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Interpolation with trigonometric polynomials
In mathematics, trigonometric interpolation is interpolation with trigonometric polynomials. Interpolation is the process of finding a function which goes
Trigonometric_interpolation
Function in discrete mathematics
{\displaystyle \mathbf {X} } and Y {\displaystyle \mathbf {Y} } . The trigonometric interpolation polynomial p ( t ) = { 1 N [ X 0 + X 1 e i 2 π t + ⋯ + X N 2
Discrete_Fourier_transform
Method for estimating new data within known data points
rational functions using Padé approximant, and trigonometric interpolation is interpolation by trigonometric polynomials using Fourier series. Another possibility
Interpolation
Concept in mathematics
finite Fourier series. Trigonometric polynomials are widely used, for example in trigonometric interpolation applied to the interpolation of periodic functions
Trigonometric_polynomial
topics Trigonometric interpolation — interpolation by trigonometric polynomials Discrete Fourier transform — can be viewed as trigonometric interpolation at
List of numerical analysis topics
List_of_numerical_analysis_topics
Form of interpolation
use of interpolation polynomials was to approximate values of important transcendental functions such as natural logarithm and trigonometric functions
Polynomial_interpolation
Fundamental trigonometric functions
polynomial is known as the trigonometric polynomial. The trigonometric polynomial's ample applications may be acquired in its interpolation, and its extension
Sine_and_cosine
Lists of values of mathematical functions
mathematics, tables of trigonometric functions are useful in a number of areas. Before the existence of pocket calculators, trigonometric tables were essential
Trigonometric_table
List of values of a mathematical function
mathematics, tables of trigonometric functions are useful in a number of areas. Before the existence of pocket calculators, trigonometric tables were essential
Mathematical_table
Type of mathematical expression
finite Fourier series. Trigonometric polynomials are widely used, for example in trigonometric interpolation applied to the interpolation of periodic functions
Polynomial
solutions found by interpolation in trigonometric tables. In the 13th century, Naṣīr al-Dīn al-Ṭūsī was the first to treat trigonometry as a mathematical
History_of_trigonometry
Overview of and topical guide to trigonometry
indicates how many times one number contains another Trigonometry Trigonometric functions Trigonometric identities Euler's formula Archimedes Aristarchus
Outline_of_trigonometry
Classification scheme for mathematics
(including Fourier analysis, Fourier transforms, trigonometric approximation, trigonometric interpolation, and orthogonal functions) 43: Abstract harmonic
Mathematics Subject Classification
Mathematics_Subject_Classification
Function used in computer graphics
In geometry, spherical linear interpolation, commonly abbreviated slerp, is a function which interpolates between two points on a sphere, such that spherical
Spherical linear interpolation
Spherical_linear_interpolation
Polish-American mathematician (1900–1992)
Uppsala. Trigonometric Series (Cambridge University Press 1959, 2002) Intégrales singulières (Springer-Verlag, 1971) Trigonometric Interpolation (University
Antoni_Zygmund
Topics referred to by the same term
(boundary value problem), an identity in calculus Lagrange's trigonometric identities, two trigonometric identities Lagrange's four-square theorem, a theorem
Lagrange's identity (disambiguation)
Lagrange's_identity_(disambiguation)
and Trigonometric Functions: BKM algorithm: computes elementary functions using a table of logarithms CORDIC: computes hyperbolic and trigonometric functions
List_of_algorithms
Numerical integration method
{\displaystyle a_{2k}} are the amplitudes of the unique bandlimited trigonometric interpolation polynomial passing through the N+1 points where f(cos θ) is evaluated
Clenshaw–Curtis_quadrature
Number with an integer power equal to 1
(This fact was first noted by Gauss when solving the problem of trigonometric interpolation.) The straightforward application of U or its inverse to a given
Root_of_unity
German polymath and scholar (1777–1855)
found their similar Cooley–Tukey algorithm. He developed it as a trigonometric interpolation method, but the paper Theoria Interpolationis Methodo Nova Tractata
