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By N. K. Bary

ISBN-10: 1483199169

ISBN-13: 9781483199160

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Additional info for A Treatise on Trigonometric Series. Volume 1

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Let/(x)eZA If π o(x) = jf(t)g(t-x)dt, —Tt then M*)llp<{f\f\g(t-x)dtY π l\f{t)\{g{t-x)Y"{g{t-x)}^dt -Tt PlQ < J l/(0l p g(t -x)dtl jg(t -x)dt\ n = whence —n j\f(0\'g(t-x)dt, π n j\a(x)\'dxj —π f n ~\ \ j\f(t)\'g(t-x)dt\dx J —π \—π and changing the order of the integration j \o{x)\>dx < j |/(0I"| j g(t -x)dx\ dt. 12) f \o(x)\>dx < f \f{t)\>dt — Tt —Tt and raising to the power \\p we have the desired inequality. 12) MINKOWSKl'S INEQUALITY 21 § 10.

VI. T H E O R Y O F A P P R O X I M A T I O N O F F U N C T I O N S BY T R I G O N O M E T R I C P O L Y N O M I A L S § 22. Elementary properties of trigonometric polynomials Trigonometric polynomial is the name given to an expression of the form n T(x) = a 0 + £(a fe cos/;* + ßksinkx), whilst, if \oc„\ + \ßn\ > 0, then the number n is known as the order of the polynomial. We refer to the following simple properties of trigonometric polynomials. 1. The product of two trigonometric polynomials is itself a trigonometric polynomial (see Natanson, réf.

A 23, p. 254). § 18. Fubini's theorem If fix > y) is summable in the rectangle R[a < x

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A Treatise on Trigonometric Series. Volume 1 by N. K. Bary


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