Pointwise convergence of wavelet expansions

Abstract

In this note we announce that under general hypotheses, wavelet-type expansions (of functions in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript p"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{L^p}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1 less-than-or-equal-to p less-than-or-equal-to normal infinity"> <mml:semantics> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mi>p</mml:mi> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mi mathvariant="normal">∞<!-- ∞ --></mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">1 \leq p \leq \infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, in one or more dimensions) converge pointwise almost everywhere, and identify the Lebesgue set of a function as a set of full measure on which they converge. It is shown that unlike the Fourier summation kernel, wavelet summation kernels <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P Subscript j"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>P</mml:mi> <mml:mi>j</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{P_j}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are bounded by radial decreasing <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript 1"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{L^1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> convolution kernels. As a corollary it follows that best <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L squared"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{L^2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> spline approximations on uniform meshes converge pointwise almost everywhere. Moreover, summation of wavelet expansions is partially insensitive to order of summation. We also give necessary and sufficient conditions for given rates of convergence of wavelet expansions in the sup norm. Such expansions have order of convergence <italic>s</italic> if and only if the basic wavelet <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="psi"> <mml:semantics> <mml:mi>ψ<!-- ψ --></mml:mi> <mml:annotation encoding="application/x-tex">\psi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is in the homogeneous Sobolev space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Subscript h Superscript negative s minus d slash 2"> <mml:semantics> <mml:msubsup> <mml:mi>H</mml:mi> <mml:mi>h</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>−<!-- − --></mml:mo> <mml:mi>s</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mi>d</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msubsup> <mml:annotation encoding="application/x-tex">H_h^{ - s - d/2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We also present equivalent necessary and sufficient conditions on the scaling function. The above results hold in one and in multiple dimensions.

Locations

• arXiv (Cornell University) - View - PDF
• Bulletin of the American Mathematical Society - View - PDF