Carl_Friedrich_Gauss
Branch of mathematics
transform); a true cosine+sine DFT was used by Gauss in 1805 for trigonometric interpolation of asteroid orbits. Euler and Lagrange both discretized the vibrating
Fourier_analysis
Vector function in optics
{\boldsymbol {u}})} is usually not on the 4-D grid, DFST adopts trigonometric interpolation to compute the non-grid values. The algorithm consists of these
Light_field
Second-order polynomial interpolation formula
Brahmagupta's interpolation formula is a second-order polynomial interpolation formula developed by the Indian mathematician and astronomer Brahmagupta
Brahmagupta's interpolation formula
Brahmagupta's_interpolation_formula
Polish mathematician (1910–1940)
Marcinkiewicz obtained his doctorate based on his thesis entitled Trigonometric interpolation of absolutely continuous functions written under the supervision
Józef_Marcinkiewicz
Surveying technique
standard method of levelling in construction and surveying is called trigonometric levelling, which is preferred when levelling "out" to a number of points
Levelling
Lagrange's theorem (number theory) Lagrange's four-square theorem Lagrange's trigonometric identities Lagrange point colonization Lagrange (crater) Lagrange Island
List of things named after Joseph-Louis Lagrange
List_of_things_named_after_Joseph-Louis_Lagrange
can define L ¯ f d {\displaystyle {\bar {L}}_{f}^{d}} using trigonometric interpolation with these sample points: L ¯ f d ( s ^ , t ^ , u ^ , v ^ ) =
Light_field_microscopy
Hungarian mathematician
latter with G.H. Hardy. In 1916, he introduced the Riesz interpolation formula for trigonometric polynomials, which allowed him to give a new proof of Bernstein's
Marcel_Riesz
Mathematical analysis of frequency content of signals
the method to be convergent, a choice similar to that in the trigonometric interpolation section above should be used.) A linear differential equation
Multidimensional_transform
Algorithm for computing polynomial coefficients
algorithm, historically used for computing tables of logarithms and trigonometric functions.[citation needed] Charles Babbage's difference engine, an
Divided_differences
Overview of and topical guide to geometry
Ray Plane Bearing Angle Degree Minute Radian Circumference Diameter Trigonometric function Asymptotes Circular functions Periodic functions Law of cosines
Outline_of_geometry
Array that replaces runtime computation with a simpler array indexing operation
tables is to obtain the result of a trigonometry calculation, such as the sine of a value. Calculating trigonometric functions can substantially slow a
Lookup_table
Act of performing an initial computation before run time
implementations of digital trigonometric functions often use precomputed lookup tables to either provide coefficients for interpolation algorithms or to initialise
Precomputation
Algorithm for computing trigonometric, hyperbolic, logarithmic and exponential functions
digital computer, is a simple and efficient algorithm to calculate trigonometric functions, hyperbolic functions, square roots, multiplications, divisions
CORDIC
Pair of polynomial sequences
Chebyshev nodes because they are used as nodes in polynomial interpolation. Using the trigonometric definition and the fact that cos ( ( 2 k + 1 ) π 2 ) =
Chebyshev_polynomials
Simplification of the basic trigonometric functions
For small angles, the trigonometric functions sine, cosine, and tangent can be calculated with reasonable accuracy by the following simple approximations:
Small-angle_approximation
Approximate multiplication and division using formulas from trigonometry
logarithms that would supplant it. The trigonometric identities exploited by prosthaphaeresis relate products of trigonometric functions to sums. They include
Prosthaphaeresis
Method for estimating new data outside known data points
the basis of its relationship with another variable. It is similar to interpolation, which produces estimates between known observations, but extrapolation
Extrapolation
Mathematical function, inverse of an exponential function
{1}{d}}\log _{10}c}.} Trigonometric calculations were facilitated by tables that contained the common logarithms of trigonometric functions. Another critical
Logarithm
Formula to estimate the sine function
in one variable for the computation of the approximate values of the trigonometric sines discovered by Bhāskara I (c. 600 – c. 680), a seventh-century
