Unconditional convergence of the differences of Fejér kernels on L^2(\mathbb{R})
UUnconditional convergence of the differences ofFej´er kernels on L ( T ) Sakin DemirJanuary 14, 2021
Abstract
Let K n ( x ) denote the Fej´er kernel given by K n ( x ) = n (cid:88) j = − n (cid:18) − | j | n + 1 (cid:19) e − ijx and let σ n f ( x ) = ( K n ∗ f )( x ), where as usual f ∗ g denotes the convo-lution of f and g .Let the sequences { n k } be lacunary. Then the series G f ( x ) = ∞ (cid:88) k =1 (cid:0) σ n k +1 f ( x ) − σ n k f ( x ) (cid:1) converges unconditionally for all f ∈ L ( T ). Mathematics Subject Classifications:
Key Words:
Unconditional Convergence, Fej´er Kernel.
Definition 1.
The series (cid:80) ∞ n =1 x n in a Banach space X is said to convergeunconditionally if the series (cid:80) ∞ n =1 (cid:15) n x n converges for all (cid:15) n with (cid:15) n = ± n = 1 , , , . . . .The series (cid:80) ∞ n =1 x n in a Banach space X is said to be weakly unconditionallyconvergent if for every functional x ∗ ∈ X ∗ the scalar series (cid:80) ∞ n =1 x ∗ ( x n ) isunconditionally convergent. 1 a r X i v : . [ m a t h . C A ] J a n roposition 1. For a series (cid:80) ∞ n =1 x n in a Banach space X the followingconditions are equivalent:(a) The series (cid:80) ∞ n =1 x n is weakly unconditionally convergent;(b) There exists a constant C such that for every { c n } ∞ n =1 ∈ l ∞ sup N (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) N (cid:88) n =1 c n x n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ C (cid:107){ c n }(cid:107) ∞ . Proof.
See page 59 P. Wojtaszczyk [1].
Corollary 1.
Let X be a Banach space. If (cid:80) ∞ n =1 f n is a series in L p ( X ) , < p < ∞ , the following are equivalent:(a) The series (cid:80) ∞ n =1 f n is unconditionally convergent;(b) There exists a constant C such that for every { c n } ∞ n =1 ∈ l ∞ sup N (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) N (cid:88) n =1 c n f n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p ≤ C (cid:107){ c n }(cid:107) ∞ . Proof.
It is known (see page 66 in P. Wojtaszczyk [1]) that every weaklyunconditionally convergent series in a weakly sequentially complete space isunconditionally convergent. Since L p ( X ) is a weakly sequentially completespace for 1 < p < ∞ , the corollary follows from Proposition 1. Definition 2.
A sequence ( n k ) of integers is called lacunary if there is aconstant α > n k +1 n k ≥ α for all k = 1 , , , . . . .Let T denote the interval [ π, π ), thought of as the unit circle’ with nor-malized Lebesgue measure. For a function f ∈ L ( T ), we haveˆ f ( n ) = 12 π (cid:90) π − π f ( t ) e inx dx.
2e denote by K n ( x ) the Fej´er kernel given by K n ( x ) = n (cid:88) j = − n (cid:18) − | j | n + 1 (cid:19) e − ijx . We let σ n f ( x ) = ( K n ∗ f )( x ), where as usual f ∗ g denotes the convolutionof f and g .Our main result is the following: Theorem 2.
Let the sequences { n k } be lacunary. Then the series G f ( x ) = ∞ (cid:88) k =1 (cid:0) σ n k +1 f ( x ) − σ n k f ( x ) (cid:1) converges unconditionally for all f ∈ L ( T ) .Proof. Let { c k } ∞ k =1 ∈ l ∞ and define T N f ( x ) = N (cid:88) k =1 c k (cid:0) σ n k +1 f ( x ) − σ n k f ( x ) (cid:1) . In order to prove that G f converges unconditionally for all f ∈ L ( T ) wehave to show that for every { c n } ∞ n =1 ∈ l ∞ there exists a constant C > N (cid:107) T N f (cid:107) ≤ C (cid:107){ c n }(cid:107) ∞ for all f ∈ L ( T ) since this will verify the condition of Corollary 1 for G f .Let S N = N (cid:88) k =1 (cid:0) K n k +1 ( x ) − K n k ( x ) (cid:1) . We have | (cid:98) S N ( x ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N (cid:88) k =1 (cid:16) (cid:98) K n k +1 ( x ) − (cid:98) K n k ( x ) (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ N (cid:88) k =1 (cid:12)(cid:12)(cid:12) (cid:98) K n k +1 ( x ) − (cid:98) K n k ( x ) (cid:12)(cid:12)(cid:12) .
