We examine regularity and basis properties of the family of rescaled p-cosine functions. We find sharp estimates for their Fourier coefficients. We then determine two thresholds, p 0 < 2 and p 1 > 2, such that this family is a Schauder basis of L s (0, 1) for all s > 1 and p ∈ [p 0 , p 1 ].
In this paper we formulate a concrete method for determining whether a system of dilated periodic functions forms a Riesz basis in L 2 (0, 1). This method relies on a general framework developed by Hedenmalm, Lindqvist and Seip about 20 years ago, which turns the basis question into one about the localisation of the zeros and poles of a corresponding analytic multiplier. Our results improve upon various criteria formulated previously, which give sufficient conditions for invertibility of the multiplier in terms of sharp estimates on the Fourier coefficients. Our focus is on the concrete verification of the hypotheses by means of analytical or accurate numerical approximations. We then examine the basis question for profiles in a neighbourhood of a non-basis family generated by periodic jump functions. For one of these profiles, the p-sine functions, we determine a threshold for positive answer to the basis question which improves upon those found recently.Mathematics subject classification. 41A30, 34C25.
The basis and regularity properties of the generalized trigonometric functions
sin
p
,
q
and
cos
p
,
q
are investigated. Upper bounds for the Fourier coefficients of these functions are given. Conditions are obtained under which the functions
cos
p
,
q
generate a basis of every Lebesgue space
L
r
(0,1) with
1
<
r
<
∞
; when
q
is the conjugate of
p
, it is sufficient to require that
p
∈[
p
1
,
p
2
], where
p
1
<2 and
p
2
>2 are calculable numbers. A comparison is made of the speed of decay of the Fourier sine coefficients of a function in Lebesgue and Lorentz sequence spaces with that of the corresponding coefficients with respect to the functions
sin
p
,
q
.
These results sharpen previously known ones.
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