A Generalized Fejrs Theorem for Locally Compact Groups Huichi Huang - PowerPoint PPT Presentation

A Generalized . . . . . . . . . . . . . . . . . . . . . . A Generalized Fejér’s Theorem for Locally Compact Groups Huichi Huang April 22, 2016 . . Fejér’s . Theorem for Locally Compact Groups Motivation Preliminaries The Main Theorem Some Special Cases . . . . . . . . . . . . . . . Shanghai Jiao Tong University 黄辉斥

A Generalized . . . . . . . . . . . . . . . . . . . . . . Outline Motivation Preliminaries The Main Theorem . . Fejér’s . Theorem for Locally Compact Groups Motivation Preliminaries The Main Theorem Some Special Cases . . . . . . . . . . . . . . . Some Special Cases 黄辉斥

A Generalized . Fejér’s . . . . . . . . . . . . . . . . . . . . . Motivation Theorem (Fejér, 1900) . . . . Theorem for Locally Compact Groups Motivation Preliminaries The Main Theorem . Cases . Some Special . . . . . . . . . . . . 黄辉斥 Let T = [0 , 1) be the unit circle. For f in L 1 ( T ) , if both the left limit and the right limit of f ( x ) exist at x 0 in T (denoted by f ( x 0 +) and f ( x 0 − ) respectively), then n →∞ K n ∗ f ( x 0 ) = 1 lim 2[ f ( x 0 +) + f ( x 0 − )] . Here K n ( x ) = sin 2 ( n +1) π x ( n +1) sin 2 π x is the Fejér’s kernel .

A Generalized . . . . . . . . . . . . Fejér’s . . . . . . . . . . . . . . . Theorem for Locally Compact Groups Motivation Preliminaries The Main Theorem Some Special . Cases . . . . . . . . . . . . 黄辉斥 f ( x 0 − ) f ( x 0 +) x 0 Question: Could Fejér’s theorem be generalized to T d ?

A Generalized . . . . . Fejér’s . . . . . . . . . . . . . . . . . A partial answer: Theorem (A multivariable Fejér’s theorem) . . . . Theorem for Locally Compact Groups . Preliminaries The Main Theorem Some Special Cases . Motivation . . . . . . kernel. . . . . . . 黄辉斥 Assume that f belongs to L ∞ ( T d ) with d ≥ 2 . If for x ∈ T d , the limit f ( x , k ) exists for every k in { 0 , 1 } d , then n ∗ f ( x ) = 1 n →∞ K d ∑ lim f ( x , k ) . 2 d k ∈{ 0 , 1 } d d Here K d ∏ n ( x 1 , · · · , x d ) = K n ( x j ) is the multivariable Fejér’s j =1

A Generalized . . . . . . . . . . . . Fejér’s . . . . . . . . . . . . . . . Theorem for Locally Compact Groups Motivation Preliminaries The Main Theorem Some Special . Cases . . . . . . . . . . . . Roughly speaking f ( x , k ) is the limit of f ( y ) when y approaches x via one of the 2 d directions. 黄辉斥 T 2 = [0 , 1) 2 x

A Generalized . . . . . . Fejér’s . . . . . . . . . . . . . . . . Observations So it is possible to get a generalized Fejér’s theorem for locally . . . . Theorem for Locally Compact Groups . Motivation Preliminaries The Main Theorem Some Special Cases compact groups. . . . . . . . . . . . . . { K d n =1 is an approximate identity of L 1 ( T d ) ; n } ∞ 黄辉斥 f ( x − y ) for a Borel subset J k of T d such that f ( x , k ) ＝ lim y → 0 y ∈ J k { J k } k ∈{ 0 , 1 } d is a fjnite partition of T d and every J k ∩ N is nonempty for any neighborhood N of 0 ; ∫ n ( x ) dx = 1 K d lim 2 d . n →∞ J k

A Generalized . . . . Fejér’s . . . . . . . . . . . . . . . . . . Preliminaries is given by . . . . Theorem for Locally Compact Groups Motivation . The Main Theorem Some Special Cases . Preliminaries . . . . . . . . . . . . G -a locally compact Hausdorfg group with a fjxed left Haar measure µ . e G -the identity of G . 黄辉斥 L 1 ( G ) -the space of integrable functions (with respect to µ ) on G . L ∞ ( G ) -the space of essentially bounded functions (with respect to µ ) on G . The convolution f ∗ g for f in L 1 ( G ) ∪ L ∞ ( G ) and g in L 1 ( G ) ∫ f ( y ) g ( y − 1 x ) d µ ( y ) f ∗ g ( x ) = G for every x ∈ G .

A Generalized . . Fejér’s . . . . . . . . . . . . . . . . . . . . Defjnition (Approximate identity) 3 . . . . Theorem for Locally Compact Groups Motivation Preliminaries The Main . Some Special Cases . Theorem . . . . . . . . . . . . 黄辉斥 An approximate identity is a family of functions { F θ } θ ∈ Θ in L 1 ( G ) such that 1 ∥ F θ ∥ L 1 ( G ) ≤ C for all θ ; 2 ∫ G F θ ( x ) d µ ( x ) = 1 for all θ ; ∫ lim N c | F θ ( x ) | d µ ( x ) = 0 for any neighborhood N of e G . θ There always exists an approximate identity in L 1 ( G ) .

