Über den Stetigkeitsmodul von Fourier-Reihen mit monotonen Koeffizienten

Aljančić, S.; Tomić, M.

doi:10.1007/bf01111686

Cited by 25 publications

(7 citation statements)

References 2 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…See [1]. The properties of a slowly varying function L(t) stated above verify that these conditions are satisfied if g{k) = k~aL(k) and 1 < a + \/p < 2.…”

Section: We Let A*(apq)mentioning

confidence: 94%

On Grasia's criterion for uniform convergence of Fourier series

Oehring

1991

J Aust Math Soc A

View full text Add to dashboard Cite

Garsia's discovery that functions in the periodic Besov space A(p~', p, 1), with 1 < p < oo , have uniformly convergent Fourier series prompted him, and others, to seek a proof based on one of the standard convergence tests. We show that Lebesgue's test is adequate, whereas Garsia's criterion is independent of other classical critiera (for example, that of Dini-Lipschitz). The method of proof also produces a sharp estimate for the rate of uniform convergence for functions in A(p~', p, 1). Further, it leads to a very simple proof of the embedding theorem for these spaces, which extends (though less simply) to A(a, p, q).

show abstract

“…See [1]. The properties of a slowly varying function L(t) stated above verify that these conditions are satisfied if g{k) = k~aL(k) and 1 < a + \/p < 2.…”

Section: We Let A*(apq)mentioning

confidence: 94%

On Grasia's criterion for uniform convergence of Fourier series

Oehring

1991

J Aust Math Soc A

View full text Add to dashboard Cite

show abstract

“…If (1) is modified to posit that Q. p (h,s; K) ^ M(s) M(s)h CL L(h~1), then essentially as in the last paragraph of the proof, the conclusion becomes s n (K) = 0(n~a~1 /9 L(n)) (although this may fail if a + q' 1 = \ and ]im L(n) = 0). As a corollary, Cl p (h,s; K) ^ M(s)h^logh' 1 )-/ 1 implies that {s n (K)}el r for r~l = a + q' 1 , provided /? > r" 1 .…”

Section: Jv+1mentioning

confidence: 95%

“…As a corollary, Cl p (h,s; K) ^ M(s)h^logh' 1 )-/ 1 implies that {s n (K)}el r for r~l = a + q' 1 , provided /? > r" 1 .…”

Section: Jv+1mentioning

confidence: 95%

“…(ii) These results are sharp. For example, it can be shown by means of the criterion in Satz 4 of [1] that, for p > 1 and 0 < a < 1, the even function g for which g(n) = \n\~a~1 /Q L(\n\) (where L is slowly varying) has modulus of continuity o) p (h; g) = O(h*L{hT 1 )).…”

Section: Jv+1mentioning

confidence: 99%

“…Though elementary, it is an illuminating, as well as instrumental, fact that the singular numbers of a difference kernel, of the form K(s, t) =f(t -s) with feL 1 , are the moduli of the Fourier coefficients/(w). The associated operator is compact (see [10], p. 73).…”

mentioning

confidence: 99%

See 2 more Smart Citations

Singular numbers of smooth kernels

Oehring

1988

Math. Proc. Camb. Phil. Soc.

View full text Add to dashboard Cite

In [12] we elaborate the vague principle that the behaviour at infinity of the decreasing sequence of singular numbers sn(K) of a Hilbert–Schmidt kernel K is at least as good as that of the sequence {n−1/qω(n−1;K)}, where ωp is an Lp-modulus of continuity of K and q = p/(p − 1), where 1 ≤ p ≤ 2. Despite the author's effort to justify his study of refinements of the half-century old theorem of Smithies [13], that theorem remains the central result of the subject (viz. that for 0 < a ≤ 1, K∈Lip(a, p) implies that sn(K) = O(n−α−1/q)). For example, Cochran's omnibus theorems [5, 6] that delimit the Schatten classes to which a kernel belongs are based on the blending of ‘smoothness’ conditions and emphasize the pivotal role of the principal corollary of Smithies' theorem (viz. {sn}∈lr if r−1 < α + q−1). Cochran later offered in [7] a very simple derivation of the corollary from a Fourier series theorem of Konyushkov (see [2], vol. II, p. 197), whose proof was, however, at least as intricate as Smithies' demonstration.

show abstract

On Convergence and Summability with Speed I

2017

An Introductory Course in Summability Theory

View full text Add to dashboard Cite

Über den Stetigkeitsmodul von Fourier-Reihen mit monotonen Koeffizienten

Cited by 25 publications

References 2 publications

On Grasia's criterion for uniform convergence of Fourier series

On Grasia's criterion for uniform convergence of Fourier series

Singular numbers of smooth kernels

On Convergence and Summability with Speed I

Contact Info

Product

Resources

About