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Comp."],"abstract":"<p>\n                    On the basis of a fully discrete trigonometric Galerkin method and two grid iterations we propose solvers for integral and pseudodifferential equations on closed curves which solve the problem with an optimal convergence order\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"double-vertical-bar u Subscript upper N Baseline minus u double-vertical-bar Subscript lamda Baseline less-than-or-equal-to c Subscript lamda comma mu Baseline upper N Superscript lamda minus mu Baseline double-vertical-bar u double-vertical-bar Subscript mu\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo fence=\"false\" stretchy=\"false\">\n                              \u2016\n                              \n                            <\/mml:mo>\n                            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                  <mml:annotation encoding=\"application\/x-tex\">\\|u_N-u\\|_\\lambda \\leq c_{\\lambda ,\\mu }N^{\\lambda -\\mu }\\|u\\|_\\mu<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ,\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"lamda less-than-or-equal-to mu\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>\n                              \u03bb\n                              \n                            <\/mml:mi>\n                            <mml:mo>\n                              \u2264\n                              \n                            <\/mml:mo>\n                            <mml:mi>\n                              \u03bc\n                              \n                            <\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\lambda \\leq \\mu<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    (Sobolev norms of periodic functions) in\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"normal upper O left-parenthesis upper N log upper N right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"normal\">O<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>N<\/mml:mi>\n                         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