The degree of piecewise monotone interpolation

Authors:
Eli Passow and Louis Raymon

Journal:
Proc. Amer. Math. Soc. **48** (1975), 409-412

MSC:
Primary 41A15

MathSciNet review:
0430608

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Abstract: Let and let be real numbers such that . Estimates are obtained on the degree of an algebraic polynomial that interpolates the given data piecewise monotonely; i.e., such that (i) , and such that (ii) is increasing on , and decreasing on . if . The problem is seen to be related to the problem of monotone approximation.

**[1]**Wayne T. Ford and John A. Roulier,*On interpolation and approximation by polynomials with monotone derivatives*, J. Approximation Theory**10**(1974), 123–130. MR**0350259****[2]**William J. Kammerer,*Polynomial approximations to finitely oscillating functions*, Math. Comp.**15**(1961), 115–119. MR**0123865**, 10.1090/S0025-5718-1961-0123865-3**[3]**G. G. Lorentz and K. L. Zeller,*Degree of approximation by monotone polynomials. I*, J. Approximation Theory**1**(1968), 501–504. MR**0239342****[4]**E. Passow and L. Raymon,*Monotone and comonotone approximation*, Proc. Amer. Math. Soc.**42**(1974), 390–394. MR**0336176**, 10.1090/S0002-9939-1974-0336176-9**[5]**Eli Passow, Louis Raymon, and John A. Roulier,*Comonotone polynomial approximation*, J. Approximation Theory**11**(1974), 221–224. MR**0352807****[6]**Zalman Rubinstein,*On polynomial 𝛿-type functions and approximation by monotonic polynomials*, J. Approximation Theory**3**(1970), 1–6. MR**0261228****[7]**W. Wolibner,*Sur un polynôme d’interpolation*, Colloquium Math.**2**(1951), 136–137 (French). MR**0043946****[8]**Sam W. Young,*Piecewise monotone polynomial interpolation*, Bull. Amer. Math. Soc.**73**(1967), 642–643. MR**0212455**, 10.1090/S0002-9904-1967-11806-8

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Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9939-1975-0430608-4

Keywords:
Monotone interpolation,
monotone approximation,
comonotone approximation

Article copyright:
© Copyright 1975
American Mathematical Society