## Linear Operators, Part 1 |

### From inside the book

Results 1-3 of 24

Page 359

( v ) e - inx dx is called the nth

1/2 Eineint is called the

with colcn | 2 < oo , then there is some f in L , whose nth

( v ) e - inx dx is called the nth

**Fourier**coefficient of 1 and the formal series ( 27 ) -1/2 Eineint is called the

**Fourier**series of f . 13 Show that if { en } is a sequencewith colcn | 2 < oo , then there is some f in L , whose nth

**Fourier**coefficient is Cn ...Page 360

26 Show that convergence of

series ( a ) ( S . ) ( x ) + f ( x ) uniformly in æ for f in CBV . ( b ) ( S - 1 ) ( x ) + f ( x ) if

f is in BV and if f is continuous at x . ( c ) ( S.f ) ( x ) ( 1/2 ) ( / ( x + ) + 1 ( x- ) ) for ...

26 Show that convergence of

**Fourier**series is localized . 27 Show that for**Fourier**series ( a ) ( S . ) ( x ) + f ( x ) uniformly in æ for f in CBV . ( b ) ( S - 1 ) ( x ) + f ( x ) if

f is in BV and if f is continuous at x . ( c ) ( S.f ) ( x ) ( 1/2 ) ( / ( x + ) + 1 ( x- ) ) for ...

Page 361

convergent

all x ; NO ( ii ) F ( n ) ( x ) = { F , ( n ) pive and \ F , ( M ) SK , n = 1 , 2 , .... Show that

then the

convergent

**Fourier**series , let ge Lj . Suppose that ( i ) lim F ( n ) ( x ) = g ( x ) forall x ; NO ( ii ) F ( n ) ( x ) = { F , ( n ) pive and \ F , ( M ) SK , n = 1 , 2 , .... Show that

then the

**Fourier**series of g ( x ) converges absolutely and that g is continuous .### What people are saying - Write a review

We haven't found any reviews in the usual places.

### Contents

Preliminary Concepts | 1 |

The VitaliHahnSaks Theorem and Spaces of Measures | 7 |

B Topological Preliminaries | 10 |

Copyright | |

87 other sections not shown

### Other editions - View all

### Common terms and phrases

Akad algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed complex Consequently contains converges convex Corollary defined DEFINITION denote dense determined differential dimensional disjoint domain element equation equivalent everywhere Exercise exists extension field finite follows function defined function f given Hence Hilbert space identity implies inequality integral interval isomorphism Lebesgue Lemma limit linear functional linear operator linear space Math measure space metric space neighborhood norm open set positive measure problem projection Proof properties proved range reflexive representation respect Russian satisfies scalar seen separable sequence set function Show shown sphere statement subset Suppose Theorem theory topological space topology u-measurable uniform uniformly unique unit valued vector weak weakly compact zero