Further Examples
The following table shows some common examples of multiplier operators on the unit circle .
| Name | Multiplier | Operator | Kernel |
|---|---|---|---|
| Identity operator | 1 | f(t) | Dirac delta function |
| Multiplication by a constant c | c | cf(t) | |
| Translation by s | f(t − s) | ||
| Differentiation | in | f '(t) | |
| k-fold differentiation | |||
| Constant coefficient differential operator | |||
| Fractional derivative of order | |||
| Mean value | 1 | ||
| Mean-free component | |||
| Integration (of mean-free component) | Sawtooth function | ||
| Periodic Hilbert transform H | |||
| Dirichlet summation | Dirichlet kernel | ||
| Fejér summation | Fejér kernel | ||
| General multiplier | |||
| General convolution operator |
The following table shows some common examples of multiplier operators on Euclidean space .
| Name | Multiplier | Operator | Kernel |
|---|---|---|---|
| Identity operator | 1 | f(x) | |
| Multiplication by a constant c | c | cf(x) | |
| Translation by y | f(x − y) | ||
| Derivative (one dimension only) | |||
| Partial derivative | |||
| Laplacian | |||
| Constant coefficient differential operator | |||
| Fractional derivative of order | |||
| Riesz potential of order | |||
| Bessel potential of order | |||
| Heat flow operator | Heat kernel | ||
| Schrödinger equation evolution operator | Schrödinger kernel | ||
| Hilbert transform H (one dimension only) | |||
| Riesz transforms Rj | |||
| Partial Fourier integral (one dimension only) | |||
| Disk multiplier | (J is a Bessel function) | ||
| Bochner–Riesz operators | |||
| General multiplier | |||
| General convolution operator |
Read more about this topic: Multiplier (Fourier Analysis)
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“No rules exist, and examples are simply life-savers answering the appeals of rules making vain attempts to exist.”
—André Breton (1896–1966)
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—Alexander Pope (1688–1744)