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Shape preserving approximation of periodic functions -- Conclusion

D. Leviatan, I. O. Shevchuk

Abstract

We give here the final results about the validity of Jackson-type estimates in comonotone approximation of $2π$-periodic functions by trigonometric polynomials. For coconvex and the so called co-$q$-monotone, $q>2$, approximations, everything is known by now. Thus, this paper concludes the research on Jackson type estimates of Shape Preserving Approximation of periodic functions by trigonometric polynomials. It is interesting to point out that the results for comonotone approximation of a periodic function are substantially different than the analogous results for comonotone approximation, by algebraic polynomials, of a continuous function on a finite interval.

Shape preserving approximation of periodic functions -- Conclusion

Abstract

We give here the final results about the validity of Jackson-type estimates in comonotone approximation of -periodic functions by trigonometric polynomials. For coconvex and the so called co--monotone, , approximations, everything is known by now. Thus, this paper concludes the research on Jackson type estimates of Shape Preserving Approximation of periodic functions by trigonometric polynomials. It is interesting to point out that the results for comonotone approximation of a periodic function are substantially different than the analogous results for comonotone approximation, by algebraic polynomials, of a continuous function on a finite interval.
Paper Structure (6 sections, 15 theorems, 130 equations)

This paper contains 6 sections, 15 theorems, 130 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Divided differences
  4. Auxiliary lemmas
  5. Proofs of the positive results
  6. Counterexamples

Key Result

Theorem 2.1

Let $r=2s-2$. If $f\in C^r\cap\Delta^{(1)}(Y_s)$, then $(m)$ with $k=2$ holds.

Theorems & Definitions (30)

  • Remark 1.1
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Remark 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Remark 2.7
  • Theorem 2.8
  • Lemma 3.1
  • ...and 20 more