ 73.2, based on the second Chebyshev function ψ(x). Rossiter, Margaret W. (1995), Women Scientists in America: Before
Lowell_Schoenfeld
Summability method for a class of divergent series
{\displaystyle \psi (x)\sim x} where ψ {\displaystyle \psi } is the second Chebyshev function. Lambert series Abel–Plana formula Abelian and tauberian theorems
Lambert_summation
Numerical technique
1 < t 1 < … < t p < 1 {\displaystyle -1<t_{1}<\ldots <t_{p}<1} be the Chebyshev nodes of order p ≥ 2 , {\displaystyle p\geq 2,} and let u 1 ( y ) , …
Fast_multipole_method
Type of analog linear filter in electronics
allpass filter, to implement fractional delays. Bessel function Butterworth filter Chebyshev filter Comb filter Elliptic filter Group delay and phase
Bessel_filter
Collection of random variables
Pierre-Simon Laplace, Abraham de Moivre, Carl Gauss, Siméon Poisson and Pafnuty Chebyshev, most of the mathematical community did not consider probability theory
Stochastic_process
Mathematical transform that expresses a function of time as a function of frequency
takes a function as input and outputs another function that describes the extent to which various frequencies are present in the original function. The output
Fourier_transform
Polynomials used in approximation theory
the Chebyshev polynomials where accuracy of approximation near the origin is of less importance. Zolotarev polynomials differ from the Chebyshev polynomials
Zolotarev_polynomials
Approximation of the definite integral of a function
weights include 1 1 − x 2 {\textstyle {\frac {1}{\sqrt {1-x^{2}}}}} (Chebyshev–Gauss) and 1 − x 2 {\textstyle {\sqrt {1-x^{2}}}} . One may also want
Gaussian_quadrature
Type of curve in geometry
refined versions include error terms, weighted counting functions analogous to the Chebyshev functions, and arithmetic refinements for special surfaces such
Prime_geodesic
Prime fulfilling an inequality related to the prime-counting function
prime-counting function. In 1919, Ramanujan published a new proof of Bertrand's postulate which, as he notes, was first proved by Chebyshev. At the end of
Ramanujan_prime
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