Download e-book for kindle: A Treatise on Trigonometric Series. Volume 1 by N. K. Bary

By N. K. Bary

ISBN-10: 1483199169

ISBN-13: 9781483199160

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43 44 BASIC THEORY OF TRIGONOMETRIC SERIES and 00 v(r, x) = Σ (— bn cosnx + an sinnx)rn are conjugate harmonic functions; whence is derived the name "conjugate series". The study of the behaviour of conjugate series is no different from an investigation of the behaviour of conjugate harmonic functions on the circle \z\ = 1. § 2. 1) a different form. 1) takes the form « = + 00 Σ Ce'"*. (2-3) This is the so-called complex form of the trigonometric series. 1), that is, ün n Sn(x) = — + J^fakCoskx + bksinkx), now takes the form $ .

The class of functions {/(*)} belonging to LP forms an everywhere dense set in LP, p > 1, if for any e > 0 and for any yeLP a function / c a n be found such that U-

See Natanson, réf. A22, p. 107-109). 7) under the condition δ < min [a — a, ß — b], that is, so that the point x ± h for xe [a, b] does not lie outside the interval [a, ß]. 8) where x is any number. Modulus of smoothness. If instead of the first difference/(x + h) —/(x) we consider the second symmetrical difference, that isf(x + h) + f(x — h) — 2/(x), then we obtain the definition of the modulus of smoothness in a similar manner. œ2(ô,f) = sup \f(x + h) +f(x - h) - 2f(x)\. 9) It is clear that the following properties hold here: (1) ω2(δ) does not monotonically decrease, (2) for any positive λ ω2(λδ) <(λ+ 1)ω2(δ).

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

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