convergence of haar and fourier series

Upload: saide-sac

Post on 06-Apr-2018

216 views

Category:

Documents


0 download

TRANSCRIPT

  • 8/3/2019 Convergence of Haar and Fourier Series

    1/24

    ddnonisncGpndmm)cmh

    Cmi|evbeice ml Dnnv nij Lmuv`ev Sev`es

    n)

    `) Lmuv`ev Sev`es etwnis`mi

    t$z 8 z >7 1 z w

    t $z 8 t$z/hZ rdeve Z8> nij !)))6!>!7 8h

    $$ ztzt w8 lmv >7 1 z

    (> (75 7 75 > >5 67

    76

    7:

    7

    tw

    $z)

    Eievbp ml zde t$z `s

    3->

    >

    7

    6 88 jzzEt

    zkoreoz 7X!$ 8 Zr -67 8 >8Z

    918o

    w ozozztZ

    zt X!$X!$!$>

    $

    8o

    ozon X!$X_

    8Z

    zkorjzezt

    Zon 7$

    >X_

    7 75 >7

    76

    7:

    7t$z)

  • 8/3/2019 Convergence of Haar and Fourier Series

    2/24

    8>

    7

    jzze fz 6kof 8

    ff

    e

    f

    e ff >6/8

    /8

    o

    ff

    wff

    e

    f

    ezt

    >$

    6zko

    e6

    Aezzkor

    v

    vo

    v eonzp7X_$

    8

    8 nij $$ zpzt vv 8 lmv >7 1 z ) Ns v `icvenses $zpv

    nwwvmncdes zm )$zp nij $ztv nwwvmncdes zm )$zt ) Imze zdnz n_7X 8 $7 zt 8 7)5 `s zde

    n|evnbe |naue ml zde s`bina

    (> (74 (7< (7: (76 7 76 7: 7< 74 >7

    76

    7:

    7

    pv$z)

    p7

    $z)

    p>

    $z)

    p6$z)

    7 76 7: 7< 74 >7

    76

    7:

    7

    tv$z)

    t7

    $z)

    t>

    $z)

    t6

    $z)

  • 8/3/2019 Convergence of Haar and Fourier Series

    3/24

    7 7> 76 73 7: 75 7< 72 74 70 >(76

    7

    76

    7:

    7

    >6

    tv$z)

    t5

    $z)

    t>7

    $z)

    t>5

    $z)

    @z `s hmve su`znfae zm wamz $zp v `iszenj ml $ztv zm mfsev|e zde ellecz ml B`ffs Wdeimheimi

    (> (74 (7< (7: (76 7 76 7: 7< 74 >(76

    7

    76

    7:

    7

    >6

    p65

    $z)

  • 8/3/2019 Convergence of Haar and Fourier Series

    4/24

    (> (74 (7< (7: (76 7 76 7: 7< 74 >(76

    7

    76

    7:

    7

    >6

    p57

    $z)

    (> (74 (7< (7: (76 7 76 7: 7< 74 >(76

    7

    76

    7:

    7

    >6

    p25

    $z)

  • 8/3/2019 Convergence of Haar and Fourier Series

    5/24

    (> (74 (7< (7: (76 7 76 7: 7< 74 >(76

    7

    76

    7:

    7

    >6

    p>57

    $z)

    (> (74 (7< (7: (76 7 76 7: 7< 74 >(76

    7

    76

    7:

    7

    >6

    p377

    $z)

  • 8/3/2019 Convergence of Haar and Fourier Series

    6/24

    (> (74 (7< (7: (76 7 76 7: 7< 74 >(76

    7

    76

    7:

    7

    >6

    p577

    $z)

    Eievbp ml zde nwwvmt`hnz`mi

    zkorv

    vo

    v eonzp7X_$

    8

    8 nij $$ zpzt vv 8 lmv >7 1 z

    Mi zde mzdev dnij lmv vena t$z X_X_ onon8

    Aez vE fe zde eievbp ml $ztv

    8

    /8v

    o

    v onnE>

    66X_6X7_ >v

    65)77 8E 3776)7> 8E 3>64)76 8E 3632)75 8E3647)7>7 8E 337:)767 8E 33>0)757 8E 336:)7>77 8E

    336-3

    ``) Dnnv Sev`es etwnis`mi

  • 8/3/2019 Convergence of Haar and Fourier Series

    7/24

    t$z 8 z >7 1 z

    6$6$ 6-! izzhh

    ih 8

    $X_$ ! ziczt ihi

    h 8

    98 1 )$) !$X_ ! zztic ihh

    8 /

    >$6

    6

    6-6

    i

    i

    h

    h

    h

    zjz

    8 >6$66-63$ /

    i

    h

    $ ) 6$6>6$6$ 6->6

    7

    6-63$ izizt hh

    i

    h

    h

    /8

    8

    6$ Azt nij )$zt `i \ fuz `z `s `hwms`fae zm vewveseiz t$z `i \ wvncz`cnaap fuzh8>7 `s sull`c`eiz zm spizdes`{e t$z

