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gptkbp:instanceOf
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Polynomial sequence
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gptkbp:application
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gptkb:signal_processing
Numerical integration
Interpolation
Minimax approximation
Root finding
Solving differential equations
Spectral methods
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gptkbp:category
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Special functions
Orthogonal polynomials
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gptkbp:degree
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n (for T_n(x))
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gptkbp:discoveredBy
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gptkb:Pafnuty_Chebyshev
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gptkbp:domain
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[-1,1]
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gptkbp:explicitFormula
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T_n(x) = cos(n arccos x)
U_n(x) = sin((n+1) arccos x)/sin(arccos x)
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gptkbp:field
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gptkb:Mathematics
Numerical analysis
Approximation theory
Orthogonal polynomials
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gptkbp:firstPolynomial
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T_0(x) = 1
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gptkbp:hasWikipediaPage
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https://en.wikipedia.org/wiki/Chebyshev_polynomials
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https://www.w3.org/2000/01/rdf-schema#label
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Chebyshev Polynomials
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gptkbp:introducedIn
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1854
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gptkbp:kind
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First kind (T_n)
Second kind (U_n)
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gptkbp:minimalMaxDeviation
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Yes
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gptkbp:namedAfter
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gptkb:Pafnuty_Chebyshev
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gptkbp:orthogonalWithRespectTo
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weight function (1-x^2)^(-1/2) on [-1,1]
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gptkbp:recurrence
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T_{n+1}(x) = 2x T_n(x) - T_{n-1}(x)
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gptkbp:relatedTo
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gptkb:Jacobi_polynomials
gptkb:Legendre_polynomials
Fourier series
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gptkbp:roots
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x_k = cos((2k-1)π/(2n)), k=1,...,n
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gptkbp:secondPolynomial
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T_1(x) = x
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gptkbp:symbol
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T_n(x)
U_n(x)
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gptkbp:type
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Orthogonal polynomial
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gptkbp:usedIn
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gptkb:Chebyshev_filter
gptkb:Chebyshev_nodes
gptkb:Fast_Fourier_Transform_(FFT)_algorithms
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gptkbp:bfsParent
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gptkb:Chebyshev_Filter
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gptkbp:bfsLayer
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6
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