Math 2.0

Approximations and expansions Definition

Rational approximation

Builds on Rational functionBest approximationContinuous function

Definition

Let a<ba < b be real numbers and let m,n≥0m, n \ge 0 be integers. Let Rm,nR_{m,n} be the set of rational functions r=p/qr = p/q with real polynomials pp and qq such that deg⁡p≤m\deg p \le m and deg⁡q≤n\deg q \le n. In addition, qq has no zero on [a,b][a, b]. Let f ⁣:[a,b]→ℝf \colon [a, b] \to \R be a continuous function. A rational approximation of type (m,n)(m, n) to ff is an element rr of Rm,nR_{m,n} used in place of ff. Its error is ∥f−r∥∞=max⁡x∈[a,b]∣f(x)−r(x)∣\norm{f - r}_\infty = \max_{x \in [a, b]} \abs{f(x) - r(x)}. The error of best rational approximation is Em,n(f)=inf⁡r∈Rm,n∥f−r∥∞.E_{m,n}(f) = \inf_{r \in R_{m,n}} \norm{f - r}_\infty.

Remarks

For n=0n = 0 this is polynomial approximation, and Em,0(f)E_{m,0}(f) is the error of best approximation by polynomials of degree at most mm. For continuous ff on [a,b][a, b], a best rational approximation in Rm,nR_{m,n} exists and is unique, and Chebyshev's equioscillation theorem characterizes it. Rm,nR_{m,n} is not a vector space for n≥1n \ge 1, since a sum of two of its elements may need numerator degree m+nm + n and denominator degree 2n2n. This is why the theory differs from the linear case. Example: r(x)=1/(1+x2)r(x) = 1/(1 + x^2) lies in R0,2R_{0,2} on any interval, and E0,2(r)=0E_{0,2}(r) = 0. Non-example: on [−1,1][-1, 1], 1/x1/x is not an element of R0,1R_{0,1}, since its denominator vanishes at 0∈[−1,1]0 \in [-1, 1].

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Best approximation, concept, no evidence levelBest approximationconceptContinuous function, concept, no evidence levelContinuous functionconceptLimit point of a set, definition, no evidence levelLimit point of a setdefinitionSequence, definition, no evidence levelSequencedefinitionLimit of a function, definition, no evidence levelLimit of a functiondefinitionReal sequence, definition, no evidence levelReal sequencedefinitionContinuity at a point, definition, no evidence levelContinuity at a pointdefinitionBounded sequence, definition, no evidence levelBounded sequencedefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionContinuous function, definition, no evidence levelContinuous functiondefinitionEpsilon-delta definition of continuity, statement, no evidence levelEpsilon-delta definit…statementConvergent sequence, definition, no evidence levelConvergent sequencedefinitionEpsilon-N definition of the limit of a sequence, statement, no evidence levelEpsilon-N definition …statementSequential continuity, definition, no evidence levelSequential continuitydefinitionContinuity is equivalent to sequential continuity, statement, no evidence levelContinuity is equival…statementConvergent sequences are bounded, statement, no evidence levelConvergent sequences …statementLimit of a product of sequences, statement, no evidence levelLimit of a product of…statementProduct of continuous functions is continuous, statement, no evidence levelProduct of continuous…statementLimit of a quotient of sequences, statement, no evidence levelLimit of a quotient o…statementPolynomial functions are continuous, statement, no evidence levelPolynomial functions …statementQuotient of continuous functions is continuous, statement, no evidence levelQuotient of continuou…statementRational function, definition, no evidence levelRational functiondefinitionRational approximation, definition, no evidence levelRational approximationdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Best approximation concept
  2. Continuous function concept
  3. Limit point of a set definition
  4. Sequence definition
  5. Limit of a function definition
  6. Real sequence definition
  7. Continuity at a point definition
  8. Bounded sequence definition
  9. Limit of a sequence definition
  10. Continuous function definition
  11. Epsilon-delta definition of continuity statement
  12. Convergent sequence definition
  13. Epsilon-N definition of the limit of a sequence statement
  14. Sequential continuity definition
  15. Continuity is equivalent to sequential continuity statement
  16. Convergent sequences are bounded statement
  17. Limit of a product of sequences statement
  18. Product of continuous functions is continuous statement
  19. Limit of a quotient of sequences statement
  20. Polynomial functions are continuous statement
  21. Quotient of continuous functions is continuous statement
  22. Rational function definition
  23. Rational approximation definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Rational_function
About this page
Tag76JD0ZG3B1
URIm2://76JD0ZG3B1
Kinddefinition
WritingW0 stub
MSC41A20, 41A21, 41A50
DefinesRational approximation
UsesRational function, Best approximation, Continuous function

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