    )$)$)$

    0

    >

    Y

    7 ztztzt

    h

    h8/8

    )$X_)$ 7

    Y

    7 iziczt

    i

    8 nij $X_$ ! zijzt ihi hh 8

    Imze zdnz

    )$)$)$

    Y

    >

    Y

    ztztzt hhh /8/

    7 7 5 >7

    7 5

    >

    $z )

    7 75 >7

    76

    7:

    7t$z)

  • 8/3/2019 Convergence of Haar and Fourier Series

    8/24

    )$)$Y

    ztzt o o8

  • 8/3/2019 Convergence of Haar and Fourier Series

    9/24

    7 7> 76 73 7: 75 7< 72 74 70 >7

    7>

    76

    73

    7:

    75

    7t

  • 8/3/2019 Convergence of Haar and Fourier Series

    10/24

    7 7> 76 73 7: 75 7< 72 74 70 >7

    7>

    76

    73

    7:

    75

    7t2$z)

    7 7> 76 73 7: 75 7< 72 74 70 >

    7

    7>

    76

    73

    7:

    75

    7t4$z)

  • 8/3/2019 Convergence of Haar and Fourier Series

    11/24

    7 7> 76 73 7: 75 7< 72 74 70 >7

    7>

    76

    73

    7:

    75

    7t0$z)

    7 7> 76 73 7: 75 7< 72 74 70 >7

    7>

    76

    73

    7:

    75

    7t>7$z)

    Eievbp ml zde nwwvmt`hnz`mi

  • 8/3/2019 Convergence of Haar and Fourier Series

    12/24

    Aez tE ! toE nij otE fe zde eievbp ml t$z ! )$Y

    zto

    nij $zto

    8

    /80

    7

    7

    o

    ttt oEEE

    Mi zde mzdev dnij

    )$)$)$Y

    >

    Y

    ztztzth

    hh /8/

    htththEEE /8/>

    65)77 8tE 3>65)7> 8tE 364>6)76 8tE 33673)73 8tE

    3337>)7:8

    tE

    33365)7

    58

    t

    E

    3333>)778

    tE

    Imze zdnz ns v `icvenses eievbp ml zde $ztv nwwvmncdes zm zdnz ml )$zt rd`cd `s >-3

    f)

    `) Lmuv`ev Sev`es etwnis`mi

    t$z 8 6z >7 1 z : 3> 1 z:$:(z :3 1 z

    wt $z 8 t$z/hZ rdeve Z8: nij !)))6!>!7 8h

    $$ ztzt w8 lmv :7 1 z

  • 8/3/2019 Convergence of Haar and Fourier Series

    13/24

    7 > 6 3 :7

    >

    6

    3

    :

    5t$z)

    (: (6 7 6 : < 47

    75

    >

    >5

    6

    65

    3

    35

    :

    :5

    5

    tw

    $z)

    Eievbp ml zde t$z `s

    $

    8o

    ozon X!$X_

    8

    Z

    zkorjzezt

    Zon 7$

    >X_

    8 //:

    3

    3

    >

    >

    7

    :$::>:

    :>6

    :> jzezjzejzze czczcz 6-koc 8

  • 8/3/2019 Convergence of Haar and Fourier Series

    14/24

    _ X _ X cccccccc

    ec

    ce

    c

    cee

    cee

    ccc

    e

    c

    e 36

    :

    6

    3:3

    6

    >3$>:$:>>

    6

    >

    ///

    /8

    _ X _ X6-3

    6:

    63:3

    6>3$>:$:>>

    6>$ zko

    o

    cccccc

    cc

    w eecce

    ccee

    cee

    ccce

    cezt /// /8

    Aezzkor

    v

    vo

    v eonzp7X_$

    8

    8 nij $$ zpzt vv 8 lmv :7 1 z ) Ns v `icvenses $zp v

    nwwvmncdes zm )$zp nij $ztv nwwvmncdes zm )$zt ) Imze zdnz n_7X 8 $7 zt 8 6)25 `s zde

    n|evnbe |naue ml zde s`bina

    (: (3 (6 (> 7 > 6 3 :7

    75

    >

    >5

    6

    65

    3

    35

    :