Bhāskara I's sine approximation formula
Bhāskara_I's_sine_approximation_formula
Analytic function that does not satisfy a polynomial equation
are the exponential, trigonometric, and hyperbolic functions, and their inverses, such as the logarithm and inverse trigonometric functions. All special
Transcendental_function
Development of the mathematical function
of tables of trigonometric functions and their natural logarithms. These tables greatly simplified calculations in spherical trigonometry, which are central
History_of_logarithms
Association of one output to each input
image I. This is how inverse trigonometric functions are defined in terms of trigonometric functions, where the trigonometric functions are monotonic. Another
Function_(mathematics)
Indian mathematician (c. 598–c. 668)
sine-value of 225 (although the rest of its sine-table is lost), implying a trigonometric radius of R = 3438 approx= C(')/2π: a tradition followed, as we have
Brahmagupta
Digital signal generator
reduce the amount of memory required. This include various trigonometric expansions, trigonometric approximations and methods which take advantage of the
Numerically controlled oscillator
Numerically_controlled_oscillator
2nd century AD trigonometric table
geometer, and geographer Ptolemy in Egypt during the 2nd century AD, is a trigonometric table in Book I, chapter 11 of Ptolemy's Almagest, a treatise on mathematical
Ptolemy's_table_of_chords
Element of a basis for a function space
example, monomials are convenient for elementary polynomial calculations, trigonometric functions are useful for Fourier analysis, and locally supported functions
Basis_function
Field of mathematical analysis
for some 0 < α < 1 if and only if for every natural n there exists a trigonometric polynomial Pn of degree n such that max 0 ≤ x ≤ 2 π | f ( x ) − P n
Constructive_function_theory
Examples of curve-fitting compaction consisting of discretization and then interpolation are: Breaking of a continuous curve into a series of straight line segments
Curve-fitting_compaction
Polynomial equation of degree 3
of trigonometric functions of angles related to 2 π / 7 {\displaystyle 2\pi /7} satisfy cubic equations. Given the cosine (or other trigonometric function)
Cubic_equation
Limiting a signal to contain only low-frequency components
of trigonometric functions, and since f(t) is time-limited, this sum will be finite, so F 2 {\displaystyle F_{2}} will be actually a trigonometric polynomial
Bandlimiting
Polynomial equation of degree two
require using a different trigonometric form. To illustrate, let us assume we had available seven-place logarithm and trigonometric tables, and wished to
Quadratic_equation
unique rank Hermite-Sobolev spaces Hermite's cotangent identity, a trigonometric identity Hermite's criterion Hermite's identity, an identity on fractional
List of things named after Charles Hermite
List_of_things_named_after_Charles_Hermite
Indian mathematician-astronomer-astrologer (505–587)
spherical trigonometry by systematizing the use of jyā (half-chord/sine) and koṭi-jyā (cosine) functions. He formulated several fundamental trigonometric identities
Varāhamihira
Indian mathematician and astronomer (1114–1185)
knowledge of trigonometry, including the sine table and relationships between different trigonometric functions. He also developed spherical trigonometry, along
Bhāskara_II
Plimpton 322 and Maor, Eli (1993), "Plimpton 322: The Earliest Trigonometric Table?", Trigonometric Delights, Princeton University Press, pp. 30–34, ISBN 978-0-691-09541-7
Timeline_of_mathematics
Mathematics used in Ancient China
spherical trigonometry in calendar science and astronomical calculations. The polymath and official Shen Kuo (1031–1095) used trigonometric functions
Chinese_mathematics
Characterizes spherical triangles with fixed base and area
and Euler (1778) included trigonometric proofs in their papers, and several later mathematicians have presented trigonometric proofs, including Adrien-Marie
Lexell's_theorem
First publication of complete tables of logarithms, 1614
of tables of trigonometric functions and their Napierian logarithms. These tables greatly simplified calculations in spherical trigonometry, which are central
Mirifici Logarithmorum Canonis Descriptio
Mirifici_Logarithmorum_Canonis_Descriptio
First published description of a stored-program computer
