Next: Equivalence of the order Up: 1: WHAT IS A Previous: Functions with discontinuities

## Fourier transformations

Very simple examples of Fourier transformations for functions with discontinuities are given Table 2.
 2|c|Table 2: Fourier Transformations e-| t | e-t.H(t) H(t) t e-t.H(t)

It is very simple to determine the Fourier transforms of all the functions in Table 1 using these simple properties of the Fourier transformation:

We can see from the Table 2 that the Fourier transforms of functions with discontinuities of order 0 behave at as .Meanwhile, in case of a discontinuity of order 1, behaves asymptotically as . Let us take, for instance, the third function from Table 1: . In this case, where Then

and when

More generally, it is simple to prove that, if (i.e., is continuous), and if (i.e., is summable), then .Consequently, when , ,this convergence being not slower than .

We shall prove the following statement. Let f(t) be a right-sided function, and, let's suppose that, for all , (when ) in the same way as exp . Then:
 (2)

As f(t) is a right-sided function, we have:
 (3)
Let us integrate this integral by parts:
 (4) (5) (6)

Repeating this N times we obtain
 (7)

By definition of the order of discontinuity . From the condition concerning the behavior of f(k)(t) at , the integral on the righthand side of equation (7) converges to at . This means that the last equation is equivalent to equation (2). So the asymptotic behavior of the Fourier-transforms for high frequencies reflects the sharpness of the discontinuities of the (original) functions. Both notions (asymptotic behavior and discontinuities) are equivalent ones.

Next: Equivalence of the order Up: 1: WHAT IS A Previous: Functions with discontinuities
Stanford Exploration Project
1/13/1998
{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9874923229217529, "perplexity": 1056.687132684205}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187824225.41/warc/CC-MAIN-20171020135519-20171020155519-00594.warc.gz"}