3e first want to show that there exits a constant
C > | (cid:98) S N ( x ) | ≤ C for all x ∈ T .The Fej´er kernel has a Fourier transform given by (cid:98) K n ( x ) = (cid:26) − | x | n +1 if | x | ≤ n ;0 if | x | > n. Fix x ∈ T , and let k be the first k such that | x | ≤ n k and let I = N (cid:88) k =1 (cid:12)(cid:12)(cid:12) (cid:98) K n k +1 ( x ) − (cid:98) K n k ( x ) (cid:12)(cid:12)(cid:12) . Then we have I = n k − (cid:88) k =1 (cid:12)(cid:12)(cid:12) (cid:98) K n k +1 ( x ) − (cid:98) K n k ( x ) (cid:12)(cid:12)(cid:12) + N (cid:88) k = n k (cid:12)(cid:12)(cid:12) (cid:98) K n k +1 ( x ) − (cid:98) K n k ( x ) (cid:12)(cid:12)(cid:12) = I + I . We clearly have I = 0 since (cid:98) K n ( x ) = 0 for | x | > n so in order to control | (cid:98) S N ( x ) | it suffices to control I = N (cid:88) k = n k (cid:12)(cid:12)(cid:12) (cid:98) K n k +1 ( x ) − (cid:98) K n k ( x ) (cid:12)(cid:12)(cid:12) .
4e have I = N (cid:88) k = n k (cid:12)(cid:12)(cid:12) (cid:98) K n k +1 ( x ) − (cid:98) K n k ( x ) (cid:12)(cid:12)(cid:12) = N (cid:88) k = n k (cid:12)(cid:12)(cid:12)(cid:12) − | x | n k +1 + 1 + | x | n k + 1 − (cid:12)(cid:12)(cid:12)(cid:12) = N (cid:88) k = n k (cid:12)(cid:12)(cid:12)(cid:12) − | x | n k +1 + 1 + | x | n k + 1 (cid:12)(cid:12)(cid:12)(cid:12) ≤ N (cid:88) k = n k | x | n k +1 + 1 + N (cid:88) k = n k | x | n k + 1 ≤ N (cid:88) k = n k | x | n k +1 + N (cid:88) k = n k | x | n k ≤ N (cid:88) k = n k n k n k +1 + N (cid:88) k = n k n k n k . On the other hand, since the sequence { n k } is lacunary there is a real number α > n k +1 n k ≥ α for all k ∈ N . Hence we have n k n k = n k n k +1 · n k +1 n k +2 · n k +2 n k +3 · · · n k − n k ≤ α k . Thus we get N (cid:88) k = n k n k n k ≤ N (cid:88) k = n k α k ≤ αα − . and similarly, we have ≤ N (cid:88) k = n k n k n k +1 ≤ αα − I ≤ αα − . x ∈ T what we have justproved is true for all x ∈ T .We conclude that there exits a constant C > | (cid:98) S N ( x ) | ≤ C ( ∗ )for all x ∈ T and N ∈ N .We now have (cid:107) T N f (cid:107) = (cid:90) T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N (cid:88) k =1 c k (cid:0) σ n k +1 f ( x ) − σ n k f ( x ) (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dx = (cid:90) T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N (cid:88) k =1 c k (cid:0) K n k +1 ∗ f ( x ) − K n k ∗ f ( x ) (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dx ≤ (cid:107){ c n }(cid:107) ∞ (cid:90) T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N (cid:88) k =1 (cid:0) K n k +1 ∗ f ( x ) − K n k ∗ f ( x ) (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dx = (cid:107){ c n }(cid:107) ∞ (cid:90) T | (cid:98) S N ( x ) | · | ˆ f ( x ) | dx (by Plancherel’s theorem) ≤ C (cid:107){ c n }(cid:107) ∞ (cid:90) T | ˆ f ( x ) | dx (by ( ∗ ))= C (cid:107){ c n }(cid:107) ∞ (cid:90) T | f ( x ) | dx (by Plancherel’s theorem)= C (cid:107){ c n }(cid:107) ∞ (cid:107) f (cid:107) and we thus we get sup N (cid:107) T N f (cid:107) ≤ √ C (cid:107){ c n }(cid:107) ∞ (cid:107) f (cid:107) which completes our proof. Remark . Our argument can easily be modified to see that the operator G f ( x ) = ∞ (cid:88) k =1 c k (cid:0) σ n k +1 f ( x ) − σ n k f ( x ) (cid:1) satisfies a strong type (2 ,
2) inequality. i.e., there exists a constant