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gptkbp:instanceOf
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gptkb:mathematical_concept
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gptkbp:application
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Fourier analysis
interpolation
numerical integration
minimax approximation
spectral methods
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gptkbp:category
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special functions
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gptkbp:definedIn
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real numbers
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gptkbp:degree
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n
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gptkbp:differential
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(1-x^2)y'' - x y' + n^2 y = 0
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gptkbp:explicitFormula
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T_n(x) = cos(n arccos x)
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gptkbp:family
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gptkb:Chebyshev_polynomials_of_the_first_kind
gptkb:Chebyshev_polynomials_of_the_second_kind
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gptkbp:field
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gptkb:mathematics
numerical analysis
approximation theory
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gptkbp:firstPolynomial
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T_0(x) = 1
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https://www.w3.org/2000/01/rdf-schema#label
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Chebyshev polynomial
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gptkbp:namedAfter
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gptkb:Pafnuty_Chebyshev
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gptkbp:notation
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T_n(x)
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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:property
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extremal property (minimizes maximum deviation from zero)
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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:rootWord
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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:type
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orthogonal polynomials
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gptkbp:usedIn
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gptkb:Chebyshev_filter
gptkb:Chebyshev_approximation
gptkb:Chebyshev_nodes
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gptkbp:variant
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x
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gptkbp:bfsParent
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gptkb:Hermite_polynomial
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gptkbp:bfsLayer
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7
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