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Page 359
... Fourier coefficient of f and the formal series 00 ( 27 ) -1 / 2Σfeinx ∞- is called the Fourier series of f . ∞ 13 Show that if { c } is a sequence with n2 < ∞o , then there is some f in L2 whose nth Fourier coefficient is c . Show ...
... Fourier coefficient of f and the formal series 00 ( 27 ) -1 / 2Σfeinx ∞- is called the Fourier series of f . ∞ 13 Show that if { c } is a sequence with n2 < ∞o , then there is some f in L2 whose nth Fourier coefficient is c . Show ...
Page 360
... Fourier series ( a ) ( Snf ) ( x ) → f ( x ) uniformly in a for f in CBV . ( b ) ( Snf ) ( x ) → f ( x ) if ƒ is ... Fourier series of a function in L1 converges to the value of the function at every point where the function is ...
... Fourier series ( a ) ( Snf ) ( x ) → f ( x ) uniformly in a for f in CBV . ( b ) ( Snf ) ( x ) → f ( x ) if ƒ is ... Fourier series of a function in L1 converges to the value of the function at every point where the function is ...
Page 361
... Fourier series of g ( x ) converges absolutely and that g is continuous . ( A. Beurling , " Sur les intégrales de Fourier abso- lument convergentes " , 9th Scandinavian Mathematics Congress , 1938. ) 33 Let ( a , ẞ ) be any subinterval ...
... Fourier series of g ( x ) converges absolutely and that g is continuous . ( A. Beurling , " Sur les intégrales de Fourier abso- lument convergentes " , 9th Scandinavian Mathematics Congress , 1938. ) 33 Let ( a , ẞ ) be any subinterval ...
Contents
Preliminary Concepts | 1 |
B Topological Preliminaries | 10 |
Algebraic Preliminaries | 34 |
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A₁ Acad additive set function algebra Amer analytic arbitrary B-space B₁ ba(S Banach spaces Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions converges convex set Corollary countably additive DEFINITION dense disjoint Doklady Akad E₁ element ergodic exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism K₁ L₁ L₁(S Lebesgue Lemma Let f linear manifold linear map linear operator linear topological space measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative o-field o-finite open set operator topology positive measure space Proc PROOF properties proved real numbers reflexive Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory topological space u-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