mathematical operations, such as logarithms and trigonometric functions are to be done with table look up and interpolation, possibly biquadratic. He notes that
First Draft of a Report on the EDVAC
First_Draft_of_a_Report_on_the_EDVAC
Hindu astronomy, mathematics, science school in India
mathematical concepts. Their most important results—series expansion for trigonometric functions—were described in Sanskrit verse in a book by Neelakanta called
Kerala school of astronomy and mathematics
Kerala_school_of_astronomy_and_mathematics
Automatic mechanical calculator
engineering, science and navigation are built from logarithmic and trigonometric functions, which can be approximated by polynomials, so a difference
Difference_engine
Process of constructing a curve that has the best fit to a series of data points
possibly subject to constraints. Curve fitting can involve either interpolation, where an exact fit to the data is required, or smoothing, in which
Curve_fitting
lookup table and interpolation. It is a fast and efficient method for generating values of functions like the exponential or the trigonometric functions to
Gal's_accurate_tables
Special mathematical function defined as sin(x)/x
Sinc numerical methods Trigonometric functions of matrices – Important functions in solving differential equations Trigonometric integral – Special function
Sinc_function
Mathematical approximation of a function
astronomy and mathematics suggest that he found the Taylor series for the trigonometric functions of sine, cosine, and arctangent; see Madhava series. During
Taylor_series
modern fundamental trigonometric functions, sine and cosine, are described in the Siddhantas of India. This formulation of trigonometry is an improvement
Timeline of scientific discoveries
Timeline_of_scientific_discoveries
System of writing numbers using Greek letters
same papyrus. In Ptolemy's table of chords, the first fairly extensive trigonometric table, there were 360 rows, portions of which looked as follows: π ε
Greek_numerals
This is a list of polynomial topics, by Wikipedia page. See also trigonometric polynomial, list of algebraic geometry topics. Degree: The maximum exponents
List_of_polynomial_topics
Indian historian of mathematics (1935–2024)
age of 89. In 1969 Gupta addressed interpolation in Indian mathematics. He wrote on Govindasvamin and his interpolation of sine tables. Furthermore, he contributed
Radha_Charan_Gupta
second-order interpolation for computing the positions of the sun and the moon. 665 - Brahmagupta discovers a second order Newton-Stirling interpolation for sin
Timeline of calculus and mathematical analysis
Timeline_of_calculus_and_mathematical_analysis
second order derivatives can be computed more correctly using cubic or trigonometric splines. Efficient derivative filters need to be of odd length so that
Image_derivative
Perpendicular line segment from a triangle's side to opposite vertex
altitudes are also related to the sides of the triangle through the trigonometric functions. In an isosceles triangle (a triangle with two congruent sides)
Altitude_(triangle)
Extension of the factorial function
combinatorics. The gamma function can be seen as a solution to the interpolation problem of finding a smooth curve y = f ( x ) {\displaystyle y=f(x)}
Gamma_function
Product of numbers from 1 to n
provides a continuous interpolation of the factorials, offset by one, the digamma function provides a continuous interpolation of the harmonic numbers
Factorial
Audio process
)})^{n}}{2}}} Finally, use the binomial formula to transform back to trigonometric form and find coefficients for each harmonic. a 0 + ∑ n = 1 N [ a n
Waveshaper
Mathematical problem in classical harmonic analysis
12 Teschl, Theorem 8.14 Follows from the Weierstrass M-test Zygmund, Trigonometric Series, vol. 1, Chapter 8, Theorem 1.13, p. 300 Teschl Example 8.6 or
Convergence_of_Fourier_series
Red supergiant star in the constellation Orion
330 ly. The second was the Hipparcos Input Catalogue (1993) with a trigonometric parallax of 5±4 mas, a distance of 200 pc or 650 ly. Given this uncertainty
Betelgeuse
Mathematical curve outputted from a specific pair of parametric equations
which a function may be sampled in order to compute either a bivariate interpolation or quadrature of the function over the domain [−1,1] × [−1,1]. The relation
Lissajous_curve
Process of creating a screw thread