    :5

    5

    pv$z)

    p7

    $z)

    p>

    $z)

    p6

    $z)

  • 8/3/2019 Convergence of Haar and Fourier Series

    15/24

    7 75 > >5 6 65 3 35 :7

    75

    >

    >5

    6

    65

    3

    35

    :

    :5

    5

    tv$z)

    t7

    $z)

    t>

    $z)

    t6$z)

    7 75 > >5 6 65 3 35 :7

    75

    >

    >5

    6

    65

    3

    35

    :

    :5

    tv$z)

    t5

    $z)

    t>7$z)

    t>5

    $z)

  • 8/3/2019 Convergence of Haar and Fourier Series

    16/24

    7 75 > >5 6 65 3 35 :7

    75

    >

    >5

    6

    65

    3

    35

    :

    :5

    t65

    $z)

    7 75 > >5 6 65 3 35 :7

    75

    >

    >5

    6

    65

    3

    35

    :

    :5

    t57

    $z)

  • 8/3/2019 Convergence of Haar and Fourier Series

    17/24

    7 75 > >5 6 65 3 35 :7

    75

    >

    >5

    6

    65

    3

    35

    :

    :5

    t25

    $z)

    7 75 > >5 6 65 3 35 :7

    75

    >

    >5

    6

    65

    3

    35

    :

    :5

    t>57

    $z)

  • 8/3/2019 Convergence of Haar and Fourier Series

    18/24

    7 75 > >5 6 65 3 35 :7

    75

    >

    >5

    6

    65

    3

    35

    :

    :5

    t377

    $z)

    7 75 > >5 6 65 3 35 :7

    75

    >

    >5

    6

    65

    3

    35

    :

    :5

    t577

    $z)

  • 8/3/2019 Convergence of Haar and Fourier Series

    19/24

    Eievbp ml zde nwwvmt`hnz`mi

    zkorv

    vo

    v eonzp7X_$

    8

    8 nij $$ zpzt vv 8 lmv :7 1 z

    Aez vE fe eievbp ml $ztv

    /8

    8

    v

    o

    v onnZE>

    66X_6X7_ >v Z8:

    655:)377 8E :: 8E 340:)346 8E 5>43)345 8E5052)34>7 8E

  • 8/3/2019 Convergence of Haar and Fourier Series

    20/24

    :6 6-h hh ti 636 1

    664$6>6->$ hhh i hh tit 6:63 1

    $ ) 6$6>6$6$6-

    >6

    7

    6-3

    izizthh

    i

    h

    h

    /8

    8

    $ ) 6$6:6 6->63

    6

    6-izhh

    t

    i

    h

    h

    h

    /

    8

    6$6664$66-

    >6:

    63

    >6->$ izi hht

    ti

    hhh

    h

    h

    /

    8

    6$ Azt nij )$zt `i \ fuz `z `s `hwms`fae zm vewveseiz t$z `i \ wvncz`cnaap fuzh8>7 `s sull`c`eiz zm spizdes`{e t$z

    )$)$)$0

    >

    Y

    7 ztztzt

    h

    h8

    /8

    )$X_)$7

    Y

    7iziczt

    i

    8 nij $X_$ ! zijzt ihi hh 8

    Imze zdnz )$)$)$Y

    >

    Y

    ztztzth

    hh /8/

    )$)$Y

    ztzt o o8

  • 8/3/2019 Convergence of Haar and Fourier Series

    21/24

  • 8/3/2019 Convergence of Haar and Fourier Series

    22/24

    7 75 > >5 6 65 3 35 :7

    75

    >

    >5

    6

    65

    3

    35

    :

    :5t4$z)

  • 8/3/2019 Convergence of Haar and Fourier Series

    23/24

    7 75 > >5 6 65 3 35 :7

    75

    >

    >5

    6

    65

    3

    35

    :

    :5t0$z)

    7 75 > >5 6 65 3 35 :7

    75

    >

    >5

    6

    65

    3

    35

    :

    :5t>7$z)

  • 8/3/2019 Convergence of Haar and Fourier Series

    24/24

    Aez tE ! toE nij otE fe zde eievbp ml t$z ! )$Y

    zto

    nij $zto

    8

    /80

    7

    7

    o

    ttt oEEE

    Mi zde mzdev dnij

    )$)$)$Y

    >

    Y

    ztztzth

    hh /8/

    htththEEE /8/>

    65)377 8tE >65)33> 8tE 326 8tE 65)343 8tE

    5