Displaying similar documents to “On a gap series of Mark Kac”

Non-similarity of Walsh and trigonometric systems

P. Wojtaszczyk (2000)

Studia Mathematica

Similarity:

We show that in L p for p ≠ 2 the constants of equivalence between finite initial segments of the Walsh and trigonometric systems have power type growth. We also show that the Riemann ideal norms connected with those systems have power type growth.

Integrability theorems for trigonometric series

Bruce Aubertin, John Fournier (1993)

Studia Mathematica

Similarity:

We show that, if the coefficients (an) in a series a 0 / 2 + ∑ n = 1 ∞ a n c o s ( n t ) tend to 0 as n → ∞ and satisfy the regularity condition that ∑ m = 0 ∞ ∑ j = 1 ∞ [ ∑ n = j 2 m ( j + 1 ) 2 m - 1 | a n - a n + 1 | ] ² 1 / 2 < ∞ , then the cosine series represents an integrable function on the interval [-π,π]. We also show that, if the coefficients (bn) in a series ∑ n = 1 ∞ b n s i n ( n t ) tend to 0 and satisfy the corresponding regularity condition, then the sine series represents an integrable function on [-π,π] if and only if ∑ n = 1 ∞ | b n | / n < ∞ . These conclusions were previously known to hold under stronger restrictions on the sizes...