either using helical interpolation (which is circular interpolation in one plane [typically XY] with simultaneous linear interpolation along a third axis
Threading_(manufacturing)
formulated the root formula for cubics in modern language and applied trigonometric methods to root-solving, believed that his methods generalize to a closed-form
Polynomial_root-finding
Divergent sum of positive unit fractions
function provides a continuous interpolation of the factorials, the digamma function provides a continuous interpolation of the harmonic numbers, in the
Harmonic_series_(mathematics)
Study of Earth's spatial information
two different locations, al-Biruni developed a new method of using trigonometric calculations based on the angle between a plain and mountain top, which
Geography
Correspondence between quaternions and 3D rotations
}{2}}\sin {\frac {\alpha }{2}}\end{aligned}}} Or with angle addition trigonometric substitutions... γ = 2 cos − 1 ( ( 1 − A ⋅ B ) cos β − α 2 + ( 1
Quaternions and spatial rotation
Quaternions_and_spatial_rotation
Analytic function in mathematics
MR 0670132. Korobov, Nikolai Mikhailovich (1958). "Estimates of trigonometric sums and their applications". Usp. Mat. Nauk. 13 (4): 185–192. Vinogradov
Riemann_zeta_function
Describes approximate behavior of a function
(1914). "Some problems of diophantine approximation: Part II. The trigonometrical series associated with the elliptic θ functions". Acta Mathematica
Big_O_notation
Mathematical function
integral representations closely relate as the definite integral of trigonometric functions with product of its power and multiple-angle: ∫ 0 π sin x
Beta_function
Machine learning technique
it suffices to prove the case of D = 1 {\displaystyle D=1} . By the trigonometric identity cos ( a − b ) = cos ( a ) cos ( b ) + sin ( a ) sin
Random_feature
Polyhedron with four faces
2006. Vondran, Gary L. (April 1998). "Radial and Pruned Tetrahedral Interpolation Techniques" (PDF). HP Technical Report. HPL-98-95: 1–32. Archived from
Tetrahedron
(c. 100 AD) pioneered spherical trigonometry through Menelaus' theorem. The most complete and influential trigonometric work of antiquity is the Almagest
History_of_mathematics
Unique positive real number which when multiplied by itself gives 2
finite number of terms, 2 {\displaystyle {\sqrt {2}}} appears in various trigonometric constants: sin π 32 = 1 2 2 − 2 + 2 + 2 sin 3 π 16 = 1 2 2 − 2 −
Square_root_of_2
Discrete Fourier transform algorithm
sensitive to the accuracy of the twiddle factors used in the FFT (i.e. the trigonometric function values), and it is not unusual for incautious FFT implementations
Fast_Fourier_transform
Indian mathematician and astronomer (1444–1544)
this Bhasya, Nilakantha has discussed infinite series expansions of trigonometric functions, planet theory, and problems of algebra and spherical geometry
Nilakantha_Somayaji
Mathematical concept
integrand is a trigonometric polynomial in z and ζ and so the integral is a trigonometric polynomial in ζ. It tends in L2 to the trigonometric polynomial
Singular integral operators of convolution type
Singular_integral_operators_of_convolution_type
German polymath (1646–1716)
metaphysics. Although the mathematical notion of function was implicit in trigonometric and logarithmic tables, which existed in his day, Leibniz was the first
Gottfried_Wilhelm_Leibniz
Rule of thumb for rifle firing
to determine the bullet drop over that equivalent horizontal range (interpolation is likely to be required) Compute the bore angle correction that is
Rifleman's_rule
Shape between a square and a circle
FG squircle, while bounded between 0 and 1, results in a nonlinear interpolation of the squircle "corner" between the inner circle and the square corner
Squircle
Function that maps matrices to matrices
section 1.2.2, pages 4-7, of the book Higham (2008) by the Hermite interpolation with the use of confluent Vandermonde matrix. Consider these blocks
Analytic_function_of_a_matrix
Oscillatory error in Fourier series
(x-x_{0}))}}=0} where the 2nd equality is from one of Lagrange's trigonometric identities. Solving this condition gives x − x 0 = k π / ( N ω ) = k
Gibbs_phenomenon
Mathematical transform that expresses a function of time as a function of frequency
)\sin(2\pi \lambda t){\bigr )}\,d\lambda .} This is called an expansion as a trigonometric integral, or a Fourier integral expansion. The coefficient functions
Fourier_transform
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