Lagrange Interpolation

Lagrange interpolation is a well known, classical technique for
interpolation [194]. It is also called Waring-Lagrange
interpolation, since Waring actually published it 16 years before
Lagrange [312, p. 323]. More generically, the term
*polynomial interpolation* normally refers to Lagrange interpolation.
In the first-order case, it reduces to *linear interpolation*.

Given a set of
known samples
,
, the
problem is to find the *unique* order
polynomial
which
*interpolates* the samples.^{5.2}The solution can be expressed as a linear combination of elementary
th order polynomials:

where

From the numerator of the above definition, we see that is an order polynomial having zeros at all of the samples except the th. The denominator is simply the constant which normalizes to give . Thus, we have

The polynomial can be interpreted as the th

- Interpolation of Uniformly Spaced Samples
- Fractional Delay Filters
- Lagrange Interpolation Optimality
- Explicit Lagrange Coefficient Formulas
- Lagrange Interpolation Coefficient Symmetry
- Matlab Code for Lagrange Interpolation
- Maxima Code for Lagrange Interpolation
- Faust Code for Lagrange Interpolation
- Lagrange Frequency Response Examples
- Orders 1 to 5 on a fractional delay of 0.4 samples
- Order 4 over a range of fractional delays
- Order 5 over a range of fractional delays

- Avoiding Discontinuities When Changing Delay
- Lagrange Frequency Response Magnitude Bound
- Even-Order Lagrange Interpolation Summary
- Odd-Order Lagrange Interpolation Summary
- Proof of Maximum Flatness at DC
- Variable
Filter Parametrizations
- Table Look-Up
- Polynomials in the Delay
- Farrow Structure
- Farrow Structure Coefficients
- Differentiator Filter Bank

- Recent Developments in Lagrange Interpolation
- Relation of Lagrange to Sinc Interpolation

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