A Generalized . . Fejér’s . . . . . . . . . . . . . . . . . . . . Defjnition (Local partition) 1 . . . . Theorem for Locally Compact Groups Motivation Preliminaries The Main . Some Special Cases . Theorem . . . . . . . . . . . . 黄辉斥 A fjnite collection { A 1 , A 2 , · · · , A k } of Borel subsets of G is called a local partition (at e G ) if the following are true: A i ∩ A j = ∅ for 1 ≤ i ̸ = j ≤ k ; k ∪ 2 µ ( G \ A i ) = 0 ; i =1 3 each A j ∩ N ̸ = ∅ for any neighborhood N of e G .

A Generalized . . . . . . . . . . . . . Fejér’s . . . . . . . . . . Picture of a local partition . . . . Theorem for Locally Compact Groups Motivation Preliminaries The Main Theorem Some Special Cases . . . . . . . . . . . . . G 黄辉斥 N e G

A Generalized . . . . . . . Fejér’s . . . . . . . . . . . . . . A generalized Fejér’s theorem for locally compact groups Theorem (H. 2015) then . . . . Locally Compact Groups . Motivation Preliminaries The Main Theorem Some Special Cases . . Theorem for . . . . . . . . . . . . Consider a locally compact group G with a fjxed left Haar measure µ . Let { F θ } θ ∈ Θ be an approximate identity of L 1 ( G ) . 黄辉斥 Assume that there exists a local partition { A 1 , A 2 , · · · , A k } of ∫ G such that lim F θ ( y ) d µ ( y ) = λ j for every 1 ≤ j ≤ k . θ A j For f in L ∞ ( G ) , if there exists x in G such that f ( y − 1 x ) (denoted by f ( x , A j ) ) exists for every 1 ≤ j ≤ k , lim y → e G y ∈ A j k ∑ lim θ F θ ∗ f ( x ) = λ j f ( x , A j ) . j =1

A Generalized . . . . Fejér’s . . . . . . . . . . . . . . . . . . . Theorem (Continued) . . . . Theorem for Locally Compact Groups Motivation . The Main Theorem Some Special Cases Preliminaries . . . . . . . . . . . . 黄辉斥 Moreover if lim y ∈N c | F θ ( y ) | = 0 for any neighborhood N of sup θ e G , then for every f in L 1 ( G ) ∪ L ∞ ( G ) such that each f ( x , A j ) exists at some x in G , we have k ∑ lim θ F θ ∗ f ( x ) = λ j f ( x , A j ) . j =1

A Generalized Fejér’s . . . . . . . . . . . . . . . . . . . . . . . Then . . . . Locally Compact Groups Theorem for Motivation Preliminaries The Main Theorem Some Special Cases . . . . . . . . . . . . . d-torus T d d For k = ( k 1 , · · · , k d ) in { 0 , 1 } d , defjne I k = ∏ I k j with j =1 黄辉斥 I 0 = (0 , 1 2 ) and I 1 = ( 1 2 , 1) . ∫ 1 ∫ 1 2 K n ( t ) dt = 1 0 K n ( t ) dt = 2 for all n ≥ 0 . 2 1 So { I k } k ∈{ 0 , 1 } d is a local partition of T d = [0 , 1) d with d ∫ ∫ K n ( x j ) dx j = 1 K d ∏ n ( x ) dx = 2 d I k I kj j =1 for all k ∈ { 0 , 1 } d and n ≥ 0 .

A Generalized . . . . Fejér’s . . . . . . . . . . . . . . . . . . Corollary exists, then . . . . Theorem for Locally Compact Groups Motivation . The Main Theorem Some Special Cases . Preliminaries . . . . . . . . . . . . For f in L 1 ( T d ) , defjne f ( x , I k ) = lim f ( x − y ) for every y → 0 y ∈ I k 黄辉斥 k ∈ { 0 , 1 } d . Let d ≥ 2 . For f in L ∞ ( T d ) and x in T d , if each f ( x , I k ) n ∗ f ( x ) = 1 n →∞ K d ∑ lim f ( x , I k ) . 2 d k ∈{ 0 , 1 } d

A Generalized . . . . . . . Fejér’s . . . . . . . . . . . . . . . . . . . . Theorem for Locally Compact . Motivation Preliminaries The Main Theorem Some Special Cases Groups . . . . . . . . . . . . Euclidean spaces R d The Poisson kernel P θ ( t ) is given by 1 黄辉斥 P θ ( t ) = πθ (1 + t 2 θ 2 ) for all t ∈ R and θ > 0 . Then { P θ ( t ) } θ> 0 is an approximate identity of L 1 ( R ) and θ → 0 sup lim t ∈N c | P θ ( t ) | = 0 for every neighborhood N of 0 in R . d Defjne P d ∏ θ ( x ) = P θ ( x j ) For any positive integer d , then j =1 { P d θ ( x ) } θ> 0 is an approximate identity of L 1 ( R d ) .

A Generalized . . . . Fejér’s . . . . . . . . . . . . . . . . . . . Note that . . . . Theorem for Locally Compact Groups Motivation . The Main Theorem Some Special Cases Preliminaries . . . . . . . . . . . . For k = ( k 1 , · · · , k d ) in { 0 , 1 } d , defjne 黄辉斥 d ∏ J k = J k l l =1 with J 0 = ( −∞ , 0) and J 1 = (0 , ∞ ) . ∫ 0 ∫ ∞ 0 P θ ( t ) dt = 1 −∞ P θ ( t ) dt = 2 for all θ > 0 .