Vol. 37, No. 3, 1971

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ISSN: 0030-8730
An estimate for Wiener integrals connected with squared error in a Fourier series approximation

Frederick Stern

Vol. 37 (1971), No. 3, 813–815
Abstract

If a function x(σ),0 σ t, is in Lipα,0 < α < 1,x(0) = 0 and if ck(k = 0,1,2,) are its Fourier coefficients with respect to the functions ∘2∕tsin[π(k + 1
2)σ∕t], then it is known [1, pp. 171–172] that

∑  c2≦  ---A----,  n ≧ 0
k   (n + 12)2α
k≧n
(1)

where A is a positive number not depending on n. We will show a connection between this estimate and an estimate for Wiener integrals. Let Ew{ } denote expectation on a Wiener process, that is, a Gaussian process with mean function zero, covariance function min(σ,τ),0 σ,τ t and sample functions z(σ) with z(0) = 0.

Theorem: Let x(σ) be in C[0,t] and let ck be the Fourier coefficients of x(σ) with respect to the normalized eigenfunctions associated with min(σ,τ). That is

    ∘ --∫ t
ck =   2   x(σ )sin[π(k + 1)σ∕t]dσ.
t 0             2

Let 0 < α < 1. Then estimate (1) is a necessary and sufficient condition for the estimate

                      ∫
−(B ∕2)ν1−α   EW {e−(ν∕2) 0t[z(σ)− x(σ)]2dσ}
e         ≦ ---------−(ν∕2)∫t-2----------
EW {e      0 z (σ)dσ}
(2)

for all positive ν, where B is a positive number not depending on ν.

Mathematical Subject Classification 2000
Primary: 42A16
Secondary: 28A40
Milestones
Received: 25 September 1970
Published: 1 June 1971
Authors
Frederick Stern